Question 1

Paper 1Medium6 marks

A nutrient-solution pump feeds a tank. During the first 6 minutes, its delivery rate is modelled by

q(t)=12tt2+9 mL min−1,q(t)=\frac{12t}{t^2+9}\text{ mL min}^{-1},

where tt is the time in minutes after the pump starts. Let V(t)V(t) be the volume, in mL, delivered by time tt with V(0)=0V(0)=0

(a)

By evaluating the definite integral for V(t)V(t) show that

V(t)=6ln⁡(t2+99).V(t)=6\ln\left(\frac{t^2+9}{9}\right).
[2]
(b)

Calculate V(6)V(6) giving your answer to 3 significant figures.

[1]
(c)

A warning light activates when 8 mL has been delivered. Use your answer to part (b) to establish that the light activates before 6 minutes, and hence determine the time at which it activates. Give your answer to 3 significant figures.

[3]

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