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Date May 2021 Marks available 2 Reference code 21M.1.SL.TZ2.13
Level Standard Level Paper Paper 1 Time zone Time zone 2
Command term Justify and Describe Question number 13 Adapted from N/A

Question

A company produces and sells electric cars. The company’s profit, P, in thousands of dollars, changes based on the number of cars, x, they produce per month.

The rate of change of their profit from producing x electric cars is modelled by

dPdx=1.6x+48, x0.

The company makes a profit of 260 (thousand dollars) when they produce 15 electric cars.

Find an expression for P in terms of x.

[5]
a.

The company regularly increases the number of cars it produces.

Describe how their profit changes if they increase production to over 30 cars per month and up to 50 cars per month. Justify your answer.

[2]
b.

Markscheme

recognition of need to integrate (eg reverse power rule or integral symbol)             (M1)

Px=-0.8x2+48x+c             A1A1

260=-0.8×152+48×15+c             (M1)


Note: Award M1 for correct substitution of x=15 and P=260. A constant of integration must be seen (can be implied by a correct answer).


c=-280

Px=-0.8x2+48x-280          A1


[5 marks]

a.

profit will decrease (with each new car produced)             A1

EITHER

because the profit function is decreasing / the gradient is negative / the rate of change of P is negative             R1


OR

3050-1.6x+48dx=-320             R1


OR

evidence of finding P30=440 and P50=120             R1


Note:
Award at most R1A0 if P30 or P50 or both have incorrect values.


[2 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis
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