Three variables p,q, and r are such that p varies directly as q and inversly as the square of r.(a)when p =9,q=12 and r=5.
(b)express q in terms of p and r.
(c)if p is increased by 10% and r decreased by 10%,find;
(1)a simplified expression for the change in q in terms of p and r.
(2)the percantage change in q
p = kq/r^2
(a) plug in your values to find k
(c) q = pr^2/k
replacing p by 1.1p and r by -.9r, we have a new q, which is
q' = (1.1p)(0.9r)^2/k = (1.1 * 0.81) pr^2/k = 0.891q
now just figure the % change q'/q - 1
(a) We are given that p varies directly as q and inversely as the square of r. This can be written as:
p = k * (q / r^2)
where k is the constant of variation.
Substituting the given values p = 9, q = 12, and r = 5, we can solve for k:
9 = k * (12 / 5^2)
9 = k * (12 / 25)
k = 225 / 12
k = 18.75
So the equation becomes:
p = 18.75 * (q / r^2)
(b) To express q in terms of p and r, we can rearrange the equation:
p = 18.75 * (q / r^2)
Multiplying both sides by r^2, we get:
p * r^2 = 18.75 * q
Dividing both sides by 18.75, we obtain:
q = (p * r^2) / 18.75
Therefore, q is expressed in terms of p and r as:
q = (p * r^2) / 18.75
(c) Now, let's consider the changes in p and r, and determine the corresponding change in q.
(1) When p is increased by 10% and r is decreased by 10%, the new values become:
p + (10% of p) = p + 0.1p = 1.1p
r - (10% of r) = r - 0.1r = 0.9r
Substituting these new values into our expression for q:
q = (1.1p * (0.9r)^2) / 18.75
Simplifying, we get:
q = (1.1p * 0.81r^2) / 18.75
q = (0.891p * r^2) / 18.75
So, the simplified expression for the change in q in terms of p and r is:
Change in q = (0.891p * r^2) / 18.75
(2) To find the percentage change in q, we can use the following formula:
Percentage Change = (New Value - Original Value) / Original Value * 100%
In this case, the original value of q is given by:
Original Value = (p * r^2) / 18.75
And the new value (after the changes in p and r) is given by:
New Value = (0.891p * r^2) / 18.75
Substituting these values into the percentage change formula:
Percentage Change = ((0.891p * r^2) / 18.75 - (p * r^2) / 18.75) / ((p * r^2) / 18.75) * 100%
Simplifying, we get:
Percentage Change = (0.891p * r^2 - p * r^2) / (p * r^2) * 100%
Percentage Change = (0.891 - 1) * 100%
Percentage Change = - 0.109 * 100%
Percentage Change ≈ -10.9%
Therefore, the percentage change in q is approximately -10.9%.
To solve this problem, let's break it down step by step:
(a) When p = 9, q = 12, and r = 5, we are given the initial values of the variables. We can use these values to determine the relationship between p, q, and r.
The given statement tells us that p varies directly as q and inversely as the square of r.
So, we can write the equation as:
p = k * (q / r^2)
To find the value of k, we substitute the given values into the equation and solve for k:
9 = k * (12 / 5^2)
9 = k * (12 / 25)
9 = (12k / 25)
Now, we can solve for k by multiplying both sides of the equation by 25/12:
9 * 25/12 = k
k = 18.75
Therefore, the equation relating p, q, and r is:
p = 18.75 * (q / r^2)
(b) To express q in terms of p and r, we rearrange the equation:
p = 18.75 * (q / r^2)
Multiply both sides of the equation by r^2:
p * r^2 = 18.75 * q
Now, divide both sides of the equation by 18.75 to solve for q:
q = (p * r^2) / 18.75
So, q is expressed in terms of p and r as:
q = (p * r^2) / 18.75
(c) If p is increased by 10% and r is decreased by 10%, we need to find the changes in q and the percentage change in q.
(1) To find the change in q, we substitute the new values of p and r into the equation and subtract the original value of q from the new value of q:
Change in q = [(p + 0.1p) * (r - 0.1r)^2] / 18.75 - q
Simplifying the expression, we get:
Change in q = [(1.1p) * (0.9r)^2] / 18.75 - q
(2) Percentage change in q can be calculated by dividing the change in q by the original value of q and multiplying by 100:
Percentage change in q = (Change in q / q) * 100
So, the percentage change in q is given by:
Percentage change in q = [(1.1p * 0.9^2r^2) / 18.75 - q] / q * 100