Chapter 3, Problem 63RPP

Using proportions A proportion is defined as an equality between two ratios; for instance, a/b = c/d. Proportions can be used to determine the expected change in one quantity when another quantity changes Suppose, for example, that the speed of a car doubles By what factor does the stopping distance of the car change? Proportions can also be used to answer everyday questions, such as whether a large container or a small container of a product is a better buy on a cost-per-unit-mass basis.

Suppose that a small pizza costs a certain amount. How much should a larger pizza of the same thickness cost? If the cost depends on the amount of ingredients used, then the cost should increase in proportion to the pizza s area and not in proportion to its diameter:

$Cost=k\left(Area\right)=k \left(\pi r^2\right) \left(3.10\right)$ where r is the radius of the pizza and k is a constant that depends on the price of the ingredients per unit area. If the area of the pizza doubles, the cost should double, but k remains unchanged.

Let us rearrange Eq. (3.10) so the two variable quantities (cost and radius) are on the right side of the equation and the constants are on the left:

$k\pi =\frac{Cost}{r^2}$

This equation should apply to any size pizza. If r increases, the cost should increase so that the ratio Cost/r² remains constant. Thus, we can write a proportion for pizzas of different sizes:

$k\pi =\frac{Cost}{r^2}=\frac{Cost\text{'}}{r^2}$

For example, if a 3.5-in. -radius pizza costs $4.00, then a 5.0-in. radius pizza should cost.

$Cost\text{'}=\frac{r^2}{r^2}Cost=\frac{{\left(5.0\text{in}\right)}^2}{{\left(3.5\text{in}\right)}^2}\left(\$4.00\right)=\$8.20$

This process can be used for most equations relating two quantities that change while all other quantities remain constant.

A circular wool quilt of 1.2 m diameter costs $200. What should the price of a 1.6-m-diameter quilt be if it is to have the same cost per unit area?

a. $110

b. $150

c. $270