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 15 - 2017 - Paper 2

Question 15

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 15 - 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 find the total surface area of the can, we start with the formula for the volume of a cylinder, which is given by:

$V = ext{π}r^2h$

The can has a capacity of 500 ml, therefore:

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

Next, the total surface area S of the can can be expressed in terms of r using the formula:

$S = 2 ext{π}r^2 + 2 ext{π}rh$

Substituting for h, we have:

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

This simplifies to:

$S = 2 ext{π}r^2 + \frac{1000}{r}$

Thus, we have proved that

$S = 2 ext{π}r^2 + \frac{1000}{r}$

Step 2

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

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Answer

To find the dimensions that minimize the surface area, we need to differentiate S with respect to r:

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

Setting the derivative to zero for optimization:

$0 = 4 ext{π}r - \frac{1000}{r^2} \\ \\ 4 ext{π}r^3 = 1000 \\ \\ r^3 = \frac{1000}{4 ext{π}} \\ \\ r = \sqrt[3]{\frac{1000}{4 ext{π}}} \approx 4.30 ext{ cm}$

Now substituting back to find h:

$h = \frac{500}{ ext{π}(4.30)^2} \approx 8.60 ext{ cm}$

Therefore, the dimensions of the can that minimize the 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 choose not to manufacture a can with minimum surface area for several reasons:

Handling Convenience: The radius of 4.30 cm and height of 8.60 cm may make the can difficult to hold for consumers, as it could be too tall or narrow.
Profile Shape: A can that is too tall compared to its width might not fit comfortably in cup holders or standard packaging, unlike more traditionally shaped cans.
Stability and Shelf Stacking: The dimensions might not provide optimal stability when stacked with other cans, affecting logistics and shelf presentation.

A company decides to manufacture a soft drinks can with a capacity of 500 ml - Edexcel - A-Level Maths Pure - Question 15 - 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 15 - 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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