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Date May 2016 Marks available 3 Reference code 16M.1.sl.TZ1.15
Level SL only Paper 1 Time zone TZ1
Command term Calculate Question number 15 Adapted from N/A

Question

A company sells fruit juices in cylindrical cans, each of which has a volume of 340cm3. The surface area of a can is Acm2 and is given by the formula

A=2πr2+680r ,

where r is the radius of the can, in cm.

To reduce the cost of a can, its surface area must be minimized.

Find dAdr

[3]
a.

Calculate the value of r that minimizes the surface area of a can.

[3]
b.

Markscheme

(dAdr)=4πr680r2        (A1)(A1)(A1)      (C3)

Note: Award (A1) for 4πr (accept 12.6r), (A1) for 680, (A1) for 1r2 or r2

Award at most (A1)(A1)(A0) if additional terms are seen.

a.

4πr680r2=0          (M1)

Note: Award (M1) for equating their dAdr to zero.

4πr3680=0          (M1)

Note: Award (M1) for initial correct rearrangement of the equation. This may be assumed if r3=6804π or r=36804π seen.

OR

sketch of A with some indication of minimum point         (M1)(M1)

Note: Award (M1) for sketch of A, (M1) for indication of minimum point.

OR

sketch of dAdr with some indication of zero      (M1)(M1)

Note: Award (M1) for sketch of dAdr(M1) for indication of zero.

(r=)3.78(cm)(3.78239...)                (A1)(ft) (C3)  

Note: Follow through from part (a).

b.

Examiners report

Question 15: Optimization
Many candidates were able to differentiate in part (a), but then were unable to relate this to part (b). However, it seemed that many more had not studied the calculus at all.

a.

Question 15: Optimization
Many candidates were able to differentiate in part (a), but then were unable to relate this to part (b). However, it seemed that many more had not studied the calculus at all.

b.

Syllabus sections

Topic 7 - Introduction to differential calculus » 7.2
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