Sketching Exponential Graphs

Edexcel IAL WMA12/01/P2/June 2022/Q9 (Sketching​​ Exponentials &​​ Logarithm Decay Graphs)

A scientist is using carbon‑14 dating to determine the age of some wooden items.​​ 

The equation for carbon‑14 dating an item is given by​​ 

N = kλt

where​​ 

• N grams is the amount of carbon‑14 currently present in the item​​ 

• k grams was the initial amount of carbon‑14 present in the item​​ 

• t is the number of years since the item was made​​ 

• λ is a constant, with 0 < λ < 1​​ 

• Sketch the graph of N against t for k = 1​​ 

(2)​​ 

Given that it takes 5700 years for the amount of carbon‑14 to reduce to half its​​ initial value,​​ 

• show that the value of the constant λ is 0.999878 to 6 decimal places.​​ 

(2)

Given that Item A​​ 

• is known to have had 15 grams of carbon‑14 present initially​​ 

• is thought to be 3250 years old​​ 

• calculate, to 3 significant figures, how much carbon‑14 the equation predicts is currently in Item A.​​ 

(2)

Item B is known to have initially had 25 grams of carbon‑14 present, but only 18 grams now remain.​​ 

• Use algebra to calculate the age of Item B to the nearest 100 years.​​ 

(3)

SOLUTION​​ 

a- ​​ It is given that​​ 

N=k λt

Where,​​ k=1

So, we get​​ 

 N=λt

Since​​ 0<λ<1, it has to be a fraction. So let us suppose​​ λ=12. This would give the equation to be​​ 

N=12t

N=12t

N=2-1t

N=2-t

So, we need to sketch the​​ graph of​​ N=2-t​​ or y=2-x, which is the reflection of​​ y=2x.

b- When​​ t=5700 and the amount of carbon-14 present is half of the initial amount, meaning thereby that the new value of N is​​ 12k.

N=k λt

12k=k λt

k​​ gets cancelled out on both sides.​​ 

λt=12

λ5700 =12

λ=125700=0.99987840…

λ=0.999878406 dp

c- Now, when​​ k=15g,​​ t=3250 g, and​​ λ= 0.999878, the amount of carbon 14​​ present is​​ 

N=150.9998783250

N=10.103…. 

N=10.1 g

d- Using the same equation.​​ 

N=kλt

18=250.999878t

1825= 0.999878t

log0.999878⁡1825=t

t=2701.407…

t=2700 years