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A soft drink can has a volume of 340 cm³, a height of h cm and a radius of r cm - NSC Mathematics - Question 9 - 2016 - Paper 1
Question 9
A soft drink can has a volume of 340 cm³, a height of h cm and a radius of r cm. 9.1 Express h in terms of r. 9.2 Show that the surface area of the can is given by... show full transcript
Worked Solution & Example Answer:A soft drink can has a volume of 340 cm³, a height of h cm and a radius of r cm - NSC Mathematics - Question 9 - 2016 - Paper 1
Step 1
Express h in terms of r.
Answer
To express the height h in terms of the radius r, we can use the formula for the volume of a cylinder, given by:
V = ext{Base Area} imes ext{Height} = rac{22}{7}r^2h
Setting this equal to the given volume of 340 cm³, we have:
340 = rac{22}{7} r^2 h
Rearranging for h gives:
h = rac{340 imes 7}{22 r^2} = rac{2380}{22 r^2} = rac{109.09}{r^2}.
Step 2
Show that the surface area of the can is given by A(r) = 2πr² + 680r⁻¹.
Answer
The surface area A of a cylinder can be given by:
A = 2 ext{Base Area} + ext{Lateral Surface Area} = 2 rac{22}{7}r^2 + 2 rac{22}{7}rh
Substituting h from the previous part, we have:
A = 2 rac{22}{7}r^2 + 2 rac{22}{7}r\left(\frac{2380}{22 r^2}\right)
This simplifies to:
A = 2 rac{22}{7}r^2 + rac{47600}{22 r}
Thus,
A(r) = 2 rac{22}{7}r^2 + 680r^{-1}.
Step 3
Determine the radius of the can that will ensure the surface area is a minimum.
Answer
To find the radius that minimizes the surface area, we differentiate A with respect to r:
A'(r) = 4 rac{22}{7}r - 680r^{-2}
Setting the derivative equal to zero:
4 rac{22}{7}r - 680r^{-2} = 0
This leads to:
4 rac{22}{7}r^3 = 680
Solving for r:
Calculating this gives:
.