Question

1. shoe boxes are often used in other capacities once the shopper has bought the shoes. sometimes the boxes are used to hold other items, so it is helpful to know the volume of the box.

a. write the equation to solve for the volume of the shoe box.
b. if the area of the base of the box is 112 square inches and the height is 3.5 inches, what is the volume of the box?
c. rewrite the equation to solve for height.
d. a box has a volume of 456 cubic inches, with a length of 1 foot and a height of 4 inches. determine the width of the box.

Explanation:

Step1: Volume formula for rectangular prism

The volume \(V\) of a rectangular prism (shoe - box) is given by \(V = B\times h\), where \(B\) is the area of the base and \(h\) is the height.

Step2: Calculate volume for part b

Given \(B = 112\) square inches and \(h=3.5\) inches. Substitute into \(V = B\times h\), we get \(V=112\times3.5\).

$$V = 112\times3.5=(100 + 12)\times3.5=100\times3.5+12\times3.5=350+42 = 392$$

Step3: Solve for height in part c

Starting from \(V = B\times h\), divide both sides by \(B\). The formula for height is \(h=\frac{V}{B}\)

Step4: Convert length for part d

Since \(1\) foot \( = 12\) inches. Let the width be \(w\). The area of the base \(B=l\times w\), and \(V = B\times h=(l\times w)\times h\). We know \(V = 456\) cubic inches, \(l = 12\) inches, \(h = 4\) inches. Substitute into \(V=(l\times w)\times h\), we get \(456=(12\times w)\times4\). First, simplify the right - hand side: \((12\times w)\times4 = 48w\). Then solve for \(w\): \(w=\frac{456}{48}\)

$$w=\frac{456}{48}=\frac{456\div12}{48\div12}=\frac{38}{4} = 9.5$$

Answer:

a. \(V = B\times h\)
b. \(392\) cubic inches
c. \(h=\frac{V}{B}\)
d. \(9.5\) inches