WMA12/01 June 2022 Question 9: Solving Exponential and Log Equations

Question 1

Sketch the graph of N against t when k = 1, showing exponential decay.

Accepted Answer

N = λᵗ with 0 < λ < 1 is exponential decay At t = 0, N = λ⁰ = 1, so the curve cuts the N-axis at 1 As t increases N decreases towards 0 but never reaches it, so the t-axis is an asymptote Answer: Exponential decay curve: starts at (0, 1) on the N-axis, decreases as t increases, always above the t-axis and approaching it (N = 0) as an asymptote, with no turning points.

Question 2

Using a half-life of 5700 years, show that λ = 0.999878 to 6 decimal places.

Accepted Answer

After 5700 years N = k/2: (1/2)k = kλ⁵⁷⁰⁰ λ⁵⁷⁰⁰ = 1/2 λ = (1/2)^(1/5700) = 0.9998784… = 0.999878 to 6 d.p. Answer: Shown: λ⁵⁷⁰⁰ = 1/2, so λ = (1/2)^(1/5700) = 0.999878 (6 d.p.)

Question 3

For Item A with initial 15 grams and age 3250 years, find the current carbon-14 to 3 significant figures.

Accepted Answer

N = 15 × 0.999878³²⁵⁰ N = 10.09… ≈ 10.1 grams Answer: 10.1 grams

Question 4

For Item B with initial 25 grams and 18 grams remaining, calculate its age to the nearest 100 years.

Accepted Answer

18 = 25 × 0.999878ᵗ 0.999878ᵗ = 18/25 t = log(18/25)/log(0.999878) = 2692.5… Age ≈ 2700 years (to the nearest 100 years) Answer: 2700 years

WMA12/01 Jun 2022, Question 9: Solving Exponential and Log Equations

The question, in short

Part (a) · 2 marks

Part (b) · 2 marks

Part (c) · 2 marks

Part (d) · 3 marks

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