The owner of a photocopy store charges 8 cents per

Questions: The owner of a photocopy store charges 8 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100. In addition, there is a setup fee of 2.60 for each photocopying job. (a) Determine R(x), the revenue from doing one photocopying job consisting of x copies. (b) If it costs the store owner 4 cents per copy, what is the profit from doing one photocopying job consisting of x copies? (Recall that profit is revenue minus cost.) (a) Choose the correct answer below. A. R(x)=.08x+2.60 for 1≤x≤100 .06x for 100<x B. R(x)=.08x+2.60 for 1≤x≤100 .06x+2.60 for 100<x C. R(x)=.08x+2.60 for 1≤x≤100 .14x+2.60 for 100<x D. R(x)=.08x+2.60 for 1≤x≤100 .06x+4.60 for 100<x

Transcript text: The owner of a photocopy store charges 8 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100 . In addition, there is a setup fee of $\$ 2.60$ for each photocopying job. (a) Determine $\mathrm{R}(\mathrm{x})$, the revenue from doing one photocopying job consisting of x copies. (b) If it costs the store owner 4 cents per copy, what is the profit from doing one photocopying job consisting of x copies? (Recall that profit is revenue minus cost.) (a) Choose the correct answer below. A. $R(x)=\left\{\begin{array}{ll}.08 x+2.60 & 1 \leq x \leq 100 \\ .06 x & 100

Solution

Solution Steps

Step 1: Understand the pricing structure

The revenue $ R(x) $ depends on the number of copies $ x $. For the first 100 copies, the charge is 8 cents per copy plus a setup fee of \$2.60. For each copy exceeding 100, the charge is 6 cents per copy plus the same setup fee.

Step 2: Write the revenue function for $ 1 \leq x \leq 100 $

For $ 1 \leq x \leq 100 $, the revenue is calculated as: \[ R(x) = 0.08x + 2.60 \] This includes the cost per copy (8 cents) and the setup fee (\$2.60).

Step 3: Write the revenue function for $ x > 100 $

For $ x > 100 $, the revenue is calculated as: \[ R(x) = 0.08 \times 100 + 0.06(x - 100) + 2.60 \] Simplify the expression: \[ R(x) = 8 + 0.06x - 6 + 2.60 = 0.06x + 4.60 \]

Step 4: Compare with the given options

The correct revenue function is: \[ R(x) = \begin{cases} 0.08x + 2.60 & \text{if } 1 \leq x \leq 100 \\ 0.06x + 4.60 & \text{if } x > 100 \end{cases} \] This matches option D.

Step 1: Understand the profit calculation

Profit is calculated as revenue minus cost. The cost per copy is 4 cents, so the total cost for $ x $ copies is $ 0.04x $.

Step 2: Write the profit function for $ 1 \leq x \leq 100 $

For $ 1 \leq x \leq 100 $, the profit is: \[ P(x) = R(x) - \text{Cost} = (0.08x + 2.60) - 0.04x = 0.04x + 2.60 \]

Step 3: Write the profit function for $ x > 100 $

For $ x > 100 $, the profit is: \[ P(x) = R(x) - \text{Cost} = (0.06x + 4.60) - 0.04x = 0.02x + 4.60 \]

Step 1: Identify the correct revenue function

From part (a), the correct revenue function is option D: \[ R(x) = \begin{cases} 0.08x + 2.60 & \text{if } 1 \leq x \leq 100 \\ 0.06x + 4.60 & \text{if } x > 100 \end{cases} \]