Further Applications of Differentiation (Edexcel A-Level Mathematics): Revision Notes

1. Identify the quantities involved: Determine which quantities are changing with respect to time (or another variable).
2. Write down the relationship: Express the relationship between the quantities using an equation.
3. Differentiate the equation with respect to time: Use implicit differentiation to find the rates of change of the quantities with respect to time (or the other variable).
4. Substitute known values: Plug in the given values (including the known rate of change) and solve for the unknown rate of change.
5. Interpret the result: Make sure the solution makes sense in the context of the problem.

Example 1: Expanding Circle

Problem: A circle's radius is increasing at a rate of $2$ $cm$ per second. How fast is the area of the circle increasing when the radius is $5$ $cm$?

• Step 1: Identify the quantities involved:
• Let $r(t)$ be the radius of the circle at time $t$ .
• Let $A(t)$ be the area of the circle at time $t$ .
• Given: $\frac{dr}{dt}$ = $2$ $cm/s.$
• Find: $\frac{dA}{dt}$ when $r = 5 cm.$

• Step 2: Write down the relationship:
• The area of a circle is given by $A = \pi r^2 .$

• Step 3: Differentiate the equation with respect to time: $\frac{dA}{dt} = \frac{d}{dt} \left(\pi r^2\right) = 2\pi r \frac{dr}{dt}$

• Step 4: Substitute known values:
• When $r = 5 cm\ and\ \frac{dr}{dt} = 2 cm/s:$ $\frac{dA}{dt} = 2\pi(5)(2) = 20\pi \text{ cm}^2/\text{s}$

• Step 5: Interpret the result:
• The area of the circle is increasing at a rate of $20\pi$ square centimetres per second when the radius is $5$ $cm$.

Example 2: Draining Water from a Conical Tank

Problem: Water is draining from a conical tank at a rate of $3$ cubic meters per minute. If the tank has a height of $12$ meters and a base radius of 6 meters, how fast is the water level falling when the water is $4$ meters deep?

• Step 1: Identify the quantities involved:
• Let V(t) be the volume of water in the tank at time t .
• Let h(t) be the height of the water in the tank at time t .
• Given: $\frac{dV}{dt}$= -3 $m^3/min$ (negative because the volume is decreasing).
• Find: $\frac{dh}{dt}$ when h = 4 meters.
• Step 2: Write down the relationship:
• The volume of a cone is $V = \frac{1}{3} \pi r^2 h .$
• The radius $r$ of the water surface and the height h of the water are related by similar triangles: $\frac{r}{h} = \frac{6}{12} = \frac{1}{2} .$
• So, $r = \frac{h}{2} .$
• Step 3: Substitute $r = \frac{h}{2}$ into the volume formula:

$\ V = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h = \frac{1}{3} \pi \frac{h^3}{4} = \frac{\pi h^3}{12}$

• Step 4: Differentiate the equation with respect to time: $\frac{dV}{dt} = \frac{d}{dt} \left(\frac{\pi h^3}{12}\right) = \frac{\pi}{4} h^2 \frac{dh}{dt}$

• Step 5: Substitute known values:

• Given $\frac{dV}{dt} = -3 m^3/min\ and\ h = 4$ meters:

• $-3 = \frac{\pi}{4} (4)^2 \frac{dh}{dt} \\$ $-3 = \frac{\pi}{4} \times 16 \times \frac{dh}{dt} \\$ $-3 = 4\pi \frac{dh}{dt} \\$ $\frac{dh}{dt} = -\frac{3}{4\pi} \text{ meters/minute}$

• Step 6: Interpret the result:

• The water level is falling at a rate of $\frac{3}{4\pi}$ meters per minute when the water is $4$ meters deep.

• Always start by identifying the relationship between the quantities involved.
• Use implicit differentiation when necessary, and always differentiate with respect to the independent variable, typically time.
• Pay attention to the signs of the rates (positive for increasing quantities, negative for decreasing quantities).

Connected Rates of Change

The radius of a circle is increasing at a constant rate of $0.2 \, \text{cm} \, \text{s}^{-1}$.

a) Show that the perimeter of the circle is increasing at the rate of $0.4\pi \, \text{cm} \, \text{s}^{-1}$.

b) Find the rate at which the area of the circle is increasing when the radius is $10 \, \text{cm}$.

c) Find the radius of the circle when its area is increasing at the rate of $20 \, \text{cm}^2 \, \text{s}^{-1}$.

The area of a circle is decreasing at a constant rate of $0.5 \, \text{cm}^2 \, \text{s}^{-1}$. a) Find the rate at which the radius of the circle is decreasing when the radius is $8 \, \text{cm}$.

b) Find the rate at which the perimeter of the circle is decreasing when the radius is $8 \, \text{cm}$.