Question

The basketball shown is packaged in a box that is in the shape of a cube. The edge length of the box is equal to the diameter of the basketball. What is the surface area and the volume of the box?

Volume= 4500 π in.²

Answer 1

/* Cross section of the figure */

$Let \: Edge \:of \:a \: box = a$

$Diameter \: of \:the \: basketball (d) = a$

$Radius \:of \:the \: basketball (r) = \frac{d}{2} \\= \frac{a}{2}$

$Volume \: of \: the \: Basketball \\= Volume \:of \:the \: sphere \\= \frac{4}{3} \pi r^{3} \\= \frac{4}{3} \pi \Big(\frac{a}{2}\Big)^{3} \: --(1)$

$Volume \: of \: the \: Basketball = 4500 \pi \:--(2)$

/* From (1) and (2) , we get */

$\frac{4}{3} \pi \Big(\frac{a}{2}\Big)^{3}= 4500 \pi$

$\implies a^{3} = 4500 \times 8\times \frac{3}{4}$

$\implies a^{3} = 1500\times 8 \times 3$

$\implies a = 30$

$Now, Surface \:area \:of \:the \:box = 6a^{2} \\= 6 \times (30)^{2} \\= 6 \times 900 \\= 5400 \:square \: units$

$Volume \: of \:the \:box = a^{3} \\= (30)^{3} \\= 27000 \:cubic \:units$

Therefore.,

$\red {Surface \:area \:of \:the \:box}\\ \green{ = 5400 \:square \: units}$

$\red {Volume \: of \:the \:box }\\\green {= 27000 \:cubic \:units}$

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