IB Mathematics SL 5.8 Approximating areas AI SL Paper 1- Exam Style Questions- New Syllabus
Question
Inspectors are investigating the carbon dioxide emissions of a power plant. Let R be the rate, in tonnes per hour, at which carbon dioxide is being emitted, and t be the time in hours since the inspection began.
When R is plotted against t, the total amount of carbon dioxide produced is represented by the area between the graph and the horizontal t-axis.
The rate, R, is measured over the course of two hours. The results are shown in the following table.
| \( t, \text{hours} \) | 0 | 0.4 | 0.8 | 1.2 | 1.6 | 2 |
|---|---|---|---|---|---|---|
| \( R, \text{tonnes per hour} \) | 30 | 50 | 60 | 40 | 20 | 50 |
(a) Use the trapezoidal rule with an interval width of 0.4 to estimate the total amount of carbon dioxide emitted over the two-hour period. [3]
The real amount of carbon dioxide emitted during these two hours was 72 tonnes.
(b) Determine the percentage error of the estimate from part (a). [2]
▶️ Answer/Explanation
Markscheme
(a)
Use the trapezoidal rule with \( h = 0.4 \), \( n = 5 \):
Formula: \( \text{Area} \approx \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + y_3 + y_4)] \). M1
Data: \( t = [0, 0.4, 0.8, 1.2, 1.6, 2] \), \( R = [30, 50, 60, 40, 20, 50] \).
Substitute: \( \frac{0.4}{2} [30 + 50 + 2(50 + 60 + 40 + 20)] \). A1
Compute: \( 2(50 + 60 + 40 + 20) = 2 \times 170 = 340 \), \( 30 + 50 + 340 = 420 \).
Thus: \( 0.2 \times 420 = 84 \).
Total carbon dioxide: 84 tonnes. A1
[3 marks]
Use the trapezoidal rule with \( h = 0.4 \), \( n = 5 \):
Formula: \( \text{Area} \approx \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + y_3 + y_4)] \). M1
Data: \( t = [0, 0.4, 0.8, 1.2, 1.6, 2] \), \( R = [30, 50, 60, 40, 20, 50] \).
Substitute: \( \frac{0.4}{2} [30 + 50 + 2(50 + 60 + 40 + 20)] \). A1
Compute: \( 2(50 + 60 + 40 + 20) = 2 \times 170 = 340 \), \( 30 + 50 + 340 = 420 \).
Thus: \( 0.2 \times 420 = 84 \).
Total carbon dioxide: 84 tonnes. A1
[3 marks]
(b)
Calculate percentage error: \( \left| \frac{\text{estimate} – \text{actual}}{\text{actual}} \right| \times 100\% \). M1
Substitute: \( \left| \frac{84 – 72}{72} \right| \times 100\% = \left| \frac{12}{72} \right| \times 100\% = \frac{1}{6} \times 100\% \approx 16.6666\ldots\% \).
Rounded: \( 16.7\% \). A1
[2 marks]
Calculate percentage error: \( \left| \frac{\text{estimate} – \text{actual}}{\text{actual}} \right| \times 100\% \). M1
Substitute: \( \left| \frac{84 – 72}{72} \right| \times 100\% = \left| \frac{12}{72} \right| \times 100\% = \frac{1}{6} \times 100\% \approx 16.6666\ldots\% \).
Rounded: \( 16.7\% \). A1
[2 marks]
Total Marks: 5