Pearl’s Biking Company manufactures and sells bikes. Each bike costs £40 to make, and the company’s fixed costs are £5000. In addition, Pearl knows that the price of each bike comes from the price function P(x) = 300 – 2x where x is the total number of bikes manufactured and sold. Questions
a) Find the company’s revenue function R(x), the company’s cost function C(x) and find the breakeven points. b) By plotting the revenue and cost functions, obtain the Break-even output levels and corresponding price (use graph paper or graphical tools). c) Generate the profit function for the company and find, i. The level production that maximizes the profit. ii. The maximum profit.
Answers
Given : Pearl’s Biking Company manufactures and sells bikes. Each bike costs £40 to make, and the company’s fixed costs are £5000. price function P(x) = 300 – 2x where x is the total number of bikes manufactured and sold.
To find : company’s revenue function R(x) the company’s cost function Break-even , profit function
Solution:
Cost function of Bike
C (x) = 5000 + 40x
Price
P(x) = 300 – 2x
Revenue function = unit sold * Price = x * P(x)
R ((x) = x (300 - 2x) = 300x - 2x²
Break even point
C(x) = R(x)
=> 5000 + 40x = 300x - 2x²
=> 2x² - 260x + 5000 = 0
=> x² - 130x + 2500 = 0
=> x = 23.5 = 24 or 106.5 = 106
24 bikes need to be sold for break even
P(24) = 300 -2(24) = 252 £
P(106) = 300 -2(106) = 88 £
Profit = R(x) - C(x) = 300x - 2x² - 5000 - 40x
P(x) = - 2x² + 260x - 5000
dP/dx = - 4x + 260
=> x = 65
d²P/dx² = - 4
Selling 65 bike will give maximum profit
Maximum profit = - 2x² + 260x - 5000 = 3450 £
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