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 by using the volume formula. Given that the volume of the cylinder is:

$V = ext{π}r^2h$

And it is stated that the company wants the can to have a capacity of 500 ml, we set:

$500 = ext{π}r^2h$

From this, we can express height h in terms of r:

$h = \frac{500}{ ext{π}r^2}$

Now, we can express the total surface area S of the cylinder as:

$S = 2 ext{π}rh + 2 ext{π}r^2$

Substituting for h:

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

This simplifies to:

$S = \frac{1000}{r} + 2 ext{π}r^2$

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 surface area, we differentiate S with respect to r:

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

Setting the derivative equal to zero to find critical points:

$4 ext{π}r - \frac{1000}{r^2} = 0$

Rearranging gives us:

$4 ext{π}r^3 = 1000$

Thus:

$r^3 = \frac{1000}{4 ext{π}}$

Calculating gives:

$r \approx 4.30 ext{ cm}$

Next, we substitute back to find h:

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

Therefore, the dimensions for minimum surface area are:

Radius, r = 4.30 cm
Height, h = 8.60 cm

Step 3

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 opt not to manufacture a can with minimum surface area for several practical reasons:

Ergonomics: A radius of 4.30 cm and height of 8.60 cm may be too narrow or tall, making it awkward to hold or drink from comfortably.
Market Preferences: Consumers generally prefer cans that are easier to grip, possibly resulting in taller and wider designs.
Storage and Compatibility: Cans with smaller radii may not stack well with other beverages in retail environments, affecting shelf space and visibility.

These factors can significantly impact consumer satisfaction and marketing strategies.

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.

suggest a reason why the company may choose not to manufacture a can with minimum surface area.

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