Respuesta :
Answer:
Both boxes have the same amount of empty space
Explanation:
Given
Cube Box
[tex]Length = 10cm[/tex]
Robert:
1 glass marble
[tex]Diameter = 10cm[/tex]
Susan:
1000 glass marble
[tex]Diameter = 1cm[/tex]
Required
Whose box has more empty space?
First, we need to calculate the volume of the cubic box.
[tex]Volume = Length^3[/tex]
Substitute 10cm for Length
[tex]Volume = 10^3[/tex]
[tex]Volume = 1000cm^3[/tex]
Next, calculate the volume of Robert's glass
Volume is calculated as:
[tex]Volume = \frac{4}{3}\pi r^3[/tex] ---- Volume of a sphere
Where
r = radius
[tex]r = \frac{1}{2} * Diameter[/tex]
[tex]Diameter = 10cm[/tex] --- Given
Substitute 10 for diameter
[tex]r = \frac{1}{2} * 10cm[/tex]
[tex]r = 5cm[/tex]
So:
[tex]Volume = \frac{4}{3}\pi r^3[/tex]
[tex]Volume = \frac{4}{3} * \frac{22}{7} * 5^3[/tex]
[tex]Volume = \frac{4}{3} * \frac{22}{7} * 125[/tex]
[tex]Volume = 523.8cm^3[/tex]
Next, we determine the volume of Susan's 1000 glasses
Volume is calculated as:
[tex]Volume = \frac{4}{3}\pi r^3[/tex] ---- Volume of a sphere
Where
r = radius
[tex]r = \frac{1}{2} * Diameter[/tex]
[tex]Diameter = 1cm[/tex] --- Given
Substitute 1cm for diameter
[tex]r = \frac{1}{2} * 1cm[/tex]
[tex]r = 0.5cm[/tex]
Substitute 0.5cm for radius in [tex]Volume = \frac{4}{3}\pi r^3[/tex]
[tex]Volume = \frac{4}{3} * \frac{22}{7} * 0.5^3[/tex]
[tex]Volume = \frac{4}{3} * \frac{22}{7} * 0.125[/tex]
[tex]Volume = 0.5238[/tex]
But there are 1000 glasses.
So, the volume of 1000 glasses is:
[tex]Volume = 0.5238 * 1000[/tex]
[tex]Volume = 523.8cm^3[/tex]
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For Robert:
[tex]Volume = 523.8cm^3[/tex]
For Susan:
[tex]Volume = 523.8cm^3[/tex]
Since, they have the same volume.
Both boxes have the same amount of empty space
From the dimensions of glass marble for robert and susan, we can say that; none of the given boxes have more empty space.
We are told that the cubic boxes have a measure of 10 cm for its' sides.
Formula for volume of a cube is;
V = lwh
where;
l is length
w is width
h is height
Thus; V = 10 × 10 × 10
V = 1000 cm³
For Robert;
We are told that he puts a glass marble with a diameter of 10 cm in the box. The glass is usually spherical in shape and the volume of a sphere is;
V = ⁴/₃πr³
Thus;
V_rob = ⁴/₃π × (10/2)³
V_rob = 523.6 cm³
Volume left for rob = 1000 - 523.6
Volume of empty space left for rob = 476.4 cm³
For Susan;
V = ⁴/₃πr³
Thus;
V_sus = ⁴/₃π × (1/2)³ × 1000
V_sus = 523.6 cm³
Volume left for susan = 1000 - 523.6
Volume of empty space left for susan = 476.4 cm³
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