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Date May 2021 Marks available 3 Reference code 21M.2.SL.TZ2.6
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Show that Question number 6 Adapted from N/A

Question

All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death of a plant can be determined by measuring the amount of carbon-14 still present in the remains.

The amount, A, of carbon-14 present in a plant t years after its death can be modelled by A=A0e-kt where t0 and A0, k are positive constants.

At the time of death, a plant is defined to have 100 units of carbon-14.

The time taken for half the original amount of carbon-14 to decay is known to be 5730 years.

Show that A0=100.

[1]
a.

Show that k=ln25730.

[3]
b.

Find, correct to the nearest 10 years, the time taken after the plant’s death for 25% of the carbon-14 to decay.

[3]
c.

Markscheme

100=A0e0             A1

A0=100             AG

 

[1 mark]

a.

correct substitution of values into exponential equation             (M1)

50=100e-5730k  OR  e-5730k=12


EITHER

-5730k=ln12             A1

ln12=-ln2  OR  -ln12=ln2             A1


OR

e5730k=2             A1

5730k=ln2             A1


THEN

k=ln25730             AG


Note:
There are many different ways of showing that k=ln25730 which involve showing different steps. Award full marks for at least two correct algebraic steps seen.

 

[3 marks]

b.

if 25% of the carbon-14 has decayed, 75% remains ie, 75 units remain                      (A1)

75=100e-ln25730t


EITHER

using an appropriate graph to attempt to solve for t                      (M1)


OR

manipulating logs to attempt to solve for t                               (M1)

ln0.75=-ln25730t

t=2378.164


THEN

t=2380 (years) (correct to the nearest 10 years)                   A1

 

[3 marks]

c.

Examiners report

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

Syllabus sections

Topic 2—Functions » SL 2.9—Exponential and logarithmic functions
Topic 2—Functions

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