Photo AI
The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B - Edexcel - A-Level Maths: Pure - Question 7 - 2014 - Paper 1
Question 7
The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B. ... show full transcript
Worked Solution & Example Answer:The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B - Edexcel - A-Level Maths: Pure - Question 7 - 2014 - Paper 1
Step 1
Step 2
Find the size of the angle DBC, giving your answer in radians to 3 decimal places.
Answer
To find angle DBC, we first use the fact that angle EBD is given as 1.4 radians, and angle EAB is (\frac{\pi}{2}) radians. Hence, we calculate:
Calculating this:
Rounding to three decimal places, we get:
Step 3
Find, in cm², the area of the shape ABCDEA, giving your answer to 3 significant figures.
Answer
To find the area of the shape ABCDEA, we need to combine the area of triangle EAB and sector BDE, along with triangle DBC.
-
Area of triangle EAB: Since EAB is a right triangle: [\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 5 = 12.5 , \text{cm}^2]
-
Area of triangle DBC: The length BC is 7.5 cm and we already calculated angle DBC as approximately 0.171 radians: [\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \times \sin(\text{Angle DBC})] To find the height, we apply the sine function: [\text{Area} = \frac{1}{2} \times 7.5 \times 5 \times \sin(0.171) \approx \frac{1}{2} \times 7.5 \times 5 \times 0.1709 = 6.55 , \text{cm}^2]
-
Total area of shape ABCDEA: [\text{Total Area} = \text{Area of triangle EAB} + \text{Area of sector BDE} + \text{Area of triangle DBC}] [\text{Total Area} = 12.5 + 17.5 + 6.55 = 36.55 , \text{cm}^2] Rounding to three significant figures gives: 36.6 cm².