Question

Write an equation of variation to represent the situation and solve for the missing information The number of revolutions made by a tire traveling over a fixed distance varies inversely tothe radius of the tire. A 12-inch radius tire makes 100 revolutions to travel a certaindistance. How many revolutions would a 16-inch radius tire require to travel the samedistance?

Answer 1

74.97 revolutions

Explanation

Step 1

The number of revolutions made by a tire traveling over a fixed distance varies inversely to

the radius of the tire

Let

The number of revolutions=n

the radius of the tire:r

then

$n=\frac{\lambda}{r}$

where

$\lambda\text{ is a constant}$

A 12-inch radius tire makes 100 revolutions to travel a certain,Hence

$\begin{gathered} n=\frac{\lambda}{r} \\ 100=\frac{\lambda}{12} \\ \text{Multiply both sides by 12} \\ 100\cdot12=\frac{\lambda}{12}\cdot12 \\ 1200=\lambda \end{gathered}$

so, the constant is 1200, the equation is

$\begin{gathered} n=\frac{\lambda}{r} \\ n=\frac{1200}{r}\text{ equation} \end{gathered}$

then, we need to find the distance

then

$\begin{gathered} a\text{ revolution=2}\cdot\pi\cdot radius \\ a\text{ revolution=2}\cdot\pi\cdot12\text{ inche( for the first tire)} \\ a\text{ revolution=75.38 inches} \\ \text{then 100 revolutions = 100}\cdot75.38\text{ inches} \\ 100\text{ revolutionss=7539 inches} \end{gathered}$

then,the distance is 7539 inches

Step 2

Let

n= unknown

radius=r=16 inch

distance=7539 inches

$\begin{gathered} 2\cdot\pi\cdot r=2\cdot\pi\cdot16 \\ 32\cdot\pi=100.53\text{ inches} \end{gathered}$

it means

$1\text{ revolution}\Rightarrow100.53\text{ inches}$

then

$\nu\text{mber of revolutions}\cdot100.553=same\text{ distance}$

replacing

$\begin{gathered} N\cdot100.553=7539\text{ inches} \\ \text{divide both sides by 100.553} \\ \frac{N\cdot100.553}{100.553}=\frac{7539}{100.553} \\ N=74.97\text{ revolutions} \end{gathered}$