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Parts for an automobile repair cost ​$172. The mechanic charges ​$32 per hour. If a person receives an estimate for at least ​$220 and at most ​$380 for fixing the​ car, what is the time interval that the mechanic will be working on the​ job?

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Part 1
Select the correct choice below and fill in the answer boxes to complete your choice.
​(Type integers or​ decimals.)
A.left bracket nothing hours comma nothing hours right bracket
left bracket nothing hours comma nothing hours right bracket
  
  
B.left bracket nothing hours comma nothing hours right parenthesis
left bracket nothing hours comma nothing hours right parenthesis
  
  
C.left parenthesis nothing hours comma nothing hours right bracket
left parenthesis nothing hours comma nothing hours right bracket
  
  
D.left parenthesis nothing hours comma nothing hours right parenthesis
left parenthesis nothing hours comma nothing hours right parenthesis

1 answer

To determine the time interval that the mechanic will be working on the job, we first need to set up the relationship between the cost of parts, the hourly labor rate, and the total estimate.

  1. Parts Cost: $172
  2. Hourly Charge: $32 per hour
  3. Estimate Range: At least $220 and at most $380

Let's denote the number of hours the mechanic works as \( h \). The total cost \( C \) can be represented by the equation:

\[ C = \text{Parts cost} + \text{Hourly charge} \times h \]

Substituting the known values into the equation, we have:

\[ C = 172 + 32h \]

Now, we need to find the values of \( h \) such that the total cost \( C \) lies within the specified range of at least $220 and at most $380.

Setting up the inequalities:

  1. For the minimum estimate:

\[ 172 + 32h \geq 220 \]

Subtracting 172 from both sides:

\[ 32h \geq 48 \]

Dividing by 32:

\[ h \geq \frac{48}{32} = 1.5 \]

  1. For the maximum estimate:

\[ 172 + 32h \leq 380 \]

Subtracting 172 from both sides:

\[ 32h \leq 208 \]

Dividing by 32:

\[ h \leq \frac{208}{32} = 6.5 \]

Conclusion:

Thus, the time interval \( h \) is:

\[ 1.5 \leq h \leq 6.5 \]

In interval notation, this is expressed as:

\[ [1.5, 6.5] \]

So the correct choice is:

A. [1.5 hours, 6.5 hours]