A company decides to manufacture a soft drinks can with a capacity of 500 ml. The company models the can in the shape of a right circular cylinder with radius r cm and height h cm. In the model they assume that the can is made from a metal of negligible thickness. (a) Prove that the total surface area, S cm², of the can is given by S = 2 ext{π}r² + rac{1000}{r} (3) Given that r can vary, (b) find the dimensions of a can that has minimum surface area. (5) (c) With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area. (1)

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A company decides to manufacture a soft drinks can with a capacity of 500 ml - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2

Question 16

A company decides to manufacture a soft drinks can with a capacity of 500 ml. The company models the can in the shape of a right circular cylinder with radius r cm ... show full transcript

Worked Solution & Example Answer:A company decides to manufacture a soft drinks can with a capacity of 500 ml - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2

Step 1

Prove that the total surface area, S cm², of the can is given by

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Answer

To derive the surface area formula for a cylinder, we start with the volume formula for a cylinder:

$V = ext{π}r^2h$

Given that the can has a capacity of 500 ml (which is equivalent to 500 cm³), we set:

$500 = ext{π}r^2h$

Solving for h, we get:

$h = \frac{500}{ ext{π}r^2}$

The total surface area, S, of a cylinder is given by:

$S = 2 ext{π}rh + 2 ext{π}r^2$

Substituting for h:

$S = 2 ext{π}r \left( \frac{500}{ ext{π}r^2} \right) + 2 ext{π}r^2$

This gives:

$S = \frac{1000}{r} + 2 ext{π}r^2$

Thus, we arrive at:

$S = 2 ext{π}r^2 + \frac{1000}{r}$

Step 2

find the dimensions of a can that has minimum surface area.

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Answer

To find the minimum surface area, we differentiate S with respect to r:

$\frac{dS}{dr} = 4 ext{π}r - \frac{1000}{r^2}$

Setting this derivative equal to zero for optimization:

$4 ext{π}r - \frac{1000}{r^2} = 0$

This implies:

$4 ext{π}r^3 = 1000$

Solving for r:

$r^3 = \frac{1000}{4 ext{π}} \\ r = \sqrt[3]{\frac{1000}{4 ext{π}}} \approx 4.30 ext{ cm}$

Substituting back to find h:

$h = \frac{500}{ ext{π}r^2} \approx 8.60 ext{ cm}$

Thus, the dimensions for minimum surface area are:

Radius: 4.30 cm
Height: 8.60 cm

Step 3

With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.

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Answer

The company may decide against manufacturing a can with minimum surface area for several practical reasons. One reason is that the radius of 4.30 cm and height of 8.60 cm may not be convenient for handling:

Comfort: The radius might be too small for comfortable grip during drinking.
Stacking Issues: Products that are too tall (like this can) can be unstable when stacked, especially when transported.
Profile: Cans that are square in profile might be more space-efficient in packaging and shelf display.

These considerations can affect marketability despite the mathematical optimization of surface area.

A company decides to manufacture a soft drinks can with a capacity of 500 ml - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2

Worked Solution & Example Answer:A company decides to manufacture a soft drinks can with a capacity of 500 ml - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2

Prove that the total surface area, S cm², of the can is given by

find the dimensions of a can that has minimum surface area.

With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.

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