Answer:
The number of Senators are 120 and number of Representatives are 440.
Step-by-step explanation:
Given,
Total number of members = 560
Solution,
Let the number of Senators be 'x'.
Since, the number of representatives is 160 less than five times the number of senators.
Framing the above sentence in equation form,we get the number of Representative.
So, number of Representative = [tex]5x-160[/tex]
Now, Total number of members is the sum of total number of Senators and total number of representative.
Total number of members = Total number of Senators + Total number of representative
On substituting the values, we get;
[tex]x+5x-160=560\\\\6x=560+160\\\\6x=720\\\\x=\frac{720}{6}\\\\x=120[/tex]
Total number of Senators = 120
So, number of Representative = [tex]5\times120-160=600-160=440[/tex]
Thus the number of Senators are 120 and number of Representatives are 440.
What would be the difference at the end of one year between the simple interest earned on a deposit of $450 at 4.5% and the compound interest earned on $450 at 4.5% compounded annually?
$22.50
$22.25
$0
$20.25
Answer: $0
Step-by-step explanation:
The formula for simple interest is expressed as
I = PRT/100
Where
P represents the principal
R represents interest rate
T represents time in years
I = interest after t years
From the information given
T = 1 year
P = $450
R = 4.5%
Therefore
I = (450 × 4.5 × 1)/100
I = 2025/100
I = 20.25
For compound interest,
Initial amount deposited into the account is $450 This means that the principal,
P = 450
It was compounded annually. This means that it was compounded once in a year. So
n = 1
The rate at which the principal was compounded is 4.5%. So
r = 4.5/100 = 0.045
It was compounded for just a year. So
t = 1
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years. Therefore
A = 450 (1+0.045/1)^1×1
A = 450(1.045) = $470.25
Compound interest = 470.25 - 450 = 20.25
The difference is 20.25 - 20.25 = 0
Answer:
$0
Step-by-step explanation:
There is no difference between the simple interest and compound interest at the end of one year.
construct an equation for the expression: the sum of a number and itself is 8. Show the solution to the equation and prove your solution to be true through your work.
PLEASE HELP ME I'M SO BEHIND ON MY SCHOOLING NEED THE ANSWER ASAP PLEASE!!!!!!!! THANK YOU
Answer:
equation : 2x = 8
solution : x = 4
Step-by-step explanation:
let the number be x
"the sum of a number and itself "
= sum of x and x
= x + x
= 2x
"the sum of a number and itself is 8"
2x = 8
solving the equation, divide both sides by 2
2x = 8
x = 8/2
x = 4
Imagine that you take a road trip from A to D, but you have to do it in segments. Let’s say the distance from A to B is 145 miles; from B to C is 160 miles, and C to D is 115 miles. It takes you 6 hours to drive from A to D. What was your average speed (in miles per hour) during your trip from A to D? (Hint: How many miles did you drive from A to D? Then divide the total miles by the number of hours).
A. 50
B. 60
C. 70
D. 80
E. 90
Answer:
C
Step-by-step explanation:
The average speed can be obtained by adding the distances together and dividing by the total amount of time.
The total distance the man traveled from A to D is the addition of the distance from A to B plus the distance from B to C plus the distance from C to D. This is mathematically equal to 145 + 160 + 115 = 420 miles.
Dividing 420 miles by 6 = 70 miles per hour
Need help this and I need both them to show work
Answer:
11. ∠ABC = 96°
12. (x – 2)² + (y + 3)² = 4
Step-by-step explanation:
11. The inscribe angle (the angle inside the circle, ∠ABC) is equal to half of the outer circle.
∠ABC = 1/2∠AC
∠ABC = 1/2(192°) = 96°
12. The general equation for a circle is: (x – h)² + (y – k)² = r², where
h and k are the center of the circle (h, k), and r is the radius.
Look at the graph, the circle is centered at (2, -3), so
h=2
k=-3
and the radius of the circle is 2, so
r=2
Plug it all back into the equation:
(x – h)² + (y – k)² = r²
(x – (2))² + (y – (-3))² = (2)²
(x – 2)² + (y + 3)² = 4
f(m) = 2.5 + 0.12m
If Natalie paid $6.82 for one call, how many minutes long was it?
Select one:
A. 28
B. 36
C. 42
D. 45
Answer:
36 minutes long
Step-by-step explanation:
6.82 = 2.5 + 0.12m
4.32 = 0.12m
36 = m; 36 minutes long.
Answer:
B. 36.
Step-by-step explanation:
Substituting 6.82 for m:
6.82 = 2.5 + 0.12m
0.12 m = 6.82 - 2.5
0.12m = 4.32
m = 4.32 / 0.12
m = 36 minutes.
Point C = (4/9,-5/9) lies on the circle with the center at S=(-2/3, 3/4). If CD is a diameter of circle S, find the coordinates for D. Answer must be given as a simplified fraction to receive full credit.
Answer:
Step-by-step explanation:
If CD is a diameter of circle S, then CD goes through circle S at point S. CS is a radius, and so is DS. That means that they are the same length. That also means that S is the midpoint of CD. We can use the midpoint formula and the 2 points we are given to find the other endpoint, D.
[tex](-\frac{2}{3},\frac{3}{4}) =(\frac{\frac{4}{9}+x }{2},\frac{-\frac{5}{9}+y }{2})[/tex]
To solve for x, we will use the x coordinate of the midpoint; likewise for y. x first:
[tex]-\frac{2}{3}=\frac{\frac{4}{9}+x }{2}[/tex]
Multiply both sides by 2 to get rid of the lowermost 2 and get
[tex]-\frac{4}{3}=\frac{4}{9}+x[/tex]
Subtract 4/9 from both sides to get
[tex]x=-\frac{16}{9}[/tex]
Now y:
[tex]\frac{3}{4}=\frac{-\frac{5}{9}+y }{2}[/tex]
Again multiply both sides by that lower 2 to get
[tex]\frac{3}{2}=-\frac{5}{9}+y[/tex]
Add 5/9 to both sides to get
[tex]y=\frac{37}{18}[/tex]
And there you go!
[tex]D(-\frac{16}{9},\frac{37}{18})[/tex]
Could someone help me with this exercise?
Answer:
5%: 66 2/3 mL6.5%: 133 1/3 mLStep-by-step explanation:
Let x represent the number of milliliters of 6.5% vinegar required. Then the total amount of acetic acid in the mix is ...
6.5%·x + 5%(200 -x) = 6%·200
1.5x = 200 . . . . . . . . . . . . . multiply by 100, subtract 1000
x = 200/1.5 = 133 1/3 . . . . mL of 6.5% vinegar
200-x = 66 2/3 . . . . . . . . . mL of 5% vinegar
I NEED THE ANSWERS TO THIS QUESTION AS SOON AS POSSIBLE CHECKING UP ON BEHIND SCHOOL WORK THANK YOU
Answer:
See all four answers with their explanation below and the graph attached.The green arrow on the number line is the solution of the inequality.Explanation:
a). Description of what Brian did wrong.
Brian did not solve the inequality correctly. It seems he made several wrong steps:
He added up 2 and 3 to get 5, which is wrong because 2 and 3 are in opposite sides of the inequalityHe use the symbol <, which is wrong because there is a negative sing in front of the varialbe (x) which changes the symbol to >.b). Your work solving the inequality:
Subtract 2 from both sides:- x < 3 - 2
-x < 1
Muliply both sides by - 1, which changes the symbol < to >:x > - 1
Then the solution of the inequality is all the real numbers greater than - 1.
c). The correct solution graphed on a number line
Since the number - 1 is not included in the solution set of the inequality you must use an oper circle around the number - 1 on the number line.Since the solution set is all the numbers greater than - 1 you draw an arrow pointing to the right of the number - 1 on the number line.See the correct graph in the diagram attached. The green line on the number line is the solution to the inequality.d). The correct solution in set notation.
Three valid forms indicating the solution in set notation are:
{x: > - 1}, which is read x such that x is greater than - 1{x | x > - 1}, which is read x such that x is greater than - 1{x ∈ R | x > - 1}, which is read, x belonging to real numbers, such that x is greater than - 1.Both the colon (:) and the straight bar (|) mean "such that".
A person rolls a standard six-sided die 9 times. In how many ways can he get 3 fours, 5 sixes, and 1 two?
The person can roll the die in 1512 different ways to get 3 fours, 5 sixes, and 1 two in 9 rolls.
Explanation:When rolling a standard six-sided die, there are 6 possible outcomes for each roll. To find the number of ways the person can get 3 fours, 5 sixes, and 1 two in 9 rolls, we can use the concept of combinations. The number of combinations of getting 3 fours, 5 sixes, and 1 two from 9 rolls is calculated by multiplying the number of ways to choose the positions of the fours, sixes, and two, and then multiplying it by the probability of each outcome.
To calculate this, we can use the formula for combinations:
C(n, r) = n! / ((n - r)! x r!)
Using this formula, we can find the number of ways to choose the positions of the fours, sixes, and two:
Number of ways to choose the positions of the fours: C(9, 3) = 9! / ((9 - 3)! x3!) = 84Number of ways to choose the positions of the sixes: C(6, 5) = 6! / ((6 - 5)! x 5!) = 6Number of ways to choose the positions of the two: C(3, 1) = 3! / ((3 - 1)! x 1!) = 3Finally, we can multiply these numbers together to find the total number of ways:
Total number of ways = 84 x 6 x 3 = 1512
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The useful life of a certain piece of equipment is determined by the following formula: u =(8d)/h^2, where u is the useful life of the equipment, in years, d is the density of the underlying material, in g/cm3, and h is the number of hours of daily usage of the equipment. If the density of the underlying material is doubled and the daily usage of the equipment is halved, what will be the percentage increase in the useful life of the equipment?A. 300%B. 400%C. 600%D. 700%E. 800%
Answer:
E. 800%
Step-by-step explanation:
Since,
u = 8d/h² __________ eqn (1)
Now, density (d) is doubled and usage (h) is halved.
Hence the new life (u'), becomes:
u' = 8(2d)/(0.5h)²
u' = 8(8d/h²)
using eqn (1), we get:
u' = 8u
In percentage,
u' = 800% of u
In words, the percentage increase in useful life of the equipment is 800%.
Answer: E. 800%
Step-by-step explanation:
The useful life of a certain piece of equipment is determined by the following formula: u =(8d)/h^2, where u is the useful life of the equipment, in years, d is the density of the underlying material, in g/cm3, and h is the number of hours of daily usage of the equipment.
Assuming d = 1 and h = 1, then
u = (8 × 1)/1^2 = 8
If the density of the underlying material is doubled and the daily usage of the equipment is halved, it means that
d = 2 and h = 1/2 = 0.5, therefore,
u = (8 × 2)/0.5^2 = 16/0.25 = 64
64/8 = 8
The percentage increase in the useful life of the equipment is
8 × 100 = 800%
If log_a(13)= 4, what is the value of a^4?
Answer:
Step-by-step explanation:
[tex]log_{a}(13)=4\\so~ a^{4}=13[/tex]
A machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the machine to work properly. Assume the probability of one part working does not depend on the functionality of any of the other parts. Also, assume that the probabilities of the individual parts working are P(A) = P(B) = 0.93, P(C) = 0.95, and P(D) = 0.92. Find the probability that the machine works properly.
Answer:
0.756
Step-by-step explanation:
It is given that a machine has four components, A, B, C, and D.
[tex]P(A)=P(B)=0.93, P(C)=0.95,P(D)=0.92[/tex]
If these components set up in such a manner that all four parts must work for the machine to work properly.
We need to find the probability that the machine works properly. It means we have to find the value of [tex]P(A\cap B\cap C\cap D)[/tex].
If two events X and Y are independent, then
[tex]P(X\cap Y)=P(X)\times P(Y)[/tex]
Assume the probability of one part working does not depend on the functionality of any of the other parts.
[tex]P(A\cap B\cap C\cap D)=P(A)\times P(B)\times P(C)\times P(D)[/tex]
Substitute the given values.
[tex]P(A\cap B\cap C\cap D)=0.93\times 0.93\times 0.95\times 0.92[/tex]
[tex]P(A\cap B\cap C\cap D)=0.7559226[/tex]
[tex]P(A\cap B\cap C\cap D)\approx 0.756[/tex]
Therefore, the probability that the machine works properly is 0.756.
Final answer:
The probability that the machine works properly is found by multiplying the probabilities of all four components working: P(A) * P(B) * P(C) * P(D) = 0.93 * 0.93 * 0.95 * 0.92 = 0.7513, or 75.13%.
Explanation:
To find the probability that the machine works properly, we need to calculate the probability that all four components, A, B, C, and D, are working. Since the functionality of each component is independent, we can find this combined probability by multiplying the individual probabilities together.
The probability of A working is P(A) = 0.93, B working is P(B) = 0.93, C working is P(C) = 0.95, and D working is P(D) = 0.92. So the probability of the machine working is:
P(Machine works) = P(A) * P(B) * P(C) * P(D) = 0.93 * 0.93 * 0.95 * 0.92 = 0.7513
Therefore, the probability that the machine works properly is 0.7513, which is 75.13%.
geometry, thanks if you help me! :)
Answer:
A is a function/B is not.
Step-by-step explanation:
A, there are 1 x value for 1 y value.
B, is not a function because for 1 x value, there are 2 y values.
Answer:
See below.
Step-by-step explanation:
Graph A is a function because it passes the vertical line test. You could draw a vertical line anywhere on the graph which will only pass through the graph at one point. It is a many-to-one relation.
Graph B is not a function because some vertical lines will pass through the graph at 2 points. It is a one-to-many relation.,
One-to-one and many-to-one relations are functions but one-to-many are not functions.
You are a researcher studying the lifespan of a certain species of bacteria. A preliminary sample of 35 bacteria reveals a sample mean of ¯ x = 80 x ¯ = 80 hours with a standard deviation of s = 4.8 s = 4.8 hours. You would like to estimate the mean lifespan for this species of bacteria to within a margin of error of 0.65 hours at a 98% level of confidence. What sample size should you gather to achieve a 0.65 hour margin of error? Round your answer up to the nearest whole number.
To calculate the desired sample size with a 98% confidence level and 0.65-hour margin of error, we use the appropriate formula for sample size calculation in statistics, which leads us to a required sample size of 98 bacteria.
Explanation:In this case, we are trying to determine the appropriate sample size using statistical methodology. To calculate this, we can use the formula for the sample size which is n = (Z∗σ/E)^2. Here, 'Z' is the z-value corresponding to the desired confidence level (for 98% confidence, Z score or z-value is 2.33), σ is the standard deviation (which is 4.8 hours in this case), and 'E' is the desired margin of error (0.65 hours).
Substituting these values into the formula gives: n = (2.33∗4.8/0.65)^2. After simplifying this calculation, we get n = 97.46. However, we can't have a fractional part of a sample, so we round up to the nearest whole number which is 98. So, you would need a sample size of 98 bacteria in order to achieve a 0.65-hour margin of error at a 98% confidence level.
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A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Estimate the height h of the tree to the nearest tenth of a meter.
This is right angle trig. We know that...
cos(18°) x hypotenuse = 100
hypotenuse = 100/cos(18°)
hypotenuse = 105.15 meters approx.
Because they want the height of the tree we want "sin(18°) x hypotenuse".
sin(18°) x 105.15 = 32.5 meters approx.
answer: 32.5 meters approx.
The required height of the tree is 32.5 meters.
Given that,
A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. To estimate the height h of the tree to the nearest tenth of a meter.
These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.
Here,
let the height of the tree be x, and the slant height from the foot of the person to the top of the tree be h,
according to the question,
base length = 100
cos 18 = 100 / h
h = 105.14
Now,
sin 18 = x / h
sin 18 = x / 105.14
x = 32.5 meters
Thus, the required height of the tree is 32.5 meters.
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NEED HELP NOW 30 POINTS. Which number is closest to square root of 57? 7.8 7.5 8.5 6.7
√57 = 7.549
Round to 7.5
Answer:
7.5
Step-by-step explanation:
7.549
What is the perimeter of rectangle ABCDABCD? (1) The longer side of the rectangle is 2 meters shorter than its diagonal (2) The ratio of the shorter side of the rectangle to its diagonal is 13
Answer:
(1) P=4a+4
(2) P=2a+4[tex]\sqrt{42}[/tex]a
Step-by-step explanation:
Well, let us call longer side b, shorter side a and diagonal d. The perimeter of a rectangle is 2*(a+b). Let us write this formula as P=2*(a+b)
In (1) it is stated that b=a+2. Hence, the perimeter of the rectangle is P=2*(a+b). In terms of b, let us write (a+2). So P=2*(a+a+2)=4a+4.
In (2) it is stated that d/a=13 or d=13a. From Pythagorean theorem d^2=a^2+b^2. Hence, b^2=168a^2 or b=2[tex]\sqrt{42}[/tex]a. Finally, P=2*(a+b)=2*(a+2[tex]\sqrt{42}[/tex]a)=2a+4[tex]\sqrt{42}[/tex]a
Jack accepted a job to paint a new house. He calculated that he can complete the job in 30 hours. He hired a trainee to assist him. He has seen the trainee work and estimates that it the trainee working alone would take 45 hours to complete the job. How much time should the two of them working together need to paint the house? (Round your answer to the nearest tenth.)
Answer:
20hours
Step-by-step explanation:
Jack can complete the painting in 30hours. The fraction he can paint per hour is thus 1/30
Trainee can complete the painting in 45hours, the fraction he can paint per hour is 1/45
If they are working together, the fraction that can paint in an hour is :
1/45 + 1/30 = 5/90 = 1/18
Now we know they can paint 1/18 of the room in an hour, the number of hours needed to completely paint the room is thus 1/1/18 = 18 hours
Rounding up answer to the nearest tenth is 20hours
How many positive integers less than 10,000 are such that the product of their digits is 210?A. 24
B. 30
C. 48
D. 54
E. 72
Answer:
correct option is D. 54
Step-by-step explanation:
given data
product of digits = 210
integers = 10000
to find out
How many positive integers less than 10,000
solution
we know product of digits 210 are = 1 × 2× 3×5×7
210 = 1 × 6 × 5 × 7
here 2 × 3 = 6 ( only single digit )
here 4 digit numbers with combinations of the digits are = {1,6,5,7} and {2,3,5,7}
3 digit numbers with combinations of digits are = {6,5,7}
and product of their digits = 210
so combination will be
combinations of {1,6,5,7} is 4! = 4 × 3 × 2 × 1 = 24
combinations of {2,3,5,7} is 4! = 4 × 3 × 2 × 1 = 24
combinations of {6,5,7} is 3! = 3 × 2 × 1 = 6
so total is = 24 + 24 + 6
total is = 54
so correct option is D. 54
VThe speed of sound is approximately 768768768 miles per hour. When an object travels faster than the speed of sound, it creates a sonic boom. Write an inequality that is true only for speeds (s)(s)left parenthesis, s, right parenthesis at which a moving object creates a sonic boom.
Answer:
The Inequality [tex]s > 768\ mi/hr[/tex] is true at which a moving object creates a sonic boom.
Step-by-step explanation:
Given:
Speed of Sound = 768 miles per hour
Also Given:
When an object travels faster than the speed of sound, it creates a sonic boom.
We need to find the inequality which is true only for speeds (s) at which a moving object creates a sonic boom
So We can say;
When;
[tex]s < 768\ mi/hr[/tex] ⇒ Normal sound no sonic boom (false)
[tex]s = 768\ mi/hr[/tex] ⇒ speed of sound but no sonic boom (false)
[tex]s > 768\ mi/hr[/tex] ⇒ sonic boom is created (True)
Hence The Inequality [tex]s > 768\ mi/hr[/tex] is true at which a moving object creates a sonic boom.
What type of variable is the number of gallons of gasoline pumped by a filling station during a day? Select one: a. Qualitative b. Continuous c. Attribute d. Discrete
Answer:
b. Continuous
Step-by-step explanation:
Continuous variable is a variable that can take on any value between its minimum value and its maximum value. A continuous variable is a type of quantitative variable used to describe data that is measurable while Discrete variables are countable in a finite amount of time. Now, justifying whether the gallons of gasoline pumped by a filling station during a day is continuous or discrete. The number of gallons of gasoline pumped by a filling during during a day is a continuous variable because it has a measurable volume which can take value from the minimum to maximum values of the total volume of gasoline the filling station have in the storage tank.
The number of gallons of gasoline pumped by a filling station in a day is a Continuous variable because it can take on an infinite number of values between any two given points.
Explanation:The number of gallons of gasoline pumped by a filling station during a day is a Continuous variable. Continuous variables are numerical variables that have an infinite number of values between any two values. A continuous variable can be numeric or date/time. For instance, the number of gallons pumped, which can take on possibly infinite values ranging from zero upwards, is a continuous variable.
In contrast, a Discrete variable is a variable whose value is obtained by counting. A Qualitative (or categorical) variable is a variable that can be put into categories, but the numbers placed on the categories have no numerical meaning. An Attribute is a specific domain within qualitative data, like a sub-category.
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If ABCD is congrunent to , pqrs, then AD is congrent to ?
Answer:
AD is congruent to RS
Step-by-step explanation:
we know that
If two figures are congruent, then its corresponding sides and its corresponding angles are congruent
In this problem
If
ABCD≅PQRS
then
Corresponding angles
∠A≅∠P
∠B≅∠Q
∠C≅∠R
∠D≅∠S
Corresponding sides
AB≅PQ
BC≅QR
CD≅RS
AD≅PS
Nancy knows that the perimeter of her garden is 28 feet, and the length is 8 feet. She forgot to measure the width, but was able to solve for it by subtracting 16 from 28, and then dividing by 2. Which of the equations below can be solved with these steps?
Answer:
W = (P - 2L)/2 = (28- 2*8)/2 = 6
Where W is the width, P is the perimeter and L is the length of the garden.
Step-by-step explanation:
Since the equations are not given, i will try to come up with the similar equation than the ne that was the correct option in this exercise.
You can obtain the perimeter of a rectangle by summing the length of its four sides. Thus, the perimeter of the garden, lets call it P, is 2W + 2L, where W denotes the width and L the length. Since Nancy knows the perimeter, in order to calculate the width she can substract from it 2L (which is also known), and divide by 2 to obtain W, thus
W = (P - 2L)/2
If we reemplace P by 28 and L by 8, we obtain
W = (28-8*2)/2 = (28-16)/2 ) = 12/2 = 6.
4x/ 2x + y + 2y/ 2x + y Perform the indicated operation. Be sure the answer is reduced.
Answer:
[tex]\frac{4x}{2x+y} +\frac{2y}{2x+y}=2[/tex]
Step-by-step explanation:
Given:
The expression to simplify is given as:
[tex]\frac{4x}{2x+y} +\frac{2y}{2x+y}[/tex]
Since, the denominator is same, we add the numerators and divide it by the same denominator. This gives,
[tex]\frac{4x+2y}{2x+y}[/tex]
Now, we simplify further by factoring out the common terms from the numerator and denominator if possible.
We observe that, 2 is a common factor to both [tex]4x\ and\ 2y[/tex]. So, we factor out 2 from the numerator. This gives,
[tex]\frac{2(2x+y)}{2x+y}[/tex]
Now, the term [tex]2x+y[/tex] is common in both the numerator and denominator. Hence, [tex]\frac{2x+y}{2x+y}=1[/tex]
So, the simplified form is:
[tex]=2\times \frac{2x+y}{2x+y}\\\\=2\times 1\\\\=2[/tex]
What conic section degenerates into a line?
Parabola
Hyperbola
Circle
Ellipse
I think it's Ellipse? I'm not sure though
Some degenerate parabola cases form a single straight line, while other cases form one pair of parallel lines.
A degenerate hyperbola forms two lines that intersect at the vertex of the cone. We can rule out choice B.
A degenerate circle is a single point, so we can rule out choice C.
A degenerate ellipse is also a single point. Any circle is an ellipse (but not the other way around). We can rule out choice D.
Kelly drove north for 9 miles and then east for 12 miles at an average rate of 42 miles per hour to arrive at the town of Prime. Brenda left from the same location, at the same time, and drove along a straight road to Prime at an average rate of 45 miles per hour. How many minutes earlier than Kelly did Brenda arrive?
Answer:
Step-by-step explanation:
This is one of the more interesting motion problems I've seen. I like it! If Kelly is driving north (straight up) for 9 miles, then turns east (right) and drives for 12 miles, what we have there are 2 sides of a right triangle. The hypotenuse is created by Brenda's trip, which originated from the same starting point as Kelly and went straight to the destination, no turns. We need the distance formula to solve this problem, so that means we need to find the distance that Brenda drove. Using Pythagorean's Theorem:
[tex]9^2+12^2=c^2[/tex] and
[tex]81+144=c^2[/tex] and
[tex]225=c^2[/tex] so
c = 15.
Brenda drove 15 miles. Now we can fill in a table with the info:
d = r x t
Kelly 12+9 42 t
Brenda 15 45 t
Because they both left at the same time, t represents that same time, whatever that time is. That's our unknown.
If d = rt, then for Kelly:
21 = 42t
For Brenda
15 = 45t
Solve Kelly's equation for t to get
t = 1/2 hr or 30 minutes
Solve Brenda's equations for t to get
t = 1/3 hr or 20 minutes
That means that Brenda arrived at the destination 10 minutes sooner than Kelly.
Find the length of the curve with equation $y=\dfrac{1}{3}(x^2+2)^{3/2}$ for $1\leq x\leq 4$.
To find the length of the curve defined by $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$, we must integrate the square root of the sum of 1 and the square of the derivative of our function from 1 to 4.
To begin with, we want to find the derivative of our function. In other words, we need to compute $dy/dx$.
The derivative, with the Chain Rule gives us:
$y' = \dfrac{1}{3} \cdot \dfrac{3}{2} \cdot 2x \cdot (x^2+2)^{1/2}$
Simplifying gives:
$y' = x \cdot (x^2+2)^{1/2}$
Next, we substitute $y'$ into the formula for finding the length of a curve:
$L = \int_{a}^{b}\sqrt{1+(y')^2}dx$
We should note that $a = 1$ and $b = 4$ here. We substitute $y'=x \cdot (x^2+2)^{1/2}$ and obtain:
$L = \int_{1}^{4}\sqrt{1+(x \cdot (x^2+2)^{1/2})^2}dx$
We can now evaluate the integral, where we will square the entire derivative and add 1 as being under the square root.
So, finally, evaluating this integral gives us the length of the curve, which in this case is 24.
Therefore, the length of the curve $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$ is 24.
Learn more about Curve Length here:https://brainly.com/question/32515746
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Two sides of a triangle have the following measures: 15 and 39. What is the range of possible measures for the third side?
A) 15 < x < 24
B) 24 < x < 39
C) 24 < x < 54
D) 39 < x < 54
Answer:
A) 15 < x < 24
Step-by-step explanation:
According to the law of the triangle, we know that the lengths of any two sides of a triangle are more significant than the length of the 3rd side. Here A is the answer because 15 is the length of the first side, and 39 is the length of the third side. So the length of the 2nd side is (39-15) = 24. It means that the length of the 2nd side is included between 15<x<24.
Liam and evan are mixing paint. Liam uses 2 quarts of yellow paint and adds 3 1/4 jars of blue paint. Evan uses 1/2 quart of yellow paint and adds 5 1/2 jars of red paint. They end up with the same volume of paint. Write an equation to represent the situation.
Answer:
The required equation is given by,
2 + [tex]\frac {13x}{4}[/tex] = [tex]\frac {1}{2} + \frac {11y}{2}[/tex]
Step-by-step explanation:
Let, each jar of Liam's paint contains x quarts of paint.
Then, Liam's solution contains,
2 + [tex]3\dfrac {1}{4} \times x[/tex] quarts of paint or,
2 + [tex]\frac {13x}{4}[/tex] quarts of paint
and,
let, each jar of Evan's paint contains, y quarts of paint.
Then, Evan's solution contains,
[tex]\frac {1}{2} + 5 \dfrac {1}{2} \times y[/tex] quarts of paint or,
[tex]\frac {1}{2} + \frac {11y}{2}[/tex] quarts of paint
now, according to the question,
the required equation is given by,
2 + [tex]\frac {13x}{4}[/tex] = [tex]\frac {1}{2} + \frac {11y}{2}[/tex]
To find an equation that represents the volume of paint mixed by Liam and Evan, we assume that 1 jar equals 1 quart. By adding the quarts of paint each person uses, we get the equation 2 + 3 1/4 = 1/2 + 5 1/2, meaning they both used the same total volume of paint.
Explanation:To find the equation that represents the situation, we need to equate the total volume of paint used by Liam to the total volume used by Evan. We know that Liam uses 2 quarts of yellow paint and adds 3 1/4 jars of blue paint. Evan, on the other hand, uses 1/2 quart of yellow paint and adds 5 1/2 jars of red paint. Assuming that 1 jar is equivalent to 1 quart, we can simply add the volumes for Liam and Evan:
Liam's total volume = 2 quarts (yellow) + 3 1/4 quarts (blue)
Evan's total volume = 1/2 quart (yellow) + 5 1/2 quarts (red)
Since they end up with the same volume of paint, we have the equation:
2 + 3 1/4 = 1/2 + 5 1/2
To solve for quarts, simplify both sides:
5 1/4 = 6
This equation represents the volumes of paint mixed by Liam and Evan.
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sin (cos^-1 14x) Show the triangle that is correct to write the given expression as an algebraic expression?
Answer:
[tex]Sin(Cos^{-1} (14x))=\sqrt{1-196x^2}[/tex]
Step-by-step explandation:
First of all, from the figure we can define the cosine and sine functions as
[tex]Cos(theta)=\frac{adjacent }{hypotenuse }[/tex]
[tex]Sin(theta)=\frac{Opposite}{hypotenuse }[/tex]
And by analogy with the statement:
[tex]14x=\frac{adjacent }{hypotenuse }[/tex]
Which can be rewritten as:
[tex]\frac{14x}{1}=\frac{adjacent }{hypotenuse }[/tex]
You have then that, for the given triangle, the values of the adjacent and hypotenuse sides, are then given by:
:
Adjacent=14x
Hypotenuse=1
And according to the Pythagorean theorem:
[tex] Opposite=\sqrt{1-(14x)^2}[/tex]
Finally, by doing:
[tex]Cos^-1(14x)=theta[/tex]
We have that:
[tex]Sin(Cos^{-1} (14x))=Sen(theta)=\frac{Opposite}{hypotenuse}=\frac{\sqrt{1-(14x)^2}}{1}=\sqrt{1-(14x)^2}[/tex]
The expression [tex]\( \sin(\cos^{-1}(14x)) \)[/tex] is equivalent to the expression [tex]\[ \sqrt{1 - 196(x)^2} \][/tex].
To express [tex]\( \sin(\cos^{-1}(14x)) \)[/tex] using a right triangle, we proceed as follows:
1. Understand the expression:
[tex]\( \cos^{-1}(14x) \)[/tex] denotes the angle [tex]\( \theta \)[/tex] such that [tex]\( \cos(\theta) = 14x \)[/tex].
We are required to find [tex]\( \sin(\theta) \)[/tex].
2. Use a right triangle:
Let's consider a right triangle where:
One of the acute angles is [tex]\( \theta \)[/tex].
Assume [tex]\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{14x}{1} \)[/tex]
To find the opposite side (let's call it [tex]\( \sqrt{1 - (14x)^2} \))[/tex], we use the Pythagorean identity:
[tex]\[ \sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - (14x)^2} \][/tex]
[tex]\[ \sin(\cos^{-1}(14x)) = \sqrt{1 - (14x)^2} \][/tex]
[tex]\[ \sin(\cos^{-1}(14x)) = \sqrt{1 - 196(x)^2} \][/tex]