A manufacturer of custom planners determines that the setup cost for a production run is 1… — ACT

A manufacturer of custom planners determines that the setup cost for a production run is $100$ dollars, and each planner costs $6$ dollars to produce. The planners sell for $10$ dollars each, except for the first $10$ planners sold, which are discounted by $2$ dollars each. If the manufacturer wants to achieve a net profit of exactly $300$ dollars for a single production run, how many planners must they produce and sell?

Answer: 105 planners

Answer

The manufacturer must produce and sell

105

planners to achieve a net profit of

300

dollars.

The correct answer is found by setting up the profit equation:

\text{Profit} = \text{Total Revenue} - \text{Total Cost}

. The cost is

100 + 6x

and the revenue is

10(8) + 10(x-10) = 10x - 20

. Equating their difference to

300

yields

(10x - 20) - (100 + 6x) = 300

, which simplifies to

4x - 120 = 300

. Solving for

x

results in

105

Step-by-Step Solution

Define the variable for the number of planners.

Let

x

be the number of planners produced and sold, where

x \geq 10

Establishing the variable is necessary to set up algebraic expressions for cost and revenue.

Write the total cost expression.

\text{Total Cost} = 100 + 6x

The cost combines the fixed setup fee of

100

dollars and the variable cost of

6

dollars per planner.

Write the total revenue expression.

\text{Total Revenue} = 10(8) + 10(x - 10) = 10x - 20

The first

10

planners sell for

8

dollars each, and the remaining

x - 10

planners sell for the regular price of

10

dollars each.

Set up the profit equation and solve for

x

(10x - 20) - (100 + 6x) = 300 \implies 4x - 120 = 300 \implies 4x = 420 \implies x = 105

Profit is the difference between total revenue and total cost, which must equal the target profit of

300

dollars.

Key Concept

Translating real-world pricing and cost constraints into a single-variable linear equation.

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