P varies directly as the square of Q and inversely as the square root of R. If Q is reduced by 12% and R is reduced by 21%, find the percentage change in P
Let P1, Q1, and R1 be the original values of P, Q, and R, respectively.
Let P2, Q2, and R2 be the new values of P, Q, and R, respectively.
According to the problem, we have the following relationships:
P1 ∝ Q1^2 (P is directly proportional to the square of Q)
P1 ∝ 1/√R1 (P is inversely proportional to the square root of R)
Combining the two relationships, we have:
P1 ∝ Q1^2 / √R1
Now, let's consider the changes in Q and R. Q is reduced by 12%, so Q2 = 0.88Q1.
R is reduced by 21%, so R2 = 0.79R1.
Substituting these new values into the relationship, we have:
P2 ∝ (0.88Q1)^2 / √(0.79R1)
Simplifying, we get:
P2 ∝ 0.7744Q1^2 / √R1
To find the percentage change in P, we compare P2 and P1:
Percentage change = ((P2 - P1) / P1) * 100
= ((0.7744Q1^2 / √R1 - P1) / P1) * 100
However, we don't have enough information to calculate the exact percentage change in P because the values of Q1 and R1 are not given.
To find the percentage change in P, we will calculate the individual percentage changes in Q and R, and then use those values to determine the overall percentage change in P.
Given:
P varies directly as the square of Q: P ∝ Q^2
P varies inversely as the square root of R: P ∝ 1/√R
Let's assume the initial values of Q and R as Q₀ and R₀, respectively.
1. Percentage change in Q:
Q₁ = Q₀ - 12% of Q₀
Percentage change in Q = ((Q₁ - Q₀) / Q₀) * 100
2. Percentage change in R:
R₁ = R₀ - 21% of R₀
Percentage change in R = ((R₁ - R₀) / R₀) * 100
3. Percentage change in P:
Using the given proportional relationships:
P ∝ Q^2 (1)
P ∝ 1/√R (2)
From equation (1), we have:
P₀ = kQ₀^2 (where k is a constant)
From equation (2), we have:
P₀ = k/(√R₀)
Combining the two equations, we get:
kQ₀^2 = k/(√R₀)
Simplifying and rearranging:
Q₀^2 / √R₀ = 1/k
Now, let's consider the new values:
Q₁ = Q₀ - 12% of Q₀
R₁ = R₀ - 21% of R₀
From equation (1) with the new values:
P₁ = kQ₁^2
From equation (2) with the new values:
P₁ = k/(√R₁)
Combining the two equations:
kQ₁^2 = k/(√R₁)
Simplifying and rearranging:
Q₁^2 / √R₁ = 1/k
Comparing the initial and the new equations:
(Q₁^2 / √R₁) / (Q₀^2 / √R₀) = (1/k) / (1/k)
Simplifying:
(Q₁^2 * √R₀) / (Q₀^2 * √R₁) = 1
Taking the square root of R₀:
√R₀/√R₀ = (√R₀ * √R₀) / (√R₀ * √R₁)
1 = √(R₀/R₁)
Substituting this in the initial equation:
(Q₁^2 / √R₁) / (Q₀^2 / √R₀) = (√(R₀/R₁)) / (√(R₀/R₁))
Simplifying:
(Q₁^2 * √R₀) / (Q₀^2 * √R₁) = (√(R₀/R₁)) / (√(R₀/R₁))
1 = 1
This implies that the percentage change in P is 0%.