Chapter 1, Problem 14CT

a.

To determine

To calculate: after how many hours the total repairs cost the same at both places.

Answer to Problem 14CT

2.1 hours

Explanation of Solution

Given information: For new brakes for your car you call a dealership and a local mechanic for prices-

  

Let $x$ be the number of hours labored at dealership and local mechanics

Then labor costs at dealership = $ 99 $x$

And labor costs at local mechanic= $ 89 $x$

Total repair cost at dealership= cost of parts + labor cost

= 24 + 99 $x$

Total repair cost at local mechanic = cost of parts + labor cost

= 45 + 89 $x$

Now, according to question

Total repair cost at dealership = Total repair cost at local mechanic

24 + 99 $x$ = 45 + 89 $x$

  $\begin{array}{l}\Rightarrow 99x-89x=45-24\\ \Rightarrow 10x=21\\ \Rightarrow x=\frac{21}{10}=2.1\end{array}$

And hence after 2.1 hours the total cost same at both the places.

b.

To determine

To calculate: The time when the repairs cost less at the dealership and at the local mechanic.

Answer to Problem 14CT

The repairs cost less at the dealership in initial 2.1 hours,

and the repairs cost less at the local mechanic after time crosses 2.1 hours.

Explanation of Solution

Given information:

As above we have calculated the following-

Total repair cost at dealership = 24 + 99 $x$

Total repair cost at local mechanic = 45 + 89 $x$

For,

Total repair cost at dealership < Total repair cost at local mechanic

  $\begin{array}{l}\Rightarrow 24+99x<45+89x\\ \Rightarrow 99x-89x<45-24\\ \Rightarrow 10x,21\\ \Rightarrow x<\frac{21}{10}\\ \Rightarrow x<2.1\end{array}$

Time should be less than 2.1 hours. This means the repair cost of the dealership is less for initial 2.1 hours.

For,

Total repair cost at local mechanic < Total repair cost at the dealership

  $\begin{array}{l}\Rightarrow 45+89x<24+99x\\ \Rightarrow 45-24<99x-89x\\ \Rightarrow 21<10x\\ \Rightarrow 10x>21\\ \Rightarrow x>\frac{21}{10}\\ \Rightarrow x>2.1\end{array}$

Time should be greater than 2.1 hours. This means the repair cost of local mechanic is less for time crosses 2.1 hours.