Question: A company that manufactures small canoes has a fixed cost of $24,000. It costs $100 to produce each…

A company that manufactures small canoes has a fixed cost of $24,000. It costs $100 to produce each canoe. The selling price is $200 per canoe. (In solving this exercise, let \( x \) represent the number of canoes produced and sold.)

a. Write the cost function. \( C(x) = \) (Type an expression using \( x \) as the variable.)

b. Write the revenue function. \( R(x) = \) (Type an expression using \( x \) as the variable.)

c. Determine the break-even point. \( \) (Type an ordered pair. Do not use commas in large numbers.) This means that when the company produces and sells the break-even number of canoes

• A. there is more money coming in than going out.
• B. there is less money coming in than going out.

Solution

A company that manufactures small canoes has a fixed cost of $24,000. It costs $100 to produce each canoe, and the selling price is $200 per canoe. Let’s solve the questions step by step: a. Write the cost function. The cost function \( C(x) \) includes the fixed cost and the variable cost per canoe. \[ C(x) = 24000 + 100x \] b. Write the revenue function. The revenue function \( R(x) \) is based on the selling price per canoe. \[ R(x) = 200x \] c. Determine the break-even point. The break-even point occurs when the cost equals the revenue. Set \( C(x) = R(x) \). \[ 24000 + 100x = 200x \] Subtract \( 100x \) from both sides. \[ 24000 = 100x \] Divide both sides by 100. \[ x = 240 \] Thus, the break-even point is when 240 canoes are produced and sold. The break-even ordered pair is \((240, 48000)\), because at \( x = 240 \), \( R(240) = 200 \times 240 = 48000\). At the break-even point, there is no profit or loss, so there is neither more money coming in nor going out. However, since more items are being sold to cover the costs, it means: A. there is more money coming in than going out.