Question

the figure shows the dimensions of a bottle of shampoo. the cap at the top of the shampoo bottle can be modeled by a cylinder with a diameter of 1 inch and a height of 1 inch. a student wants to mail 2 bottles of shampoo to a friend. the student packs the 2 bottles of shampoo in a box that is 6 inches long, 4 inches wide, and 10 inches tall. after placing the 2 bottles of shampoo in the box, the student will fill the empty space in the box with packing materials. how much packing material, in cubic inches, will the student need to fill the empty space in the box? show your work or explain your answer. enter your answer and your work or explanation in the space provided.

Explanation:

Step1: Calculate volume of the box

The volume of a rectangular - box is given by \(V = l\times w\times h\), where \(l = 6\) inches, \(w = 4\) inches and \(h = 10\) inches.
\[V_{box}=6\times4\times10 = 240\space\text{in}^3\]

Step2: Calculate volume of one shampoo - bottle

The main part of the shampoo - bottle is a cylinder with diameter \(d = 2.4\) inches (radius \(r=\frac{d}{2}=1.2\) inches) and height \(h = 7\) inches, and the cap is a cylinder with diameter \(d_1 = 1\) inch (radius \(r_1=\frac{d_1}{2}=0.5\) inches) and height \(h_1 = 1\) inch.
The volume of a cylinder is \(V=\pi r^{2}h\).
The volume of the main part of the shampoo - bottle \(V_{main}=\pi r^{2}h=\pi\times(1.2)^{2}\times7=\pi\times1.44\times7 = 10.08\pi\space\text{in}^3\)
The volume of the cap \(V_{cap}=\pi r_1^{2}h_1=\pi\times(0.5)^{2}\times1 = 0.25\pi\space\text{in}^3\)
The volume of one shampoo - bottle \(V_{1}=V_{main}+V_{cap}=\pi(10.08 + 0.25)=10.33\pi\space\text{in}^3\)

Step3: Calculate volume of two shampoo - bottles

The volume of two shampoo - bottles \(V_{2}=2\times V_{1}=2\times10.33\pi=20.66\pi\space\text{in}^3\approx20.66\times3.14 = 64.8724\space\text{in}^3\)

Step4: Calculate volume of the packing material

The volume of the packing material \(V_{packing}=V_{box}-V_{2}\)
\[V_{packing}=240 - 64.8724=175.1276\space\text{in}^3\]

Answer:

\(175.1276\) cubic inches