Question

The volume of a cylinder of radius r is 1/4 of the volume of a rectangular box with a square base of side length x. If cylinder and the box have equal height, then what is the value of r in terms of x ?

Answer 1

Since volume of cylinder is equal to 1/4th of that of cuboid

=>

$\pi {r}^{2} h \: = \frac{1}{4} lbh$

Since heights are equal

=>

$\pi {r}^{2} = \frac{1}{4} lb$

=>

$\frac{88}{7} {r}^{2} = lb$

Since the base is a square and length of one side is x

=>

$\frac{88}{7} {r}^{2} = {x}^{2}$

${r}^{2} = {x}^{2} \times \frac{7}{88}$

=>

$r = x \times \sqrt{7 \div 88}$

Answer 2

πr

2

h=

4

1

lbh

Since heights are equal

=>

\pi {r}^{2} = \frac{1}{4} lbπr

2

=

4

1

lb

=>

\frac{88}{7} {r}^{2} = lb

7

88

r

2

=lb

Since the base is a square and length of one side is x

=>

\frac{88}{7} {r}^{2} = {x}^{2}

7

88

r

2

=x

2

{r}^{2} = {x}^{2} \times \frac{7}{88}r

2

=x

2

×

88

7

=>

r = x \times \sqrt{7 \div 88}r=x×

7÷88