Question

Suppose the height of a pyramid with a square base is decreased by $p\%$ and the lengths of the sides of its square base are increased by $p\%$ (where p $>0$). If the volume remains the same, then.

• A. $50<p<55$
• B. $55<p<60$
• C. $60<p<65$
• D. $65<p<70$

Answer 1

Answer: (C)

$60<p<65$

Let the length of base be $x$ and height be $y$

Volume of pyramid $=\frac{1}{3}\times l\times b\times h$

When height is decreased by $p\%$, then $h=y-\frac{p}{100}\times y$

And base is increased, $b=x+\frac{p}{100}\times x$

Volume in both cases is same

$\frac{1}{3}{x}^{2}y=\frac{1}{3}{x}^2{(1+\frac{p}{100})}^2\times y(1-\frac{p}{100})\Rightarrow p^2+100p-{100}^2=0$

solving the equation by using quadratic formula

$p=\pm \sqrt{12500}-50$

$p$ can't be negative

$\therefore p=\sqrt{12500}-50\Rightarrow p\simeq 61.80$

So option $C$ is correct