The problem states that quantity P is partly constant and partly varies as the square root of Q. This can be…

Question

The problem states that quantity P is partly constant and partly varies as the square root of Q. This can be written as a variation equation:

ScanSolve AI Tutor · Accepted Answer

Here's the solution to question 4.a:

The problem states that quantity $P$ is partly constant and partly varies as the square root of $Q$ . This can be written as a variation equation:

P = k_1 + k_2\sqrt{Q}

where $k_1$ is the constant part and $k_2$ is the constant of proportionality.

Use the first set of given values ( $P=20, Q=4$ ) to form an equation.
$20 = k_1 + k_2\sqrt{4}$ $20 = k_1 + 2k_2 \quad \text{(Equation 1)}$
Use the second set of given values ( $P=60, Q=100$ ) to form another equation.
$60 = k_1 + k_2\sqrt{100}$ $60 = k_1 + 10k_2 \quad \text{(Equation 2)}$
Solve the system of linear equations for $k_1$ and $k_2$ .
Subtract Equation 1 from Equation 2:
$(60 - 20) = (k_1 - k_1) + (10k_2 - 2k_2)$ $40 = 8k_2$ $k_2 = \frac{40}{8}$ $k_2 = 5$
Substitute $k_2 = 5$ into Equation 1:
$20 = k_1 + 2(5)$ $20 = k_1 + 10$ $k_1 = 20 - 10$ $k_1 = 10$
Write the complete variation equation using the found constants.
$P = 10 + 5\sqrt{Q}$
Find $Q$ when $P=22$ .
Substitute $P=22$ into the complete equation:
$22 = 10 + 5\sqrt{Q}$
Subtract 10 from both sides:
$22 - 10 = 5\sqrt{Q}$ $12 = 5\sqrt{Q}$
Divide by 5:
$\sqrt{Q} = \frac{12}{5}$ $\sqrt{Q} = 2.4$
Square both sides to find $Q$ :
$Q = (2.4)^2$ $Q = 5.76$
The value of $Q$ when $P=22$ is $\boxed{5.76}$ .

The problem states that quantity P is partly constant and partly varies as the square root of Q. This can be written as a variation equation: