Interpret the given relation

The expression is intended as $P = a^{1/2} b^2 c^3 d^{-4}$

For a quantity of the form $P = a^x b^y c^z d^w,$ the maximum relative error is $\frac{\Delta P}{P} = |x|\frac{\Delta a}{a} + |y|\frac{\Delta b}{b} + |z|\frac{\Delta c}{c} + |w|\frac{\Delta d}{d}.$

Write the given relative errors

$\frac{\Delta a}{a} = 2\%, \quad \frac{\Delta b}{b} = 1\%, \quad \frac{\Delta c}{c} = 3\%, \quad \frac{\Delta d}{d} = 5\%$

Apply the error formula

Here the powers are:

• for $a$: $\frac{1}{2}$
• for $b$: $2$
• for $c$: $3$
• for $d$: $-4$

Therefore, $\frac{\Delta P}{P} = \frac{1}{2}(2\%) + 2(1\%) + 3(3\%) + 4(5\%)$

Calculate each term

$\frac{1}{2}(2\%) = 1\%$ $2(1\%) = 2\%$ $3(3\%) = 9\%$ $4(5\%) = 20\%$

Add them

$\frac{\Delta P}{P} = 1\% + 2\% + 9\% + 20\% = 32\%$

Choose the correct option

$\boxed{32\%}$

So the correct option is C.