Question

Extreme Protection, Inc. manufactures helmets for ski-

ing and snow boarding. The fixed costs for one model of helmet are $6600 per month. Materials and labor for each helmet of this model are $35, and the company sells this helmet to dealers for $60 each.
(a) For this helmet, write the function for monthly total costs.
(b) Write the function for total revenue.
(c) Write the function for profit.
(d) Find C(200), R(200), and P(200) and interpret each answer.
(e) Find C(300), R(300), and P(300) and interpret each answer.
(f) Find the marginal profit and write a sentence that explains its meaning.

Answer 1

a) C(x) = 6600 + 35x

b) R(x) = 60x

c) P(x) = 25x - 6600

d) The total cost of producing 200 helmets is $13600, the total revenue from selling 200 helmets is $12000 at a loss of $1600.

e) The total cost of producing 300 helmets is $17100, the total revenue from selling 300 helmets is $18000 at a profit of $900.

f) Since (P'(x) > 0), producing one more unit contributes positively to profit.

How to determine profit during production of items.

(a) The function for monthly total costs C(x) is the sum of fixed costs and variable costs.

Given that the fixed cost is $6600

the variable cost is $35 per helmet.

Let x represents the number of helmet manufactured by the company.

Total cost

C(x) = 6600 + 35x

(b) The function for total revenue R(x) is the product of the selling price per helmet and the number of helmets sold.

Given that the selling price is $60 per helmet. Thus,

R(x) = 60x

(c) The function for profit P(x)is the difference between total revenue and total costs. T

herefore,

P(x) = R(x) - C(x)

P(x) = (60x) - (6600 + 35x)

P(x) = 25x - 6600

(d) To determine C(200),R(200) and P(200) when x = 200

C(200) = 6600 + 35*200

= 6600 + 7000 =$13600

R(200) = 60 * 200 = $12000

P(200) = 25* 200 - 6600

= 5000 - 6600 = -$1600

Interpretation

The total cost of producing 200 helmets is $13600, the total revenue from selling 200 helmets is $12000 at a loss of $1600.

(e) When x = 300 helmets

C(300) = 6600 + 35*300

= 6600 + 10500 =$17100

R(300) = 60 * 300 = $18000

P(200) = 25* 300 - 6600

= 7500 - 6600 = -$900

Interpretation

The total cost of producing 300 helmets is $17100, the total revenue from selling 300 helmets is $18000 at a profit of $900.

(f) Marginal profit is the derivative of the profit function with respect to the quantity x.

P(x) = 25x - 6600

P'(x) = 25

Since (P'(x) > 0), producing one more unit contributes positively to profit.