Melaka Trial 2025 Add Maths Paper 2, Question 9Differentiation

Solved on video in English · Form 5, Differentiation · 7:28 video

The question

(a)

A company wants to build an open-top cylindrical tank with a volume of 1000 m³. The radius is r m and the height is h m. It is given that the cost to build the base is RM5 per square metre and the cost to build the cylindrical wall is RM2 per square metre.

(a) Express the cost of building the tank, C, in terms of r and π.

(b)

For the open-top cylindrical tank with cost C = 5πr² + 4000/r (volume 1000 m³):

(b) Find the radius and height that minimize the construction cost. Hence, find the minimum construction cost. [Use π = 3.142]

(c)

After the tank (radius r = 5.031 m) is built, it starts being filled with water at a rate of 20 m³ per minute.

(c) Find the rate of change of the water height in the tank when the height of the water reaches 8 metres. [Use π = 3.142]

Melaka Trial 2025 Add Maths Paper 2, Question 9: the question as drawn in Eduly's video7:28

Part (a)

Step 1

Volume: πr²h = 1000, so h = 1000/(πr²).

4 more steps, each with the reason for it, in the full solution

Final answer

C = 5πr² + 4000/r

Where students lose marks

Watch out: the tank is open-top, so count one base only, and the RM2 rate makes the wall cost 4πrh, not 2πrh.

Part (b)

5 more steps, each with the reason for it, in the full solution

Final answer

r = 5.031 m, h = 12.57 m, minimum cost = RM1192.71

Where students lose marks

Watch out: 4000/r differentiates to −4000/r². After finding r, also find h and the minimum cost, and check it is a minimum.

Part (c)

4 more steps, each with the reason for it, in the full solution

Final answer

dh/dt = 0.2515 m per minute

Where students lose marks

Watch out: the 8 m is not needed. The radius is fixed, so dV/dt = πr² dh/dt at every height.

Watch every step of this question solved on video (7:28), with the reason behind each step and a box that checks your own answer.

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