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

Step-by-step 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=(\lambda)/(r)`

where

`\lambda\text{ is a constant}`

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

`\begin{gathered} n=(\lambda)/(r) \\ 100=(\lambda)/(12) \\ \text{Multiply both sides by 12} \\ 100\cdot12=(\lambda)/(12)\cdot12 \\ 1200=\lambda \end{gathered}`

so, the constant is 1200, the equation is

`\begin{gathered} n=(\lambda)/(r) \\ n=(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

`\\u\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} \\ (N\cdot100.553)/(100.553)=(7539)/(100.553) \\ N=74.97\text{ revolutions} \end{gathered}`