Solve optimisation problems with calculus
12th grade (Year 13 in England) · 16 questions · Common Core: HSA.CED.A.2 · England: A-level Pure Mathematics
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Question 1
An open box is made from a 24 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(24 − 2x)². Find the maximum volume in cm³.
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Question 2
A farmer has 112 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(112 − 2x). Find the maximum area in m².
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Question 3
An app sells subscriptions. At price p, it sells 90 − 3p per day, so revenue R = p(90 − 3p). What price maximises revenue?
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Question 4
An open box is made from a 18 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(18 − 2x)². Find the maximum volume in cm³.
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Question 5
A farmer has 28 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(28 − 2x). Find the maximum area in m².
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Question 6
An app sells subscriptions. At price p, it sells 40 − 2p per day, so revenue R = p(40 − 2p). What price maximises revenue?
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Question 7
An open box is made from a 48 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(48 − 2x)². Find the maximum volume in cm³.
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Question 8
A farmer has 52 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(52 − 2x). Find the maximum area in m².
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Question 9
An app sells subscriptions. At price p, it sells 72 − 2p per day, so revenue R = p(72 − 2p). What price maximises revenue?
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Question 10
An open box is made from a 18 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(18 − 2x)². Find the maximum volume in cm³.
Answer
Question 11
A farmer has 52 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(52 − 2x). Find the maximum area in m².
Answer
Question 12
An app sells subscriptions. At price p, it sells 24 − 1p per day, so revenue R = p(24 − 1p). What price maximises revenue?
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Question 13
An open box is made from a 36 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(36 − 2x)². Find the maximum volume in cm³.
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Question 14
A farmer has 40 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(40 − 2x). Find the maximum area in m².
Answer
Question 15
An app sells subscriptions. At price p, it sells 56 − 4p per day, so revenue R = p(56 − 4p). What price maximises revenue?
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Question 16
An open box is made from a 18 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(18 − 2x)². Find the maximum volume in cm³.
Answer
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