Chapter 12.4, Problem 17PSB

a.

To determine

To calculate: The volume of J’s lunch box.

Answer to Problem 17PSB

The volume of J’s lunch box is $621i{n}^3$ .

Explanation of Solution

Given information:

The rectangular box dimensions are:

l = 10 in,

b = 6 in,

h = 8 in.

Diameter of cylinder = 6 in.

Formula used:

Volume of rectangle $=l\times b\times h$

l = length of rectangle, b = breadth of rectangle and h = height of rectangle

Volume of cylinder $=B\times h$

B = Base area of cylinder and h = height of cylinder

Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

Volume of rectangle $=l\times b\times h$

Volume of rectangle $=10\times 6\times 8$

Volume of rectangle $=480$

Volume of cylinder $=B\times h$

Half of cylinder is used. Base area of cylinder is of semicircle.

  $\begin{array}{l}r=\frac{d}{2}=\frac{6}{2}=3\\ B=\frac{1}{2}\pi r^2=\frac{1}{2}\pi {\left(3\right)}^2=4.5\pi \\ h=10\end{array}$

Volume of cylinder $=B\times h$

Volume of cylinder $=4.5\pi \times 10$

Volume of cylinder $=45\pi$

Volume of lunch box = Volume of rectangle + volume of cylinder

Volume of lunch box $=480+45\pi \approx 621i{n}^3$

b.

To determine

To find: The total amount of sheet metal needed to manufacture J’s lunch box.

Answer to Problem 17PSB

The total amount of sheet metal needed is $439i{n}^2$ .

Explanation of Solution

Given information:

The rectangular box dimensions are:

l = 10 in,

b = 6 in,

h = 8 in.

Diameter of cylinder = 6 in.

Formula used:

Area of cylinder $=C\times h$

C = Circumference of cylinder and h = height of cylinder

Area of rectangle: $A=l\times w$

l = length of rectangle

w= width of rectangle Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

Area of half cylinder:

  $\begin{array}{l}{A}_{HalfCylinder}=\frac{1}{2}Ch+\pi r^2\\ A_{HalfCylinder}=\frac{1}{2}2\pi rh+\pi r^2\\ r=3\\ A_{HalfCylinder}=\frac{1}{2}\left(2\pi \times 3\times 10\right)+\pi r^2\\ A_{HalfCylinder}=\frac{1}{2}60\pi +\pi {\left(3\right)}^2\\ A_{HalfCylinder}=30\pi +9\pi \\ A_{HalfCylinder}=39\pi \end{array}$

Area of five rectangles:

  $\begin{array}{l}A(▭)=2A({▭}_1)+A({▭}_2)+2A({▭}_3)\\ A(▭)=2\left(8\times 10\right)+\left(6\times 10\right)+2\left(6\times 8\right)\\ A(▭)=2\left(80\right)+\left(60\right)+2\left(48\right)\\ A(▭)=160+60+96\\ A(▭)=316\end{array}$

Total Area = Area of five rectangles + Area of half cylinder

Total Area $=316+39\pi$

Total Area $=439i{n}^3$