Question

the number of revolutions, r, made by a tire traveling over a fixed distance varies inversely with the diameter, d, of the tire. if a tire with a diameter of 12 inches makes 9 revolutions over a certain distance, find the constant of proportionality. using the numerical constant of proportionality, write the inverse - variation equation represented by this situation. how many revolutions would a tire with a diameter of 18 inches make in the same distance? provide an answer accurate to the nearest hundredth.

Explanation:

Step1: Write inverse - variation formula

The inverse - variation formula is $r=\frac{k}{d}$, where $r$ is the number of revolutions, $d$ is the diameter, and $k$ is the constant of proportionality.

Step2: Find the constant of proportionality

We know that when $d = 12$ inches and $r=9$ revolutions. Substitute these values into the formula $r=\frac{k}{d}$:
$9=\frac{k}{12}$.
Multiply both sides by 12 to solve for $k$: $k = 9\times12=108$.

Step3: Write the inverse - variation equation

Substitute $k = 108$ into the formula $r=\frac{k}{d}$, the inverse - variation equation is $r=\frac{108}{d}$.

Step4: Find the number of revolutions for $d = 18$ inches

Substitute $d = 18$ into the equation $r=\frac{108}{d}$:
$r=\frac{108}{18}=6$.

Answer:

The constant of proportionality is 108.
The inverse - variation equation is $r=\frac{108}{d}$.
The number of revolutions for a 18 - inch diameter tire is 6.