Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians. (a) Find the length of the arc AB. (b) Find the area of the sector OAB. The line AC shown in Figure 1 is perpendicular to OA, and OBC is a straight line. (c) Find the length of AC, giving your answer to 2 decimal places. (d) Find the area of H, giving your answer to 2 decimal places.

A-Level Edexcel Maths: Pure Solving Equations: Figure 1 shows the sector OAB of

Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians - Edexcel - A-Level Maths: Pure - Question 7 - 2010 - Paper 3

Question 7

Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians. (a) Find the length of the arc AB. (b) Find the area of the sector OAB.... show full transcript

Worked Solution & Example Answer:Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians - Edexcel - A-Level Maths: Pure - Question 7 - 2010 - Paper 3

Step 1

Find the length of the arc AB.

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Answer

To find the length of the arc AB, we use the formula for arc length: $L = r\theta$ where $r$ is the radius and $\theta$ is the angle in radians. Here, $r = 9$ cm and $\theta = 0.7$ radians. Thus, the length of the arc AB is: $L = 9 \times 0.7 = 6.3 \text{ cm}$

Step 2

Find the area of the sector OAB.

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Answer

The area of a sector can be calculated using the formula: $A = \frac{1}{2} r^2 \theta$ Substituting the values, we have: $A = \frac{1}{2} \times 9^2 \times 0.7 = \frac{1}{2} \times 81 \times 0.7 = 28.35 \text{ cm}^2$

Step 3

Find the length of AC, giving your answer to 2 decimal places.

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Answer

Since line AC is perpendicular to OA, we can use the tangent function: $\tan(0.7) = \frac{AC}{9}$ Solving for AC, we have: $AC = 9 \times \tan(0.7)$ Calculating this value gives approximately: $AC \approx 9 \times 0.759 = 6.83 \text{ cm}$ Thus, to 2 decimal places, AC is $6.83$ cm.

Step 4

Find the area of H, giving your answer to 2 decimal places.

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Answer

The area of triangle AOC can be calculated using the formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base is AC and the height is the radius OA. We already found AC above. Hence: $\text{Area} = \frac{1}{2} \times AC \times OA = \frac{1}{2} \times 6.83 \times 9$ This evaluates to: $\text{Area} \approx 30.67 \text{ cm}^2$ Thus, the area of H is approximately $30.67$ cm².

Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians - Edexcel - A-Level Maths: Pure - Question 7 - 2010 - Paper 3

Worked Solution & Example Answer:Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians - Edexcel - A-Level Maths: Pure - Question 7 - 2010 - Paper 3

Find the length of the arc AB.

Find the area of the sector OAB.

Find the length of AC, giving your answer to 2 decimal places.

Find the area of H, giving your answer to 2 decimal places.

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