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Figure 1 is a sketch representing the cross-section of a large tent ABCDEF - Edexcel - A-Level Maths: Pure - Question 5 - 2016 - Paper 2

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Figure 1 is a sketch representing the cross-section of a large tent ABCDEF. AB and DE are line segments of equal length. Angle FAB and angle DEF are equal. F is the ... show full transcript

Worked Solution & Example Answer:Figure 1 is a sketch representing the cross-section of a large tent ABCDEF - Edexcel - A-Level Maths: Pure - Question 5 - 2016 - Paper 2

Step 1

the length of the arc BCD in metres to 2 decimal places

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Answer

To find the length of the arc BCD, we use the formula:

extArcLength=rimesheta ext{Arc Length} = r imes heta

where ( r ) is the radius and ( \theta ) is the angle in radians.

Given that the radius ( r = 3.5 ) m and the angle ( \theta = 1.77 ) radians, we can calculate:

extArcLength=3.5imes1.77=6.195 m ext{Arc Length} = 3.5 imes 1.77 = 6.195\,\text{m}

Rounded to two decimal places, the length of the arc BCD is:

≈6.20 m\approx 6.20\,\text{m}

Step 2

the area of the sector FBCD in m² to 2 decimal places

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Answer

The area of a sector can be calculated using the formula:

extArea=12r2θ ext{Area} = \frac{1}{2} r^2 \theta

Using the values ( r = 3.5 ) m and ( \theta = 1.77 ) radians:

extArea=12×(3.5)2×1.77=12×12.25×1.77=10.84 m2 ext{Area} = \frac{1}{2} \times (3.5)^2 \times 1.77 = \frac{1}{2} \times 12.25 \times 1.77 = 10.84\,\text{m}^2

Thus, the area of the sector FBCD, rounded to two decimal places, is:

≈10.84 m2\approx 10.84\,\text{m}^2

Step 3

the total area of the cross-section of the tent in m² to 2 decimal places

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Answer

To find the total area of the cross-section, we need to consider both the area of the sector FBCD and the area of triangle BFD.

First, we calculate the area of triangle BFD:

The formula for the area of a triangle is:

extArea=12×base×height ext{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Using ( ext{base} = BF + FD = 3.5 + 3.5 = 7 ) m and the angle at F:

extAreatriangle BFD=12×7×3.5×sin⁡(1.77) ext{Area}_{\text{triangle BFD}} = \frac{1}{2} \times 7 \times 3.5 \times \sin(1.77)

Calculating, we have:

≈12×7×3.5×0.985≈10.84 m2\approx \frac{1}{2} \times 7 \times 3.5 \times 0.985 \approx 10.84\,\text{m}^2

Now, adding both areas together:

Total Area=10.84+10.84=21.68 m2\text{Total Area} = 10.84 + 10.84 = 21.68\,\text{m}^2

Thus, the total area of the cross-section of the tent, rounded to two decimal places, is:

≈21.68 m2\approx 21.68\,\text{m}^2

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