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Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40° The attempt of a student to find the area of the sector is shown below - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 2
Question 5
Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40° The attempt of a student to find the area of the sector is shown below. Area ... show full transcript
Worked Solution & Example Answer:Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40° The attempt of a student to find the area of the sector is shown below - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 2
Step 1
Explain the error made by this student.
Answer
The error made by the student is in the use of the formula for calculating the area of the sector. The student did not convert the angle AOB from degrees to radians before using it in the formula. The formula for the area of a sector is valid only when the angle is expressed in radians. The correct conversion from degrees to radians is given by ( \frac{\theta}{180} \times \pi ). In this case, the angle 40° should be converted to radians as follows:
[ \theta = 40° \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9} \text{ radians} ]
This step is crucial for obtaining the correct area.
Step 2
Write out a correct solution.
Answer
To find the correct area of the sector, we first convert the angle from degrees to radians:
[ \theta = 40° \times \frac{\pi}{180} = \frac{2\pi}{9} \text{ radians} ]
Now, we can apply the formula for the area of a sector:
[ \text{Area of sector} = \frac{1}{2} r^2 \theta ]
Substituting the values we have:
[ \text{Area of sector} = \frac{1}{2} \times 5^2 \times \frac{2\pi}{9} ]
[ = \frac{25 \pi}{9} \text{ cm}^2 \approx 8.73 ext{ cm}^2 ]
Thus, the correct area of the sector is approximately 8.73 cm².