The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years - AQA - A-Level Maths Pure - Question 9 - 2021 - Paper 1
Question 9
The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years.
Year 1980 1985 1990... show full transcript
Worked Solution & Example Answer:The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years - AQA - A-Level Maths Pure - Question 9 - 2021 - Paper 1
Step 1
Show algebraically that the graph of log10 P against t should be linear.
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Answer
Taking the logarithm of both sides of the equation gives us:
log10P=log10(A×10kt)
This can be simplified using the properties of logarithms:
log10P=log10A+kt
This equation represents a straight line with gradient k and y-intercept \log_{10} A, thus confirming that the graph of \log_{10} P against t is linear.
Step 2
Complete the table below.
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Answer
t
log10 P
0
1.88
5
1.97
10
2.08
15
2.13
20
2.19
25
2.31
Step 3
Plot log10 P against t, and draw a line of best fit for the data.
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Answer
To plot log10 P against t, use graph paper or a digital graphing tool. Ensure that:
The x-axis represents the variable t (years since 1980).
The y-axis represents log10 P (production).
Mark the points accurately according to the completed table and draw a line of best fit through the points.
Step 4
Hence show that k is approximately 0.02.
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Answer
The gradient of the line of best fit can be calculated using any two points from the line:
Gradient=x2−x1y2−y1=25−02.41−1.88≈0.0212
Thus, k is approximately 0.02.
Step 5
Find the value of A.
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Answer
From the equation P = A × 10^(kt), we can substitute t = 0 to find A:
At t = 0:
P=A×100=A
From the table, when t = 0, P = 75. Thus, A = 75.
Step 6
Using the model with k = 0.02 predict the number of tonnes of annual global production of plastics in 2030.
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Answer
For 2030, t = 50:
Using the formula:
P=75×100.02×50
Calculating this gives:
P≈75×101=750 million tonnes.
Step 7
Using the model with k = 0.02 predict the year in which P first exceeds 8000.
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Answer
We want to find the year when P > 8000:
From the formula P = A × 10^(kt):
8000=75×100.02t
Rearranging gives:
758000=100.02t
Taking logarithms:
log10758000=0.02t
Calculating:
log10≈3.903=0.02t⟹t≈0.023.903≈195.15
Thus, the year is approximately 1980 + 195 = 2175.
Step 8
Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics.
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The model may be inappropriate as it assumes a constant growth rate. Future production can be influenced by regulatory changes, increased awareness of environmental impacts, technological advancements, and shifting consumer preferences. These factors can lead to deviations from the predicted model.
