Question

callie is sending her nephew a soccer ball for his birthday. she packs the ball in a cubic box that has the same side length as the diameter of the ball. she needs to determine how much space needs to be filled with packing material.
image of box image of soccer ball
if the sides of the box are 26 cm:
(a) what is the volume of the box? □ cm³
(b) use 3.1416 for the approximate value of π. round your answer to the nearest whole number.
what is the volume of the ball? □ cm³
(c) what is the remaining volume of the box that needs to be filled with packing material? □ cm³
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Explanation:

Part (a)

Step1: Recall volume of cube formula

The volume \( V \) of a cube with side length \( s \) is given by \( V = s^3 \).

Step2: Substitute \( s = 26 \) cm

Substitute \( s = 26 \) into the formula: \( V = 26^3 \).
Calculate \( 26^3 = 26\times26\times26 = 17576 \).

Step1: Recall volume of sphere formula

The volume \( V \) of a sphere with diameter \( d \) (radius \( r=\frac{d}{2} \)) is \( V=\frac{4}{3}\pi r^3 \). Here, \( d = 26 \) cm, so \( r=\frac{26}{2}=13 \) cm.

Step2: Substitute values into formula

Substitute \( r = 13 \) and \( \pi = 3.1416 \) into \( V=\frac{4}{3}\pi r^3 \):
\( V=\frac{4}{3}\times3.1416\times13^3 \)
First, calculate \( 13^3 = 13\times13\times13 = 2197 \).
Then, \( \frac{4}{3}\times3.1416\times2197=\frac{4\times3.1416\times2197}{3} \)
Calculate numerator: \( 4\times3.1416 = 12.5664 \); \( 12.5664\times2197 = 27606.3808 \)
Divide by 3: \( \frac{27606.3808}{3}\approx9202.1269 \)
Round to nearest whole number: \( 9202 \)

Step1: Subtract sphere volume from cube volume

The remaining volume is the volume of the box minus the volume of the ball. Let \( V_{box}=17576 \) and \( V_{ball}\approx9202 \).
So, remaining volume \( = V_{box}-V_{ball}=17576 - 9202 \).

Step2: Calculate the difference

\( 17576-9202 = 8374 \)

Answer:

\( 17576 \)

Part (b)