Question

Two students are given cubic boxes, measuring 10 cm on a side. Robert puts a single glass marble with a

Answer 1

Answer:

Both boxes have the same amount of empty space

Explanation:

Given

Cube Box

$Length = 10cm$

Robert:

1 glass marble

$Diameter = 10cm$

Susan:

1000 glass marble

$Diameter = 1cm$

Required

Whose box has more empty space?

First, we need to calculate the volume of the cubic box.

$Volume = Length^3$

Substitute 10cm for Length

$Volume = 10^3$

$Volume = 1000cm^3$

Next, calculate the volume of Robert's glass

Volume is calculated as:

$Volume = \frac{4}{3}\pi r^3$ ---- Volume of a sphere

Where

r = radius

$r = \frac{1}{2} * Diameter$

$Diameter = 10cm$ --- Given

Substitute 10 for diameter

$r = \frac{1}{2} * 10cm$

$r = 5cm$

So:

$Volume = \frac{4}{3}\pi r^3$

$Volume = \frac{4}{3} * \frac{22}{7} * 5^3$

$Volume = \frac{4}{3} * \frac{22}{7} * 125$

$Volume = 523.8cm^3$

Next, we determine the volume of Susan's 1000 glasses

Volume is calculated as:

$Volume = \frac{4}{3}\pi r^3$ ---- Volume of a sphere

Where

r = radius

$r = \frac{1}{2} * Diameter$

$Diameter = 1cm$ --- Given

Substitute 1cm for diameter

$r = \frac{1}{2} * 1cm$

$r = 0.5cm$

Substitute 0.5cm for radius in $Volume = \frac{4}{3}\pi r^3$

$Volume = \frac{4}{3} * \frac{22}{7} * 0.5^3$

$Volume = \frac{4}{3} * \frac{22}{7} * 0.125$

$Volume = 0.5238$

But there are 1000 glasses.

So, the volume of 1000 glasses is:

$Volume = 0.5238 * 1000$

$Volume = 523.8cm^3$

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For Robert:

$Volume = 523.8cm^3$

For Susan:

$Volume = 523.8cm^3$

Since, they have the same volume.

Both boxes have the same amount of empty space