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.

A-Level AQA Maths Mechanics Quantities, Units & Modelling: The region R enclosed by the lin

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 Mechanics - 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 in ce... 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 Mechanics - 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 region R, we can use the trapezium rule. We evaluate the function at the endpoints:

For $x=1$ , $f(1) = ext{ln}(8-1) = ext{ln}(7) \approx 1.945910 eval)
For $x=6$ , $f(6) = ext{ln}(8-6) = ext{ln}(2) \approx 0.693147

The area can be approximated as:

$A = \frac{b-a}{2} (f(a) + f(b))$ where $a=1$ , $b=6$ . Thus, substituting: $A \approx \frac{6-1}{2} (f(1) + f(6)) = \frac{5}{2} (1.945910 + 0.693147) \approx 6.60 \, \text{cm}^2$

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

For the trapezium rule, we need to divide the interval from $x=1$ to $x=6$ into six equal sub-intervals of width:

$h = \frac{6-1}{6} = 0.8333 \text{ cm}$

Calculating the function values at each ordinate:

$x_0 = 1$ , $f(x_0) = \text{ln}(7) \approx 1.945910$
$x_1 = 1.8333$ , $f(x_1) \approx 1.718 eval$
$x_2 = 2.6667$ , $f(x_2) \approx 1.586$
$x_3 = 3.5000$ , $f(x_3) \approx 1.252...
$x_4 = 4.3333$ , $f(x_4) \approx 0.574...
$x_5 = 5.1667$ , $f(x_5) \approx 0.545...
$x_6 = 6$ , $f(x_6) = 0.693147$

Now calculating the area: $A \approx \frac{h}{2} (f(x_0) + 2(f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5)) + f(x_6))$

The area approximates to $7.205633$ cm².

Next, Volume of Shape B (thickness = 0.2 cm): $V = A \times \text{thickness} = 7.205633 \text{ cm}^2 \times 0.2 \text{ cm} = 1.4411266 \text{ cm}^3$

To find the mass: $ext{mass} = V \times \text{density} = 1.4411266 \text{ cm}^3 \times 10.5 \text{ g/cm}^3 \approx 61 \text{ g}$

Step 3

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

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Answer

The trapezia are all below the curve, which implies that the calculated area may omit some portions of the region, leading to a lower mass estimate.

Step 4

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

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Answer

Numbers in the calculation have been rounded, potentially increasing the area estimate and thus the 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 Mechanics - 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 Mechanics - 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: (i) an underestimate

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

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