A quantity P varies jointly as the square root of Q and as the inverse of the square of R.Determine the percentage change in P when Q is increased by 44 percent and R decreased by 28 percent
Let P be the quantity, Q be the square root of Q, and R be the inverse of the square of R.
We can express the relationship as:
P = kQ^(1/2) * (1/R^2),
where k is the constant of variation.
To find the percentage change in P, we can differentiate P with respect to Q and R, and then use the chain rule to calculate the percentage changes.
dP/dQ = (1/2)k(Q^(-1/2)) * (1/R^2)
dP/dR = -2kQ^(1/2) * (1/R^3)
Given that Q is increased by 44% and R is decreased by 28%, let's calculate the percentage changes in Q and R:
Percentage change in Q = (44/100) * 100% = 44%
Percentage change in R = (-28/100) * 100% = -28%
Now, let's substitute these values into the equations above:
dP/dQ = (1/2)k(Q^(-1/2)) * (1/R^2)
= (1/2)k(Q^(-1/2)) * (1/R^2) * (0.44) [multiply by 0.44 to account for the percentage change in Q]
= 0.22k(Q^(-1/2)) * (1/R^2)
dP/dR = -2kQ^(1/2) * (1/R^3)
= -2kQ^(1/2) * (1/R^3) * (-0.28) [multiply by -0.28 to account for the percentage change in R]
= 0.56kQ^(1/2) * (1/R^3)
Now, let's calculate the percentage change in P:
Percentage change in P = (dP/dQ * percentage change in Q + dP/dR * percentage change in R) / P * 100%
Percentage change in P = (0.22k(Q^(-1/2)) * (1/R^2) * 44% + 0.56kQ^(1/2) * (1/R^3) * -28%) / (kQ^(1/2) * (1/R^2)) * 100%
Simplifying:
Percentage change in P = (0.22 * 44 - 0.56 * 28) / (Q^(1/2) * R^(-2)) * 100%
Percentage change in P = (9.68 - 15.68) / (Q^(1/2) * R^(-2)) * 100%
Percentage change in P = -6 / (Q^(1/2) * R^(-2)) * 100%
Therefore, the percentage change in P when Q is increased by 44 percent and R is decreased by 28 percent is -6%.
Let's break down the problem step by step.
Step 1: Write the formula for the variation of P.
According to the given information, we can write the formula as:
P ∝ √Q * 1/R^2
Step 2: Introduce the constant of proportionality.
Let's introduce the constant of proportionality and rewrite the formula as:
P = k * √Q * 1/R^2
Step 3: Determine the initial value of P.
Let's assume the initial value of P is P0.
P0 = k * √Q0 * 1/R0^2
Step 4: Determine the new value of P.
P = k * √Q1 * 1/R1^2
Step 5: Find the percentage change in P.
Percentage change in P = ((P - P0) / P0) * 100
Step 6: Express P in terms of initial values.
P = P0 * (√Q1 / √Q0) * (1/R1^2 / 1/R0^2)
Step 7: Simplify the expression.
P = P0 * (√Q1 / √Q0) * (R0^2 / R1^2)
Step 8: Substitute the percentage changes in Q and R.
P = P0 * (√(Q0 + 0.44Q0) / √Q0) * ((R0^2) / (R0 - 0.28R0)^2)
Step 9: Perform calculations and simplify further if possible.
P = P0 * (√(1.44Q0) / √Q0) * (1 / (0.72R0)^2)
P = P0 * (1.2 / 0.72)^2
P = P0 * 1.67
Step 10: Calculate the percentage change in P.
Percentage change in P = ((P - P0) / P0) * 100
Percentage change in P = ((P0 * 1.67 - P0) / P0) * 100
Percentage change in P = 67%
Therefore, the percentage change in P when Q is increased by 44 percent and R is decreased by 28 percent is 67%.