A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is modelled so that - face ABC is a sector of a circle with radius r cm and centre A - angle BAC = 0.8 radians - faces ABC and DEF are congruent - edges AD, CF and BE are perpendicular to faces ABC and DEF - edges AD, CF and BE have length h cm Given that the volume of the toy is 240 cm³: (a) show that the surface area of the toy, S cm², is given by S = 0.8r² + \frac{1680}{r} making your method clear. Using algebraic differentiation, (b) find the value of r for which S has a stationary point. (c) Prove, by further differentiation, that this value of r gives the minimum surface area of the toy.

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A company makes toys for children - Edexcel - A-Level Maths Pure - Question 16 - 2022 - Paper 1

Question 16

A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is modelled so that - face ABC is a sector ... show full transcript

Worked Solution & Example Answer:A company makes toys for children - Edexcel - A-Level Maths Pure - Question 16 - 2022 - Paper 1

Step 1

show that the surface area of the toy, S cm², is given by

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Answer

To find the surface area, we first express the volume of the toy:

$V = \frac{1}{2} \times r^2 \times h \times 0.8 = 240$

From this, we can solve for h:

\Rightarrow h = \frac{240}{0.4r^2} = \frac{600}{r^2} $$ Next, we can express the surface area S as: 1. Area of face ABC = $\frac{1}{2} \times r^2 \times 0.8 = 0.4r^2$ 2. Area of face DEF = $\frac{1}{2} \times r^2 \times 0.8 = 0.4r^2$ 3. Area of rectangle AD, CF, and BE = $3 \times rh = 3 \times r \times \frac{600}{r^2} = \frac{1800}{r}$ Therefore, the total surface area is: $$ S = 0.4r^2 + 0.4r^2 + \frac{1800}{r} = 0.8r^2 + \frac{1680}{r} $$

Step 2

find the value of r for which S has a stationary point

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Answer

To find the stationary point, we differentiate S with respect to r:

$\frac{dS}{dr} = 1.6r - \frac{1680}{r^2}$

Setting this to zero gives:

\Rightarrow 1.6r^3 = 1680 \\ \Rightarrow r^3 = \frac{1680}{1.6} = 1050 \\ \Rightarrow r = \sqrt[3]{1050} \approx 10.2 $$

Step 3

Prove, by further differentiation, that this value of r gives the minimum surface area of the toy

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Answer

To confirm the stationary point is a minimum, we will differentiate again:

$\frac{d^2S}{dr^2} = 1.6 + \frac{3360}{r^3}$

Since $r$ is positive, both terms in the derivative are positive, hence:

$\frac{d^2S}{dr^2} > 0$

This indicates that the function S has a minimum at $r \approx 10.2$ .

A company makes toys for children - Edexcel - A-Level Maths Pure - Question 16 - 2022 - Paper 1

Worked Solution & Example Answer:A company makes toys for children - Edexcel - A-Level Maths Pure - Question 16 - 2022 - Paper 1

show that the surface area of the toy, S cm², is given by

find the value of r for which S has a stationary point

Prove, by further differentiation, that this value of r gives the minimum surface area of the toy

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