Question 1

Profit P(x) = 80x − 2x². Which value of x maximises profit?

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Question 2

A box has square base of side x and height h. Volume = 32. Surface area = 2x² + 4xh. Find h in terms of x.

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Question 3

A rectangle has perimeter 24 cm. Which side length maximises the area?

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Question 4

Cost C(x) = x² − 10x + 40. What is the minimum cost?

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Question 5

A ball is thrown upward: h(t) = 30t − 5t². Find the maximum height.

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Question 6

A garden is enclosed by 100 m of fencing on all four sides. Find the dimensions of the garden that maximise the area.

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Question 7

A company sells x units at price (50 − x) dollars. Cost = 10x + 100.

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Question 8

Two positive numbers sum to 20. Find the values that maximise their product.

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Question 9

A farmer has 60 m of fence to make a rectangular pen against a wall (3 sides only). Find the dimensions that maximise area.

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Question 10

Revenue R(x) = 50x − x² and R(0) = 0.

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Question 11

A farmer has 120 m of fence to divide a rectangular field into 3 equal pens (2 internal dividers parallel to width). Find the dimensions that maximise total area.

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Question 12

A box with no lid has a square base of side x and height h. Volume = 108 cm³. Express the surface area S in terms of x only, then find the value of x that minimises S.

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Question 13

A sports store sells x items per week at price (120 − 2x) dollars. Manufacturing cost = 20x + 500.

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Question 14

A cylindrical can has volume 500 cm³. The cost of the top/bottom is $2 per cm² and the side is $1 per cm². Find the radius that minimises cost. (Use r in cm, leave answer to 2 d.p.)

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Question 15

f(x) = x³ − 6x² + 9x + 2. Find the maximum value of f on the interval [0, 4].

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Topic 5.7: Kinematics Questions

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