Chapter 2.10, Problem 6STP

You measure a tub shaped as a rectangular prism to be 3 ft wide, 4 ft long, and 2.5 feet tall to the nearest half foot. What are the minimum and maximum volumes of the tub? What is the greatest possible percent error in calculating the volume of the tub?

To determine

To find the minimum and maximum volumes of the tub and the greatest possible percent error in calculating the volume of the tub

Answer to Problem 6STP

Maximum volume: 47.25 feet

Minimum volume: 17.5 feet

Percent error: 29.75 cubic feet

Explanation of Solution

Given:

Dimensions of the rectangular tub:

Width: 3 feet

Height: 2.5 feet

Length: 4 feet

Concept used:

A rectangular prism is a cuboid.

Volume of a cuboid is $l\times b\times h$

Calculation:

Consider the given dimensions of the rectangular prism.

Given that the dimensions of the rectangular tub is to the nearest half foot.

So, the maximum volume is half foot greater than the given dimensions.

Maximum sizes:

Height: $\text{3 ft}$

Width: $3.5\text{ ft}$

Length: $4.5\text{ ft}$

Substituting the dimensions of the rectangular tub in the volume of the cuboid,

  $\begin{array}{c}{V}_{\max }=4.5\times 3.5\times 3\\ =47.25\text{ cubic feet}\end{array}$

Also, the minimum volume is half foot lesser than the given dimensions.

Minimum sizes:

Height: $\text{2 ft}$

Width: $2.5\text{ ft}$

Length: $3.5\text{ ft}$

Substituting the dimensions of the rectangular tub in the volume of the cuboid,

  $\begin{array}{c}{V}_{\min }=3.5\times 2.5\times 2\\ =17.5\text{ cubic feet}\end{array}$

Therefore, the maximum and the minimum volumes of the rectangular tub are 47.25 feet and 17.5 feet.

So, the greatest possible percent error in calculating the volume of the tub is the difference in the maximum and minimum volumes of the rectangular tub.

  $\begin{array}{c}{V}_{\max }-V_{\min }=47.25-17.5\\ =29.75\text{ cubic feet}\end{array}$

Conclusion:

Therefore, the maximum and minimum volumes of the rectangular tub are 47.25 feet and 17.5 feetand the maximum possible error is 29.75 cubic feet.