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2.7 (AS 91262) – 2025 Answers

Quite often, there are different ways to solve a Mathematics problem.

These solutions/strategies show one possible way to solve them.

Question 3 (a)

Question 3(b)

Question 3 (c)

Question 3 (d)

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Question 3(a)

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Differentiate the function to get the gradient function.

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Solve the equation to find the x-coordinate.

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Question 3(b)

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Question 3(b)

The rate of change simply means gradient, so we need to differentiate the Area function to get a gradient function.

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Question 3(b)

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Question 3(b)

Now we differentiate the area function to get the rate of change of area.

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Question 3(b)

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Question 3(b)

Divide both sides by 2.

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Question 3(b)

Square root both sides to get two solutions, but only one of them makes sense in this context.

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Question 3(b)

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Question 3(c)

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Question 3(c)

Front and Back

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Left & Right Side

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Bottom (base)

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Question 3(c)

Simplify

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Question 3(c)

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Question 3(c)

Divide everything by 4

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Question 3(c)

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Question 3(c)

Expand brackets

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Question 3(c)

We want to maximise the container’s volume, so we need to differentiate the volume function to get a gradient function.

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Question 3(c)

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Question 3(c)

To find a maximum or minimum, we need to put the gradient function equal to 0.

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Question 3(c)

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Question 3(c)

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Question 3(c)

Square Root both sides

When we square root a number, we get both a positive and negative result. The negative result in this context makes no sense.

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Simplify

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Factorise

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Solve

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Question 3(d)

Practice multiplying out cubics and simplying to check these.