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Date November 2021 Marks available 2 Reference code 21N.1.SL.TZ0.12
Level Standard Level Paper Paper 1 Time zone Time zone 0
Command term Solve Question number 12 Adapted from N/A

Question

The surface area of an open box with a volume of 32cm3 and a square base with sides of length xcm is given by Sx=x2+128x where x>0.

Find S(x).

[3]
a.

Solve S'(x)=0.

[2]
b.i.

Interpret your answer to (b)(i) in context.

[1]
b.ii.

Markscheme

Sx= x2+128x-1             (M1)


Note: Award (M1) for expressing second term with a negative power. This may be implied by 1x2 seen as part of their answer.


2x-128x2  OR  2x-128x-2             A1A1


Note:
Award A1 for 2x and A1 for -128x2. The first A1 is for x2 differentiated correctly and is independent of the (M1).

 

[3 marks]

a.

EITHER

any correct manipulation of 2x-128x2=0  e.g. 2x3-128=0              (M1)


OR

sketch of graph of S'(x) with root indicated              (M1)

 

OR

sketch of graph of S(x) with minimum indicated              (M1)


THEN

x=4             A1


Note:
Value must be positive. Follow through from their part (a) irrespective of working.

 

[2 marks]

b.i.

the value of x that will minimize surface area of the box               A1


Note: Accept ‘optimize’ in place of minimize.

 

[1 mark]

b.ii.

Examiners report

In part (a), many candidates scored at least the mark for correctly differentiating x2 although differentiating 128x proved to be more problematic, not realizing that the term could be written as 128x-1. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.

a.

In part (a), many candidates scored at least the mark for correctly differentiating x2 although differentiating 128x proved to be more problematic, not realizing that the term could be written as 128x-1. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.

b.i.

In part (a), many candidates scored at least the mark for correctly differentiating x2 although differentiating 128x proved to be more problematic, not realizing that the term could be written as 128x-1. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.

b.ii.

Syllabus sections

Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z
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Topic 5—Calculus » SL 5.6—Stationary points, local max and min
Topic 5—Calculus » SL 5.7—Optimisation
Topic 5—Calculus

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