Answer:
Maximum 3 socks (minimum 2) to be taken out to have a pair of same color.
Step-by-step explanation:
Well, it can be assumed that for example there are 4n black and 5n brown socks in a drawer (which satisfies ratio of 4/5). If first sock is taken out the probability of this sock being black is 4/9 (4n/(4n+5n)) or being brown is 5/9 (5n/(4n+5n)). If the first sock is black, the probability of second attempt being black is (4n-1)/(9n-1), but being brown is (5n)/(9n-1). f the first sock is brown, the probability of second attempt being brown is (5n-1)/(9n-1), but being black is (4n)/(9n-1). In either case, there is a probability of taking out different colors of socks in two attempts (taking black first then brown or vice-versa). So maximum 3 (minimum 2) attempts required to make sure you have a pair of the same color of socks.
To ensure you have a pair of the same color socks, you must take out a minimum of 5 socks from the drawer containing a mix of black and brown socks in a ratio of 4 to 5.
Explanation:If you have black socks and brown socks in your drawer, mixed in a ratio of 4 to 5, you need to consider the worst-case scenario to ensure you have a pair of the same color. If you take out 4 black socks, one by one, you may still not have a matching pair if they are all black—so you need to take out another sock. If this fifth sock is brown, you now have one pair of brown socks. If it's black, then you already had a pair from the four black socks. Thus, you must take out a minimum of 5 socks to guarantee a matching pair.
If a and b are positive integers such that gcd(a,b)=210, lcm[a,b]=210^3, and a
The problem can be solved by using a theorem from number theory stating that gcd(a, b) * lcm[a, b] = a * b. Applying this theorem with the given values leads us to solution where a = 210^2 and b = 210^2.
Explanation:In this mathematics problem, we're given two positive integers, a and b, whose greatest common divisor (gcd) equals to 210, and the least common multiple (lcm) equals to 210^3.
It is a well-known theorem in number theory that for any two positive integers a and b, gcd(a, b) * lcm[a, b] = a * b. We can apply this theorem to our problem. Since we know the values for greatest common divisor and least common multiple, namely gcd = 210 and lcm = 210^3 = 210 * 210 * 210, we have:
210 * 210^3 = a * b
This can be simplified to a * b = 210^4. Since we're told that a < b, and both a and b must divide 210^4, the only possible values for a and b are a = 210^2 and b = 210^2, which respects the condition a < b.
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When Justin goes to work, he drives at an average speed of 65 miles per hour. It takes about 1 hour and 30 minutes for Justin to arrive at work. His car travels about 25 miles per gallon of gas. If gas costs $3.65 per gallon, how much money does Justin spend on gas to travel to work?
Answer:
Justin spends $14.24 on gas to travel to work.
Step-by-step explanation:
Given:
Average speed at which Justin goes to work = 65 miles/hour
Time taken by Justin to arrive at work = 1 hour and 30 minutes = 1.5 hours [As 30 minutes =0.5 hours]
Distance he can travel per gallon of gas = 25 miles.
Cost of per gallon of gas = $3.65
Solution:
We first determine the distance Justin travels to work.
Distance = [tex]Speed\times time[/tex]
Distance = [tex]65\times 1.5 = 97.5\ miles[/tex]
Using unitary method to find the amount of gas required to cover the distance.
If 25 miles is covered in 1 gallon of gas
Then 1 mile will be covered in = [tex]\frac{1}{25}[/tex] gallons of gas
So, to cover 97.5 miles gas required = [tex]\frac{1}{25}\times 97.5=3.9[/tex] gallons of gas.
Using unitary method to find the cost of 3.9 gallons of gas.
Cost of 1 gallon of gas = $3.65
So, cost of 3.9 gallons of gas will be = [tex]\$3.65\times 3.9=\$14.235\approx\$14.24[/tex] (Answer)
Sam needs to make a long-distance call from apay phone.With his prepaid phone card, he will be charged $1.00 to connect and $0.50 per minute.If he places a collect call with the operator he will be charged $3.00 to connect and $0.25 per minute.In how many minutes will the phone card and the operator-assisted call cost the same?
A.5 minutes
B. 51/3 min.
C. 8 min.
D. 16 min.
Answer:
The answer is 8
Step-by-step explanation:
Lets consider that Sam's call will last x minutes in total. Then, equations for cost will be as follows:
A Pay Phone Cost: [tex]p(x)=1+0,5*x[/tex]
Operator Cost: [tex]o(x)=3+0,25*x[/tex]
If we make both costs equal, the equation will be:
[tex]1+0,5*x=3+0,25*x\\0,25*x=2\\x=8[/tex]
Sam's call need to last 8 minutes to make payphone cost and operator cost same.
Mario's Pizza just recieved two big orders from customers throwing parties. The first customer, Hugo, bought 7 regular pizzas and 1 deluxe pizza and paid $74. The second customer, Vincent, ordered 5 regular pizzas and 1 deluxe pizza, paying a totsl of $58. What is the price of each pizza?
The price of each regular pizza is $8 and the price of each deluxe pizza is $10.
Explanation:To find the price of each pizza, we need to set up a system of equations using the given information. Let's denote the price of a regular pizza as 'r' and the price of a deluxe pizza as 'd'. Using the first customer's order, we can write the equation: 7r + d = 74. Using the second customer's order, we can write the equation: 5r + d = 58. To solve this system of equations, we can subtract the second equation from the first equation to eliminate the 'd' variable: (7r + d) - (5r + d) = 74 - 58. Simplifying, we get 2r = 16, which gives us r = 8. Plugging this value back into the first equation, we find d = 10. Therefore, the price of each regular pizza is $8 and the price of each deluxe pizza is $10.
Seattle star blends whole bean coffee worth $3.00 per pound to get 25 pounds of a coffee blend worth $3.50 per pound. How many pounds of both blends does she use?
Answer:
Blend of Whole bean coffee used = 22.5 Ib
Blend of Half bean Coffee used = 2.5 lb
Correction in statement:
The problem statement is missing information. The proper statement is as follows:
Seattle Star blends whole bean coffee worth $3 per pound with half bean coffee worth $3.5 per pound to get 25 pounds of a coffee blend worth $3.05 per pound. How many pounds of each type of coffee does she use?
Step-by-step explanation:
Whole Bean Coffee = $3
Half Bean Coffee = $ 3.5
Amount of whole bean coffee in Pounds (lbs) = Y
Mixture amount of whole bean and half bean blend = 25 lbs
Amount of half bean coffee = 25-Y (lbs)
Total blended mixture = $3.05
Cost of mixture = Cost of whole bean coffee used + Cost of half bean coffee used
3.05 (25) = 3Y + 3.5 (25-Y)
76.25 = 3Y + 87.5 - 3.5 Y
76.25 - 87.5 = 3Y - 3.5Y
- 11.25 = - 0.5 Y
or
0.5 Y = 11.25
Y=[tex]\frac{11.25}{0.5}[/tex]
Y= 22.5 lb
Which is the amount of whole bean coffee.
Amount of half bean coffee = 25-Y = 25- 22.5 = 2.5 lb
So,
Blend of Whole bean coffee used = 22.5 lb
Blend of Half bean Coffee used = 2.5 lb
Simplify the cubed root of six over the fourth root of six
six raised to the one twelfth power
six raised to the one fourth power
six raised to the four thirds power
six raised to the seven twelfths power
Answer:
six raised to the one twelfth power
Step-by-step explanation:
The cubed root of 6/the fourth root of 6 equals (6^1/3)/(6^1/4)
6^((1/3)-(1/4))
6^((4-3)/12)
6^1/12
The simplified form of the expression is six raised to the one twelfth power
Given the expression
[tex]\dfrac{\sqrt[3]{6} }{\sqrt[4]{6} }[/tex]According to indices, this expression can also be written as:
[tex]\dfrac{(6)^{1/3}}{6^{1/4}}[/tex]Using the law of indices;
[tex]\dfrac{a^m}{a^n} = a^{m-n}[/tex]
Applying this expression will give:
[tex]=\dfrac{6^{1/3}}{6^{1/4}} \\= 6^{1/3-1/4}\\=6^{4-3/12}\\=6^{1/12}[/tex]
Hence the simplified form of the expression is six raised to the one twelfth power
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Of the 645 speckled trout in a certain fishery that contains only speckled and rainbow trout, the number of males is 45 more than twice the number of females. If the ratio of female speckled trout to male rainbow trout is 4:3 and the ratio of male rainbow trout to all trout is 3:20, how many female rainbow trout are there?A. 192B. 195C. 200D. 205E. 208
Answer: D 205
Step-by-step explanation:
Let,
Number of all trouts = N
Number of speckled trouts = Ns = 645
Number of rainbow trouts = Nr
Number of male speckled trouts = Ms
Number of female speckled trouts = Fs
Number of male rainbow trouts = Mr
Number of female rainbow trouts = Fr
Since, Ms = 2Fs + 45
Also, Ms + Fs = 645
Therefore, 2Fs + 45 + Fs = 645
Fs = (645-45)/3 = 200
Female speckled trouts = 200
Since
Ms + Fs = 645
Ms = 645- 200 = 445
Since, Fs/Mr = 4/3
Mr = 3x200/4 = 150
Since,
Mr/N = 3/20
N = 20x 150/3 = 1000
Recall that,
N = Ms+Fs+Mr+Fr
Fr = N-Ms-Fs-Mr
Fr = 1000-445-200-150
Fr = 205
Therefore, the number of rainbow female trouts = 205
The number of female rainbow trout is calculated using given ratios and the total number of speckled trout. After solving a series of equations, the answer is determined to be 205 female rainbow trout, which is option D.
To find the number of female rainbow trout, we first need to unpack the information provided in the question and express it in equations.
Let n be the number of female speckled trout. We know that the number of male speckled trout is 45 + 2n. Since there are 645 speckled trout in total, we can express this as:
n + (45 + 2n) = 645
Solving for n, we get:
3n + 45 = 645
3n = 600
n = 200
Now, we have the ratio of female speckled trout to male rainbow trout as 4:3, and since we've found there are 200 female speckled trout (n = 200), we can figure out the number of male rainbow trout. Let's call this number m. We have:
200/4 = m/3
50 = m/3
m = 150
The ratio of male rainbow trout to all trout is given as 3:20. If the total number of trout is T, then:
150/T = 3/20
We can solve for T:
20 × 150 = 3T
3000 = 3T
T = 1000
So there are 1000 trout in total, of which 645 are speckled. This means there must be 1000 - 645 = 355 rainbow trout. As we have already found there are 150 male rainbow trout, the remainder must be female. So:
355 - 150 = 205 female rainbow trout
Therefore, the correct answer is D. 205.
A differential equation that is a function of y only
a.will produce a slope field with parallel tangents along the diagonal
b.will produce a slope field that does not have rows or columns of parallel tangents
c.will produce a slope field with rows of parallel tangents
d.will produce a slope field with columns of parallel tangents
Answer:
c. Will produce a slope field with rows of parallel tangents
Step-by-step explanation:
We can write a differential equation that is a function of y only as:
[tex]y'=f(y)[/tex]
So the derivative, in this particular case, of any function will be a function of the dependent variable y only, it means that the curves you will get should all be pointing in the same direction for each value of x. Therefore the sketch of the slopes field would have parallel curves for each value of x, in other words, it will produce a slope field with rows of parallel tangents.
I hope it helps you!
A section of a hiking trail begins at the coordinates (-7, 5) and follows a straight path that ends at the coordinates (3, 9). What is the rate of change of the hiking trail?
Answer:
The rate of change of the hiking trail is [tex]m=\frac{2}{5}[/tex].
Step-by-step explanation:
A section of a hiking trail begins at the coordinates (-7, 5). It does mean (x₁, y₁) ⇒ (-7, 5)And follows a straight path that ends at the coordinates (3, 9). It does mean (x₂, y₂) ⇒ (3, 9)In mathematical language, the slope m of the line is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Plugging (x₁, y₁) ⇒ (-7, 5) and (x₂, y₂) ⇒ (3, 9) into the slope equation
[tex]m=\frac{9-5}{3-(-7)}[/tex]
[tex]m=\frac{4}{10}[/tex]
[tex]m=\frac{2}{5}[/tex]
So, the rate of change of the hiking trail is [tex]m=\frac{2}{5}[/tex].
Keywords: rate of change, slope, coordinates
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PLEASE HELP
CD¯ has endpoints C and D, with C at coordinates (5,8). CD¯- has midpoint M at (3,9).
What are the coordinates of point D?
Answer:
(1, 10)
Step-by-step explanation:
Use midpoint formula.
Mₓ = (x₁ + x₂) / 2
3 = (5 + x) / 2
6 = 5 + x
x = 1
Mᵧ = (y₁ + y₂) / 2
9 = (8 + y) / 2
18 = 8 + y
y = 10
The coordinates of point D are (1, 10).
The coordinates of point D are (1, 10).
What does a midpoint mean?Midpoint, as the word suggests, means the point which lies in the middle of something.
Midpoint of a line segment means a point which lies in the mid of the given line segment.
Given; CD has endpoints C and D, with C at coordinates (5,8). CD has midpoint M at (3,9).
Using midpoint formula;
Mₓ = (x₁ + x₂) / 2
3 = (5 + x) / 2
6 = 5 + x
x = 1
Now, Mᵧ = (y₁ + y₂) / 2
9 = (8 + y) / 2
18 = 8 + y
y = 10
Hence, The coordinates of point D are (1, 10).
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A quilt is made of 8 rows of squares, and there are 6 squares in each row. Each square measures 1 foot on a side. Explain how to find the area of the quilt in a square feet. Then write the area.
Answer:
48
Step-by-step explanation:
Answer:
Step-by-step explanation:
The total number of rows of squares in the quilt is 8. Each row contains 6 squares.
Since each square measures 1 foot on a side, the area of each square would be 1^2 = 1 foot^2
This means that each row contains six 1 foot^2. Since there are 8 rows, a triangle would be formed such that one of its sides is
6 × 1 foot == 6 feets and the other side would be
8 × 1 foot = 8 feets.
The area of the quilt in a square feet would be the area of the rectangle. It becomes
6 × 8 = 48 feets
The number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900. If 30 states joined the United States between 1776and 1849 and x states joined between 1850 and 1900, which of the following equations is true?
a. 30x = 2
b. 2x = 30
c. x/2 = 30
d. x + 30 = 2
Answer: b. 2x = 30
Step-by-step explanation:
Given : 30 states joined the United States between 1776 and 1849
and x states joined between 1850 and 1900 .
If the number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900.
i.e. No. of states joined the United States between 1776 and 1849= 2 (No. of states that joined between 1850 and 1900)
i.e . 30= 2(x) [Substitute the values]
i.e . 2x=30
Hence, the true equation : 2x=30
A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times as long as it is wide. Both the banner and the poster have perimeters of 24 inches. What are the lengths and wides of the poster and the poster?
Answer:
Length of poster is 9 inches and width of the poster is 3 inches.
Length of banner is 10 inches and width of the banner is 2 inches.
Step-by-step explanation:
Given:
Perimeter of Banner =24 in.
Perimeter of poster = 24 in.
we need to find the dimensions of poster and banner.
First we will find the dimension of poster.
Now Given:
A rectangular poster is 3 times as long as it is wide.
Let the Width of poster be [tex]'p'[/tex].
Length of the poster = [tex]3p[/tex]
Perimeter of poster = 24 in.
But perimeter of poster is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(p+3p)=24\\\\2(4p)=24\\\\8p=24\\\\p=\frac{24}{8} = 3\ in.[/tex]
Now width of poster = 3 inches
Length of the poster = [tex]3p = 3\times3 =9\ inches[/tex]
Hence Length of poster is 9 inches and width of the poster is 3 inches.
Now we will find the dimension of Banner.
Now Given:
A rectangular banner is 5 times as long as it is wide.
Let the Width of Banner be [tex]'b'[/tex].
Length of the banner = [tex]5b[/tex]
Perimeter of banner = 24 in.
But perimeter of banner is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(b+5b)=24\\\\2(6b)=24\\\\12b=24\\\\b=\frac{24}{12} = 2\ in.[/tex]
Now width of banner = 2 inches
Length of the banner = [tex]5b = 5\times2 = 10\ inches[/tex]
Hence Length of banner is 10 inches and width of the banner is 2 inches.
The width and length of the rectangular poster are 3 inches and 9 inches, respectively. The width and length of the rectangular banner are 2 inches and 10 inches, respectively. Both were calculated by setting up equations using the perimeter formula for rectangles.
Explanation:The problem is to determine the lengths and widths of a rectangular poster and a rectangular banner, both having the same perimeter of 24 inches. The poster's length is 3 times its width, while the banner's length is 5 times its width. We'll set up two separate equations for their perimeters and solve for the width and length of each.
Poster:Let w be the width of the poster. Then the length is 3w. The perimeter is given by P = 2l + 2w, where P is the perimeter and l is the length. This gives us:
24 = 2(3w) + 2w -> 24 = 6w + 2w -> 24 = 8w -> w = 3 inches
Therefore, the length of the poster is 3w = 9 inches.
Banner:Let x be the width of the banner. Then the length is 5x. Again using the perimeter formula, we get:
24 = 2(5x) + 2x -> 24 = 10x + 2x -> 24 = 12x -> x = 2 inches
Therefore, the length of the banner is 5x = 10 inches.
Julia is allowed to watch no more than 5 hours of television a week. So far this week, she has watched 1.5 hours. Write and solve an inequality to show how many hours of television Julia can still watch this week.
The inequality is used to solve how many hours of television Julia can still watch this week is [tex]x + 1.5 \leq 5[/tex]
The remaining hours of TV Julia can watch this week can be expressed is 3.5 hours
Solution:Given that Julia is allowed to watch no more than 5 hours of television a week
So far this week, she has watched 1.5 hours
To find: number of hours Julia can still watch this week
Let "x" be the number of hours Julia can still watch television this week
"no more than 5" means less than or equal to 5 ( ≤ 5 )
Juila has already watched 1.5 hours. So we can add 1.5 hours and number of hours Julia can still watch television this week which is less than or equal to 5 hours
number of hours Julia can still watch television this week + already watched ≤ Total hours Juila can watch
[tex]x + 1.5 \leq5[/tex]
Thus the above inequality is used to solve how many hours of television Julia can still watch this week.
Solving the inequality,
[tex]x + 1.5 \leq5\\\\x \leq 5 - 1.5\\\\x \leq 3.5[/tex]
Thus Julia still can watch Television for 3.5 hours
(3 points)
11. A farmer buys 20 sheep, half male and half female. She was told that the annual rate of
increase for the sheep population is 60%. Assuming that none of the sheep die, when will the
farmer have 200 sheep? Write and solve an exponential equation, showing your work.
Use
to indicate an exponent. Use /to indicate a fraction.
Answer:
The exponential Function is [tex]20+12h=200[/tex].
Farmer will have 200 sheep after 15 years.
Step-by-step explanation:
Given:
Number of sheep bought = 20
Annual Rate of increase in sheep = 60%
We need to find that after how many years the farmer will have 200 sheep.
Let the number of years be 'h'
First we will find the Number of sheep increase in 1 year.
Number of sheep increase in 1 year is equal to Annual Rate of increase in sheep multiplied by Number of sheep bought and then divide by 100.
framing in equation form we get;
Number of sheep increase in 1 year = [tex]\frac{60}{100}\times20 = 12[/tex]
Now we know that the number of years farmer will have 200 sheep can be calculated by Number of sheep bought plus Number of sheep increase in 1 year multiplied by number of years is equal to 200.
Framing in equation form we get;
[tex]20+12h=200[/tex]
The exponential Function is [tex]20+12h=200[/tex].
Subtracting both side by 20 using subtraction property we get;
[tex]20+12h-20=200-20\\\\12h=180[/tex]
Now Dividing both side by 12 using Division property we get;
[tex]\frac{12h}{12} = \frac{180}{12}\\\\h =15[/tex]
Hence Farmer will have 200 sheep after 15 years.
Mr. Jones took a survey of college students and found that 60 out of 65 students are liberal arts majors. If a college has 8,943 students, what is the expected number of students who are liberal arts majors? Answer quick
Answer:
We can expect 8255 numbers of students are liberal arts majors.
Step-by-step explanation:
Given:
Total Number of students in the college = 8943
Now According to Mr. Jones Survey;
60 out of 65 students are liberal arts majors.
We need to find the number of students who are liberal arts majors out of total number of students in college.
Solution:
First we will find the Percentage number of students who are liberal arts majors according to survey.
Percentage number of students who are liberal arts majors can be calculated by dividing 60 from 65 then multiplying by 100.
framing above quote in equation form we get;
Percentage number of students who are liberal arts majors = [tex]\frac{60}{65}\times 100 = 92.31\%[/tex]
Now we will find the Total number of students who are liberal arts major.
Total number of students who are liberal arts major can be calculated by Multiplying Percentage number of students who are liberal arts majors with total number of students in the college and then dividing by 100.
framing above quote in equation form we get;
Total number of students who are liberal arts major = [tex]\frac{92.31}{100}\times 8493 \approx8255.28[/tex]
Since number of students cannot be in point, so we will round the value.
Hence We can expect 8255 numbers of students are liberal arts majors.
Vector u has its initial point at (21, 12) and its terminal point at (19, -8). Vector v has a direction opposite that of u, whose magnitude is five times the magnitude of v. Which is the correct form of vector v expressed as a linear combination of the unit vectors i and j?
[tex]\boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]
Explanation:In this exercise, we have the following facts for the vector [tex]\vec{u}[/tex]:
It has its initial point at [tex](21,12)[/tex], let's call it [tex]P_{1}[/tex] It has its terminal point at [tex](19,-8)[/tex], let's call it [tex]P_{2}[/tex]Since the vector [tex]\vec{u}[/tex] goes from point [tex]P_{1}[/tex] to [tex]P_{2}[/tex], then:
[tex]\vec{u}=(19,-8)-(21,12) \\ \\ \vec{u}=(19-21,-8-12) \\ \\ \vec{u}=(-2,-20)[/tex]
On the other hand, we have the following facts for the vector [tex]\vec{v}[/tex]:
Vector [tex]\vec{v}[/tex] has a direction opposite that of [tex]\vec{u}[/tex], The magnitude of [tex]\vec{u}[/tex] is five times the magnitude of [tex]v[/tex].So we can write this relationship as follows:
[tex]5\vec{v}=-\vec{u} \\ \\ \vec{v}=-\frac{1}{5}\vec{u} \\ \\ \vec{v}=-\frac{1}{5}(-2,-20) \\ \\ \vec{v}=(\frac{2}{5},4) \\ \\ \\ Finally: \\ \\ \boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]
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Prove that it is impossible to dissect a cube into finitely many cubes, no two of which are the same size.
explanation:
The sides of a cube are squares, and they are covered by the respective sides of the cubes covering that side of the big cube. If we can show that a sqaure cannot be descomposed in squares of different sides, then we are done.
We cover the bottom side of that square with the bottom side of smaller squares. Above each square there is at least one square. Those squares have different heights, and they can have more or less (but not equal) height than the square they have below.
There is one square, lets call it A, that has minimum height among the squares that cover the bottom line, a bigger sqaure cannot fit above A because it would overlap with A's neighbours, so the selected square, lets call it B, should have less height than A itself.
There should be a 'hole' between B and at least one of A's neighbours, this hole is a rectangle with height equal to B's height. Since we cant use squares of similar sizes, we need at least 2 squares covering the 'hole', or a big sqaure that will form another hole above B, making this problem inifnite. If we use 2 or more squares, those sqaures height's combined should be at least equal than the height of B. Lets call C the small square that is next to B and above A in the 'hole'. C has even less height than B (otherwise, C would form the 'hole' above B as we described before). There are 2 possibilities:
C has similar size than the difference between A and BC has smaller size than the difference between A and BIf the second case would be true, next to C and above A there should be another 'hole', making this problem infinite. Assuming the first case is true, then C would fit perfectly above A and between B and A's neighborhood. Leaving a small rectangle above it that was part of the original hole.
That small rectangle has base length similar than the sides of C, so it cant be covered by a single square. The small sqaure you would use to cover that rectangle that is above to C and next to B, lets call it D, would leave another 'hole' above C and between D and A's neighborhood.
As you can see, this problem recursively forces you to use smaller and smaller squares, to a never end. You cant cover a sqaure with a finite number of squares and, as a result, you cant cover a cube with finite cubes.
A drug is eliminated from the body through urine. Suppose that for a dose of 10 milligrams, the amount A(t) remaining in the body t hours later is given by A(t) = 10(0.7)t and that in order for the drug to be effective, at least 4 milligrams must be in the body.
a. Determine when 2 milligrams is left in the body.
b. What is the half-life of the drug?
Answer: a - 4.512 hours
b - 1.94 hours
Step-by-step explanation:
Given,
a) A(t) = 10 (0.7)^t
To determine when 2mg is left in the body
We would have,
A(t) = 2, therefore
2 = 10(0.7)^t
0.7^t =2÷10
0.7^t = 0.2
Take the log of both sides,
Log (0.7)^t = log 0.2
t log 0.7 = log 0.2
t = log 0.2/ 0.7
t = 4.512 hours
Thus it will take 4.512 hours for 2mg to be left in the body.
b) Half life
Let A(t) = 1/2 A(0)
Thus,
1/2 A(0) = A(0)0.7^t
Divide both sides by A(0)
1/2 = 0.7^t
0.7^t = 0.5
Take log of both sides
Log 0.7^t = log 0.5
t log 0.7 = log 0.5
t = log 0.5/log 0.7
t = 1.94 hours
Therefore, the half life of the drug is 1.94 hours
A family has two cars. During one particular week, the first car consumed 25 gallons of gas and the second consumed 40 gallons of gas. The two cars drove a combined total of 1225 miles, and the sum of their fuel efficiencies was 40 miles per gallon. What were the fuel efficiencies of each of the cars that week?
First Car: ~ miles per gallon
Second Car: ~ miles per gallon
Answer:
first car: 25 mpgsecond car: 15 mpgStep-by-step explanation:
Let x represent the fuel efficiency of the car that used the most gas. Since the sum of the fuel efficiencies is 40 mpg, the other car had a fuel efficiency of (40-x). Then the combined miles driven is ...
40x +25(40-x) = 1225
15x = 225
x = 15
(40-x) = 25
The first car got 25 miles per gallon; the second car got 15 miles per gallon.
_____
We made use of the relation ...
gallons × (miles/gallon) = miles
The envelop weighs 1/2 of the whole balloon The weight of the basket is 3/5 of the envelope if the weight of the balloon will be 210 kg what is the weight of the basket
Given the total weight of the hot air balloon as 210 kg, the envelope weighting half of this, the weight of the basket, which is 3/5 of the envelope, is calculated to be 63 kg.
To determine the weight of the basket attached to a hot air balloon, we can use the information that the envelope weighs [tex]\frac{1}{2}[/tex] of the total weight of the balloon and that the basket weighs [tex]\frac{3}{5}[/tex] of the envelope's weight. If the total weight of the hot air balloon will be 210 kg, then the envelope weighs 105 kg (which is [tex]\frac{1}{2}[/tex] of 210 kg). The weight of the basket can then be calculated as:
Weight of the balloon = 210 kg
Weight of the envelope = 210 * (1/2) = 105 kg
Weight of the basket = 105 * (3/5) = 63 kg
Therefore, the weight of the basket is 63 kg.
A residual plot has data points that are all very close to the x-axis. What does this say about the data?
A) The line of best fit will be a horizontal line.
B) A linear model is appropriate.
C) There is not enough information to determine this.
D) A non-linear model is appropriate.
Answer:
B
Step-by-step explanation:
because, the closer the data points are to the x axis on a residual plot with no definite shape means that a linear model is appropriate for this data set
A linear model is appropriate because, the closer the data points are to the x axis on a residual plot with no definite shape. So, option B is correct.
What is residual plot?A residual is a measure of how far away a point is vertically from the regression line.
It is the error between a predicted value and the observed actual value.
A residual plot is a graph that has data points that are all very close to the x-axis. It shows the residuals on the y axis and the independent variable on the x axis.
The goodness of fit of a linear model is depicted by the pattern of the graph of a residual plot. If each individual residual is independent of each other, they create a random pattern together.
When graphing the residual values you know if a linear model is an appropriate model for your data if the points in the residual plot are scattered.
A linear model is appropriate because, the closer the data points are to the x axis on a residual plot with no definite shape.
So, option B is correct.
Learn more about residual plot here;
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Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12,
what is the probability that
a. Smith will lose his first 5 bets;
b. his first win will occur on his fourth bet?
Answer:
The probability that Smith will lose his first 5 bets is 0.15
The probability that his first win will occur on his fourth bet is 0.1012
Step-by-step explanation:
Consider the provided information.
A roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. Smith always bets that the outcome will be one of the numbers 1 through 12,
It is given that Smith always bets on the numbers 1 through 12.
There are 12 numbers from 1 to 12.
Thus, the probability of success (winning) is= [tex]\frac{12}{38}[/tex]
The probability of not success (loses) is= [tex]1-\frac{12}{38}=\frac{26}{38}[/tex]
Part (A) Smith will lose his first 5 bets.
The probability that Smith loses his first 5 bets is,
[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}=(\frac{26}{38})^5\approx0.15[/tex]
Hence, the probability that Smith will lose his first 5 bets is 0.15
Part (B) His first win will occur on his fourth bet?
Smith’s first win occurring on the fourth bet means that he loses the first 3 bets and wins on the fourth bet. That is,
[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{12}{38}=(\frac{26}{38})^3\times\frac{12}{38}\approx0.1012[/tex]
Hence, the probability that his first win will occur on his fourth bet is 0.1012
If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?
Answer:
23
Step-by-step explanation:
We can check all the possibilities.
It is not necessary to consider y>16, because in this case, 16#y=16 as 16 is too small to be split in y parts.
Now, 1,2,4, 8 and 16 are factors of 16. When you divide 16 by any of the previous integers, the remainder is zero so we discard these.
When y=3, 16=5(3)+1, 16#3=1 so we add y=3. From this, 16=3(5)+1 thus 16#5=1 and we add y=5.
We discard y=6 as 16#6=4 (using that 16=6(2)+4). We also discard y=7 because 16=2(7)+2 then 16#7=2.
For y=9,10,11,12,13,14, when dividing the quotient is one so 16#y=16-y>1 and these values are discarded. However, we add y=15 because 16=15(1)+1 and 16#15=1.
Adding the y values, the sum is 3+5+15=23.
A garden hose emits 9 quarts of water in 6 seconds at this rate How long will it take the hose to emit 12 quarts how much water does the hose emit in 10 second
Answer:
8 seconds
15 quarts
Step-by-step explanation:
Givens
6 seconds emits 9 quarts of water.
x seconds emits 12 quarts of water.
Formula
6/x = 9/12
Solution
Cross multiply
9x = 12*6 Simplify the right
9x = 72 Divide by 9
9x/9 = 72/9
x = 8 seconds Answer.
In 8 seconds, the hose will emit 12 quarts
=========================
Givens
9 quarts of water are emitted in 6 second
x quarts of water are emitted in 10 seconds
Formula
9/x = 6/10 Cross multiply
Solution
9*10 = 6x Combine the left
90 = 6x Divide by 6
90/6=6x/6
15 = x
In 10 seconds the hose emits 15 quarts.
Answer:
Step-by-step explanation:
The garden hose emits 9 quarts of water in 6 seconds. It means that the number of seconds that it takes the garden hose to emit 1 quart of water would be 6/9 = 2/3 seconds
Therefore, the number of seconds that it will take the garden hose to emit 12 quarts of water would be
12 × 2/3 = 8 seconds.
Again, the number of quarts that the garden hose will emit in one second is 9/6 = 3/2 quarts.
Therefore, the number of quarts of water that the hose emits in 10 seconds would be
10 × 3/2 = 15 quarts.
Last year, sales at a book store increased from $5,000 to $10,000. This year, sales decreased to $5,000 from $10,000. What percentage did sales increase last year? What percentage did sales decrease this year? Sales increased last year, from $5,000 to $10,000. When sales dropped from $10,000 to $5,000 this year, sales decreased .
Answer:
Step-by-step explanation:
Last year, sales at a book store increased from $5,000 to $10,000. The amount by which it increased would be 10000 - 5000 = $5000
The percentage by which the sales increased would be
5000/5000 × 100 = 100%
This year, sales decreased to $5,000 from $10,000.The amount by which it decreased would be 5000 - 10000 = - $5000
The percentage by which the sales increased would be
5000/10000 × 100 = 50%
Simon has 20 quarters and 12 dimes.He wants to purchase ice cream for his friends.An ice cream cone cost $1.00.How many cones can Simon buy for his friends?
Answer:
6
Step-by-step explanation:
Each ice cream cone costs 4 quarters or 10 dimes, so Simon has ...
20/4 + 12/10 = 5 + 1.2 = 6.2
times the price of an ice cream cone. He can buy 6 cones for his friends.
A test car went 64 miles on 2 gallons in the morning, 16 miles on 1/2 gallon at noon, and 32 miles on 1 gallon at 5 PM.Is the relationship a proportional relationship? Explain.
It is proportional because the relationship between gallons of gas and distance is constant and linear. The rate is 32 miles per gallon. You can also call 32 the slope of this linear relationship.
Suppose that $p$ and $q$ are positive numbers for which \[\log_9 p = \log_{12} q = \log_{16} (p + q).\] Then $q/p$ can be expressed in the form $(x + \sqrt{y})/z$, where $x$, $y$, and $z$ are positive integers, and $y$ is not divisible by the square of a prime. Find $x + y + z$.
Value of (x + y + z) = 8
Suppose p and q are the positive numbers for which
[tex]log_(9)p=log_(12)q=log_(16)(p+q)[/tex] from the given expression,
[tex]log_(9)p=log_(12)q[/tex]
[tex](logp)/(log9)=(log(q))/(log12)[/tex]
log(p).log(12) = log(q).log(9)
log(q).2log(3) = log(p).log(12) ------(1)
[tex]Now log_(12)q=log_(16)(p+q)[/tex]
[tex](logq)/(log12)=(log(p+q))/(log(16))[/tex]
log(q).log(16) = log(p + q).log(12)
2log(4).log(q) = log(p + q).log12 -------(2)
By adding both the equations (1) and (2),
2log(3).log(q) + 2log(4).log(q) = log(12).log(p) + log(12).log(p + q)
log(q)[2log(3) + 2log(4)] = log(12)[logp + log(p + q)]
2log(q).log(12) = log(12).log[p.(p + q)]
2log(q) = log[p.(p+q)]
q² = p(p + q)
(q)/(p)=(p+q)/(q)
(q)/(p)=(p)/(q)+1
Let (q)/(p)=a
a = (1)/(a)+1
a² - a - 1 = 0 from quadratic formula,
a = [tex]\frac{1\pm \sqrt{(-1)^(2)-4 \times 1 \times (-1)}}{2}[/tex]
a = [tex](1\pm √((1+4)))/(2)[/tex]
a = [tex](1\pm √((5)))/(2)[/tex]
If the solution is represented by (x+√(y))/(z) then it will be equal to
(1+√((5)))/(2) then x = 1, y = 5 and z = 2.
Now we have to find the value of (x + y + z).
By placing the values of x, y and z,
(x + y + z) = (1 + 5 + 2) = 8
Therefore, value of (x + y + z) = 8
Cousin Edie drank all of Clark's egg nog.Edie finds a coupon for 50 cents off that can only be use at a local grocery store.Clark normally buys his egg nog at the supermarket. The tax rate at both stores is 2.25% .Which deal is better? Clark:$5:95 plus tax Eddie:6.35+coupon tax
Answer:
Edie has a better deal since his final price is lower
Step-by-step explanation:
Cousin Edie drank all of Clark's egg nog . Edie finds a coupon for 50 cents off that can only be use at a local grocery store.Clark normally buys his egg nog at the supermarket.
The tax rate at both stores is 2.25%.
Clark buys the egg nog at $5.95 plus tax .
Edie buys the egg nog at $6.35 plus coupon plus tax.
After Edie applies the coupon , the final price is $( 6.35 - 0.50 ) = $5.85
Since the percentage of tax applied is the same, the deal with lower final price is better.
Hence, Edie has a better deal.