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 4 - 2022 - Paper 1

Question 4

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 4 - 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 determine the surface area of the toy, we need to consider each of its components as described in the problem statement:

Calculate Volume: The volume of the toy is given by: $V = \frac{1}{2} \times r^2 \times 0.8 \times h = 240$ Rearranging gives: $h = \frac{480}{0.8 r^2} = \frac{600}{r^2}$
Surface Area Calculation: The total surface area, S, consists of the following parts:
- Area of face ABC: This is a sector of a circle: $\text{Area}_{ABC} = \frac{0.8}{2\pi} \times \pi r^2 = 0.4 r^2$
- Area of face DEF, which is the same as face ABC: $\text{Area}_{DEF} = 0.4 r^2$
- Area of the rectangular faces AD, CF, and BE: $\text{Area}_{rectangles} = rh + rh + rh = 3rh$
Putting these parts together:
- Total surface area becomes: $S = 0.4r^2 + 0.4r^2 + 2rh = 0.8r^2 + 2r\left(\frac{600}{r^2}\right)$
- This simplifies to: $S = 0.8r^2 + \frac{1200}{r}$
- Finally, rewriting gives: $S = 0.8r^2 + 1680/r$

Step 2

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

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Answer

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

Differentiate S: $\frac{dS}{dr} = 1.6r - \frac{1680}{r^2}$
Set the Derivative to Zero: Setting the derivative to zero gives: $1.6r - \frac{1680}{r^2} = 0$ This can be rearranged to find r: $1.6r^3 = 1680$ $r^3 = \frac{1680}{1.6} = 1050$ Thus, $r = \sqrt[3]{1050} \approx 10.2$

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 verify that the value of r gives a minimum surface area:

Second Derivative Test: Differentiate S again to find the second derivative: $\frac{d^2S}{dr^2} = 1.6 + \frac{3360}{r^3}$
Evaluate the Second Derivative: Substitute r back into the second derivative: $\frac{d^2S}{dr^2} |_{r=10.2} = 1.6 + \frac{3360}{(10.2)^3}$ Since both terms are positive, the second derivative is positive at r = 10.2: This shows that S has a local minimum at this point.

A company makes toys for children - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

Worked Solution & Example Answer:A company makes toys for children - Edexcel - A-Level Maths Pure - Question 4 - 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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