The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Step-by-step explanation:
Let us define ratio first.
"Ratio is the quantitative relationship between two quantities which tells that one quantity is how many times of another quantity"
Given
Number of fifth grade students = 8
Number of sixth grade students = 7
We have to find the ratio of sixth grade students to fifth grade students
So,
[tex]r = \frac{sixth-grade\ students}{fifth-grade\ students}\\r = \frac{7}{8}\\r = 7:8[/tex]
Hence,
The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Keywords: ratio, fraction
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A woman standing on a cliff is watching a motor boat though a telescope as the boat approaches the shoreline directly below her. If the telescope is 250 feet above the water and if the boat is approaching at 20 feet per second, at what rate is the angle of the telescope changing when the boat is 250 feet from shore
Answer:
Dα/dt = 0.079 degree/sec
Step-by-step explanation:
From problem statement, is easy to see, that if point A is ubicated at the top of the telescope, the shoreline is directly below the woman ( point B), and the point where the boat is, which is at distance x from shoreline is point C. These three point shape a right triangle with angle α (the angle of the telescope).
So we have
tan α = x/250
Differentiating both sides of the equation we get
D (tan α)/dt = ( 1/250)* Dx/dt
sec² α Dα/dt = ( 1/250)* Dx/dt
we already know that Dx/dt = 20 feet/sec
sec² α Dα/dt = 20/250 ⇒ sec² α Dα/dt = 0.08
Dα/dt = 0.08 / sec² α
Then
tan α = 20/250 = 0,08 α = arctan 0.08 α ≈ 5⁰
Dα/dt = 0.08/ sec² α
From tables we get cos 5⁰ = 0.9961 then
1/ 0.9961 = 1.003
sec α = 1.003 and sec² α = 1.0078
Dα/dt = 0.08/ sec² α ⇒ Dα/dt = 0.08/1.0078
Dα/dt = 0.079 degree/sec
The change of position of the boat results in a change of the angle of the telescope, which can be calculated using related rates. With known factors such as the speed of the boat and its distance from shore, the rate of change of the telescope's angle can be found using the principles of trigonometry and calculus.
Explanation:This question is about related rates in calculus. Related rates describe the relationship between different rates of change that are connected to each other. In this case, the changing position of the boat creates a change in the angle of the telescope.
We know the woman is watching a boat that is approaching a shoreline directly below her at 20 feet per second, and we are asked to find the rate that the angle of the telescope is changing when the boat is 250 feet from shore. Here is a way of visualizing it:
Let D be the distance of the boat from the base of the cliff and θ the angle that the telescope makes with the horizontal. We are given that D(t) decreases at 20 feet per second and that when D(t) = 250, we want to know what is dθ/dt.
By using trigonometry, we can find a relationship between D and θ. Specifically, tan(θ) = 250/D, so by implicit differentiation, (sec^2(θ)) * dθ/dt = -250/D^2 * dD/dt. From the given data, dD/dt = -20 and D = 250, so substitute them into the equation and evaluate θ using the tan–1(1) to obtain dθ/dt.
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(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.
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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If natalie and her friends decide to rent 4 lanes at reguler cost for a party ten people need to rent shoes and 4 people are members what is the total cost for the party
The question is missing a tabular data. So, it is attached below.
Answer:
The total cost for the party is $74.50.
Step-by-step explanation:
Given:
Lanes rented at regular cost = 4
Cost of 1 lane rented at regular cost = $9.75
Cost of 1 lane rented for members = $7.50
Cost of 1 shoe rental at regular cost(non members) = $3.95
Cost of 1 shoe rental for members = $2.95
Since, lanes are rented at regular cost, we use unit rate at regular cost
So, cost of 4 lanes rented = [tex]4\times 9.75= \$ 39[/tex]
Now, out of 10 people who rented shoes, 4 are members. So, the number of non-members is given as:
Non members who rented shoes = 10 - 4 = 6
So, 4 members and 6 nonmembers rented shoes.
So, cost of 6 non members renting shoes = [tex]6\times 3.95=\$ 23.70[/tex]
Cost of 4 members renting shoes = [tex]4\times 2.95=\$ 11.80[/tex]
Total cost for the party is the sum of all the costs. This gives,
= 39 + 11.80 + 23.70
= 50.80 + 23.70
= $74.50
Therefore, the total cost for the party is $74.50.
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
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)
Find angle A in the following triangle.
A. 35.54
B. 35.67
C. 36.24
D. 36.77
Tan(Angle) = Opposite leg / Adjacent leg
Tan(A) = 5/7
A = Arctan(5/7)
A = 35.54 degrees.
Answer:
A. 35.54
Step-by-step explanation:
To find the angle A in the diagram above, we will simply use angle formula, we will check and see which angle formula will be best fit to use.
The formulas are;
SOH CAH TOA
sin Ф = opposite / hypotenuse
cos Ф = adjacent / hypotenuse
tan Ф = opposite / adjacent
Looking at the figure given, since we are to find angle A, then our adjacent is 7 and the opposite is 5.
since we are given both opposite and adjacent, then the best formula to use for this is tan Ф
tan Ф = opposite / adjacent
opposite = 5 and adjacent =7
we will go ahead and insert our values
tan A = 5/7
tan A = 0.71429
To get the value of tan A, we will simply take the [tex]tan^{-1}[/tex] of both-side
[tex]tan^{-1}[/tex] tan A = [tex]tan^{-1}[/tex] 0.71429
A = 35.54
Therefore angle A is 35.54
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.
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.
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
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.
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 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.
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
PLEASE HELP
which graph represents the given inequality y < 3x-4
Answer: Here is the answer.
Step-by-step explanation:
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
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.
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 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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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.
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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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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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.
Alexander and Jo live 5 miles apart. They decide to leave their homes at 3 p.M. And travel toward each other. If Alexander bikes 14 miles per hour and Jo jogs 6 miles per hour, when will they meet?
Answer:
Step-by-step explanation:
Here is an illustration of the problem:
----------------------------->|<------------------
A t J
Alex and Jo start from their separate homes and drive towards one another. The t indicates the time at which they meet, which is the same time for both. Filling in a d = rt table:
d = r x t
Alex 14 t
Jo 6 t
The formula for motion is d = rt, so that means that Alex's distance is 14t and Jo's distance is 6t.
14t 6t
---------------------------------->|<------------------
A t J
The distance between them is 5 miles, so that means that Alex's distance plus Jo's distance equals 5 miles. In equation form:
14t + 6t = 5 and
20t = 5 so
t = .25 hours or 15 minutes.
If they leave their homes at 3 and they meet 15 minutes later, then they meet at 3:15.
Tim Has 480 Pokemon cards in his collection, which he arranges in his album. Each page of his album holds 12 cards. The album has 35 pages. Write an expression that shows how many Pokemon cards Tim Has left after he has filled his album completely?
Answer:
Step-by-step explanation:
Total number of Pokemon cards that Tim has in his collection is 480
He arranges the cards in his album and each page of his album holds 12 cards.
The total number of pages in the album is 35. After he has filled his album completely, the total number of cards that the album would contain would be 35 × 12 = 420
Let x represent the number of Pokemon cards he has left. Therefore, the expression becomes
x + 420 = 480
x = 480 - 420 = 60
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.
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.
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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
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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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%