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 = 0.8 r² + 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 15 - 2022 - Paper 1

Question 15

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 15 - 2022 - Paper 1

Step 1

show that the surface area of the toy, S cm², is given by S = 0.8 r² + 1680 / r

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Answer

To find the surface area, we start by using the given volume of the toy, which is 240 cm³. The volume V is expressed as:

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

Rearranging gives us: h = \frac{480}{r^2} $$

Now, we can express the surface area S as:

$S = 2 \times \text{Area of ABC} + ext{Area of DEF} + ext{Area of side faces}$

Calculating each area:

Area of face ABC = 0.4 r² (sector area with angle 0.8 radians)
Area of face DEF is congruent, so it is also 0.4 r².
The area of the lateral faces (AD, CF, BE) is calculated as the height times the corresponding width, which gives us: 2(rh) = 2(r \times h).

Substituting h from volume:

$S = 0.4 r² + 0.4 r² + 2 \left(r \times \frac{480}{r^2} \right)$

This simplifies to:

$S = 0.8 r² + \frac{960}{r}.$

Correctly substituting gives us:

$S = 0.8 r² + \frac{1680}{r}.$

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 first differentiate the surface area S with respect to r:

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

Setting this equal to zero gives:

$1.6 r - \frac{1680}{r^2} = 0$

Solving for r, we rearrange to find:

$1.6 r^3 = 1680$

Dividing both sides by 1.6 gives:

$r^3 = 1050 \implies r = \sqrt[3]{1050} \approx 10.2.$

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 whether we have a minimum, we differentiate S again:

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

Substituting r ≈ 10.2 into this expression:

Since both terms are positive, it follows that:

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

This indicates a local minimum at this point. Therefore, the value of r found indeed gives the minimum surface area of the toy.

A company makes toys for children - Edexcel - A-Level Maths Pure - Question 15 - 2022 - Paper 1

Worked Solution & Example Answer:A company makes toys for children - Edexcel - A-Level Maths Pure - Question 15 - 2022 - Paper 1

show that the surface area of the toy, S cm², is given by S = 0.8 r² + 1680 / r

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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