Chapter 3, Problem 59RPP

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.

$Cost=k\left(\text{Area}\right)=k\left(\pi r^2\right)$ 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.

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

This equation should apply to any size pizza If r increases, the cost should Increase so that the ratio $\begin{array}{l}\text{Cost/}{r}^{\text{2}}\\ d=\frac{1}{2}a{t}^2.\end{array}$ remains constant Thus, we can write a proportion for pizzas of different sizes:

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

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

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

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

The downward distance d that an object falls in a time interval t if starting at rest is $d=\frac{1}{2}a{t}^2.$

On the Moon, a rock falls 10.0 m in 3.50 s How far will the object fall in 5.00 s, assuming the same acceleration?

a. 14.3 m

b. 20.4 m

c. 4.90 m

d. 7.00 m

e. 100 m