4.6 Integration (A-level only)

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Question 53

A spherical drop of industrial lubricant is being injected into a precision-engineered cavity. The volume, $V$ , of the drop is increasing at a constant rate of $s−1180\pi \text{ mm}^3\text{ s}^{-1}$ . Calculate the rate of increase of the radius, $r$ , of the drop in $\text{mm s}^{-1}$ at the moment when the radius is exactly $\text{ mm}$ . [The volume $V$ of a sphere of radius $r$ is given by $\frac{4}{3}\pi r^3$ ]

[5]

ii.

The depth of sediment, $\text{ metres}$ , settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation

\frac{\text{d}y}{\text{d}t} = \frac{k}{y^2}

where $k$ is a positive constant and $t$ hours is the time after monitoring began. Given that:

at the start of monitoring ( $t = 0$ ), the sediment depth was $\text{ metres}$ .
after $5$ hours of monitoring, the sediment depth had reached $\text{ metres}$ .
after $T$ hours of monitoring, the sediment depth reached $\text{ metres}$ .

Solve the differential equation to determine the value of $T$ . Give your answer to one decimal place.

[8]

4.6 Integration (A-level only)

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Question 53

[5]

ii.

The depth of sediment, $\text{ metres}$ , settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation

\frac{\text{d}y}{\text{d}t} = \frac{k}{y^2}

where $k$ is a positive constant and $t$ hours is the time after monitoring began. Given that:

at the start of monitoring ( $t = 0$ ), the sediment depth was $\text{ metres}$ .
after $5$ hours of monitoring, the sediment depth had reached $\text{ metres}$ .
after $T$ hours of monitoring, the sediment depth reached $\text{ metres}$ .

Solve the differential equation to determine the value of $T$ . Give your answer to one decimal place.

[8]

4.6 Integration (A-level only)

Question 53

Markscheme

4.6 Integration (A-level only)

Question 53

Markscheme

4.6 Integration (A-level only)

Question 53

4.6 Integration (A-level only) Questions

4.6 Integration (A-level only)

Question 53

4.6 Integration (A-level only) Questions