If P varies directly as the square root of q and inversely as the square of r, what is the… — JAMB UTME

If $P$ varies directly as the square root of $q$ and inversely as the square of $r$ , what is the percentage change in $P$ when $q$ is increased by $44\%$ and $r$ is decreased by $20\%$ ?

A
$50\%$ increase
B
$125\%$ increase
$87.5\%$ increaseAnswer
D
$23.2\%$ decrease

Answer

An increase of

87.5\%

The variation relation is

P = k \frac{\sqrt{q}}{r^2}

. Increasing

q

44\%

scales

\sqrt{q}

\sqrt{1.44} = 1.2

. Decreasing

r

20\%

scales

r^2

(0.8)^2 = 0.64

. Dividing

1.2

0.64

gives

1.875

, meaning the new value

P'

187.5\%

of the original

P

, which corresponds to an increase of

87.5\%

Step-by-Step Solution

Set up the joint variation equation

P = k \frac{\sqrt{q}}{r^2}

, where

k

is a constant of variation.

Direct variation puts

\sqrt{q}

in the numerator, and inverse variation puts

r^2

in the denominator.

Express new values

q'

and

r'

in terms of original variables

q' = 1.44q

and

r' = 0.80r

An increase of

44\%

gives

1 + 0.44 = 1.44

, and a decrease of

20\%

gives

1 - 0.20 = 0.80

Substitute new variables into the variation formula to find the new value

P'

P' = k \frac{\sqrt{1.44q}}{(0.80r)^2} = k \frac{1.2\sqrt{q}}{0.64r^2} = \frac{1.2}{0.64} \left(k \frac{\sqrt{q}}{r^2}\right) = 1.875P

Evaluating

\sqrt{1.44} = 1.2

and

(0.80)^2 = 0.64

gives the multiplier for

P

Calculate the percentage change in

P

\text{Percentage Change} = (1.875 - 1) \times 100\% = 87.5\%

increase.

Subtracting

1

converts the multiplier to a relative increase.

Key Concept

Joint Variation with Percentage Changes

Estimated Time:1m 30s

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