22M.1.AHL.TZ1.16

pestleMathematicsAIHLPaper 122M· ahl-5-9-differentiating-standard-functions-and-derivative-rulessource ↗

The wind chill index $W$ is a measure of the temperature, in $° C$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $({km h}^{- 1})$ is given by the equation

$W = 19.34 - 7.405 v^{0.16}$

Find an expression for $\frac{d W}{d v}$ .

[2]

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5 {km h}^{- 1} {minute}^{- 1}$ .

Find the rate of change of $W$ at this time.

[5]

Markscheme / solution

use of power rule (M1)

$\frac{d W}{d v} = - 1.1848 v^{- 0.84}$ OR $- 1.18 v^{- 0.84}$ A1

[2 marks]

$\frac{d v}{d t} = 5$ (A1)

$\frac{d W}{d t} = \frac{d v}{d t} \times \frac{d W}{d v}$ (M1)

$(\frac{d W}{d t} = - 5 \times 1.1848 v^{- 0.84})$

when $v = 10$

$\frac{d W}{d t} = - 5 \times 1.1848 \times 10^{- 0.84}$ (M1)

$- 0.856 (- 0.856278 \dots) ° {C min}^{- 1}$ A2

Note: Accept a negative answer communicated in words, “decreasing at a rate of…”.
Accept a final answer of $- 0.852809 \dots ° {C min}^{- 1}$ from use of $- 1.18$ .
Accept $51.4$ (or $51.2$ ) $° {C hour}^{- 1}$ .

[5 marks]

Examiners’ report

There was some success in using the power rule to differentiate the function in part (a). Many failed to recognize that part (b) was a related rates of change problem. There was also confusion about the term “rate of change” and with the units used in this question.

[N/A]