A company produces cylindrical cans for soup. The standard can has a radius \(r\) and height \(h\).
The volume \(V\) of a can is given by \(V = \pi r^2 h\).
The surface area \(A\) is given by \(A = 2\pi r^2 + 2\pi r h\).
The company wants to produce cans with a volume of 340 cm³.
(a) Express \(h\) in terms of \(r\). [1]
(b) Show that the surface area \(A\) can be expressed as \(A = 2\pi r^2 + \frac{680}{r}\). [2]
(c) Find the value of \(r\) that minimizes the surface area of the can. [2]
(d) Find this minimum surface area. [2]
Question
Express \(h\) in terms of \(r\). [1]
Hint
Hint
Hint
Rearrange the volume equation \(\pi r^2 h = 340\) to solve for h.
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\(h = \frac{340}{\pi r^2}\)
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Question
Show that the surface area \(A\) can be expressed as \(A = 2\pi r^2 + \frac{680}{r}\). [2]
Hint
Hint
Hint
Substitute the expression for h into \(A = 2\pi r^2 + 2\pi r h\) and simplify.
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By substituting \(h = \frac{340}{\pi r^2}\), we get \(A = 2\pi r^2 + 2\pi r \times \frac{340}{\pi r^2} = 2\pi r^2 + \frac{680}{r}\).
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Find the value of \(r\) that minimizes the surface area of the can. [2]
Hint
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Hint
Differentiate \(A = 2\pi r^2 + \frac{680}{r}\) with respect to r, set the derivative equal to zero and solve for r.
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\(r = \sqrt[3]{\frac{170}{\pi}} \approx 3.78\,cm\)
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Find the minimum surface area. [2]
Hint
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Substitute the optimal radius \(r \approx 3.78\,cm\) into \(A = 2\pi r^2 + \frac{680}{r}\) to compute the minimum surface area.
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Minimum surface area \(A \approx 269\,cm^2\)
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