Exercises for Question Type 2: Optimizing profit or cost functions under a given context

Question 1

This question tests the application of differentiation in economics, specifically finding the profit-maximizing output level given linear demand and cost functions.

A firm faces a linear demand p (q) = 100 - 2 q and incurs cost C (q) = 20 q + 50 . Find the output level q that maximizes profit.

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Question 2

The revenue for a product is given by R (q) = 40 q - 0.5 q^{2} and the cost is given by C (q) = 5 q + 100, where q is the quantity produced. Find the value of q that maximizes the profit.

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Question 3

A company sells a product at a fixed price of 18 per unit and has cost function C (q) = 5 q^{2} - 22 q + 200 . Determine the quantity q that maximizes profit.

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Question 4

A product’s price depends on demand, p (q) = 200 - q, and its cost is C (q) = 0.01 q^{2} + 2 q + 500 . Determine the output level q that maximizes profit.

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Question 5

A company’s revenue is R (q) = 150 ln (q) and its cost is C (q) = 20 q + 100 . Find the output q that maximizes profit.

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Question 6

A bakery sells cakes at p (q) = 60 - 0.5 q dollars each and has cost C (q) = 10 q + 150 . Find the production level q that maximizes profit.

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Question 7

A smartphone maker earns 300 dollars per unit and has cost function C (q) = 0.5 q^{2} + 50 q + 1000 . Determine the output q that maximizes profit.

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Question 8

A car manufacturer faces demand p (q) = 50000 - 0.01 q and cost C (q) = 20000 q + 1 0^{6} . Find the output q that maximizes profit.

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Question 9

A service provider has profit function P (q) = 70 q - 0.005 q^{2} - 200, where q is the quantity of units sold. Find the value of q that maximizes profit.

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Question 10

Profit Maximization

The price-demand function is p (q) = 80 - 0.1 q and the cost function is C (q) = 2 q + 300 . Determine the value of q that maximizes profit.

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Question 11

Total revenue is R (q) = 120 q - 0.2 q^{2} and cost is C (q) = 30 q + 400 . Determine the output level q \geq 0 that maximizes profit.

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Question 12

This question assesses the application of calculus in economics, specifically the minimization of marginal cost. Candidates must differentiate the total cost function to find marginal cost and then use optimization techniques.

The total cost of producing q units is given by C (q) = q^{3} - 30 q^{2} + 300 q + 200 . Determine the output level q that minimizes the marginal cost.

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IB Mathematics Applications & Interpretation Question Type 2: Optimizing Profit or Cost Functions Under a Given Context Exercises