The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below. All distances are measured in centimetres. (a) Use a single trapezium to find an approximate value of the area of the shaded region, giving your answer in cm² to two decimal places. (b) Shape B is made from four copies of region R as shown in Figure 4 below. Shape B is cut from metal of thickness 2 mm. The metal has a density of 10.5 g/cm³. Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B. Give your answer to the nearest gram. (c) Without further calculation, give one reason why the mass found in part (b) may be: (i) an underestimate. (ii) an overestimate.

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The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below - AQA - A-Level Maths Pure - Question 11 - 2020 - Paper 1

Question 11

The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below. All distances are measured... show full transcript

Worked Solution & Example Answer:The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below - AQA - A-Level Maths Pure - Question 11 - 2020 - Paper 1

Step 1

Use a single trapezium to find an approximate value of the area of the shaded region

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Answer

To find the area of the shaded region R using the trapezium rule, we first need to evaluate the function at the endpoints and determine the height of the trapezium.

Calculate the values:
- For $x = 1$ : $f(1) = ext{ln}(8 - 1) = ext{ln}(7) \approx 1.94591$
- For $x = 6$ : $f(6) = ext{ln}(8 - 6) = ext{ln}(2) \approx 0.69315$
The trapezium rule formula for area A is:
$A = \frac{(b-a)}{2}(f(a) + f(b))$
where $a = 1$ , $b = 6$ .
Substituting the values:
$A = \frac{(6-1)}{2}(f(1) + f(6)) = \frac{5}{2}(1.94591 + 0.69315) = 5 \times 1.3195305 = 6.5976525$
Rounding to two decimal places gives $6.60$ cm².

Step 2

Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B

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Answer

To approximate the mass of Shape B, follow these steps:

Identify the six ordinates using the function $f(x) = ext{ln}(8 - x)$ for $x = 1$ to $x = 6$ . The values are:
- $f(1) \approx 1.94591$
- $f(2) \approx 1.79176$
- $f(3) \approx 1.09861$
- $f(4) \approx 1.38629$
- $f(5) \approx 0.69315$
- $f(6) \approx 0.69315$
  (repeating the last ordinate due to the nature of the trapezium rule).
Calculate the area A under the curve using the trapezium rule:
$A = \frac{(b-a)}{12}(f_0 + 2(f_1+f_2+f_3+f_4) + f_5)$
As $A = 7.205633$ cm². This area is the combined area for 4 regions, hence repeat for the shape B defined above.
The volume of Shape B, given thickness of 2 mm (0.2 cm), is:
$V = A \times \text{thickness} = 7.205633 \times 0.2 = 1.4411266 \text{ cm}^3$
To find mass $m$ , use the formula $m = \text{density} \times \text{volume}$ :
$m = 10.5 \text{ g/cm}^3 \times 1.4411266 \text{ cm}^3 = 15.105831 \text{ g}$
Rounding gives approximately $61$ g.

Step 3

Without further calculation, give one reason why the mass found in part (b) may be: an underestimate

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Answer

The trapezia are all below the curve, indicating that under the assumptions of the trapezium rule, the area calculated will be an underestimate.

Step 4

Without further calculation, give one reason why the mass found in part (b) may be: an overestimate

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Answer

The calculations round numbers, likely causing the final answer to be higher than the actual mass.

The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below - AQA - A-Level Maths Pure - Question 11 - 2020 - Paper 1

Worked Solution & Example Answer:The region R enclosed by the lines $x = 1$, $x = 6$, $y = 0$ and the curve $y = ext{ln} (8 - x)$ is shown shaded in Figure 3 below - AQA - A-Level Maths Pure - Question 11 - 2020 - Paper 1

Use a single trapezium to find an approximate value of the area of the shaded region

Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B

Without further calculation, give one reason why the mass found in part (b) may be: an underestimate

Without further calculation, give one reason why the mass found in part (b) may be: an overestimate

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