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The crescent, shown in the shaded part of the diagram, was created by removing a disc of radius 2.5 cm from a disc of radius 3 cm - Leaving Cert Mathematics - Question 5 - 2019
Question 5
The crescent, shown in the shaded part of the diagram, was created by removing a disc of radius 2.5 cm from a disc of radius 3 cm. Find the area and the perimeter o... show full transcript
Worked Solution & Example Answer:The crescent, shown in the shaded part of the diagram, was created by removing a disc of radius 2.5 cm from a disc of radius 3 cm - Leaving Cert Mathematics - Question 5 - 2019
Step 1
Find the area and the perimeter of the crescent
Answer
To calculate the area of the crescent shape, we need to subtract the area of the smaller circle from the area of the larger circle.
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Calculate the area of the larger circle (radius = 3 cm):
Area = π(3^2) = 9π ≈ 28.27 cm²
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Calculate the area of the smaller circle (radius = 2.5 cm):
Area = π(2.5^2) = 6.25π ≈ 19.63 cm²
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Find the area of the crescent:
Area of crescent = Area of larger circle - Area of smaller circle
= 9π - 6.25π = 2.75π ≈ 8.64 cm². -
Calculate the perimeter of the crescent:
The perimeter consists of the perimeter of the larger semicircle plus the straight edge of the smaller circle.
- Perimeter of larger semicircle = (1/2)(2π(3)) = 3π ≈ 9.42 cm
- Perimeter of smaller semicircle = (1/2)(2π(2.5)) = 2.5π ≈ 7.85 cm
Total Perimeter of the crescent:
= 3π + 2.5π = (3 + 2.5)π = 5.5π ≈ 17.28 cm.
Step 2
Find the time it takes to fill the cone
Answer
To find the time it takes to fill the cone, we first need to calculate the volume of the cone using the formula:
where r is the radius and h is the height of the cone.
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Substituting the values (r = 7 cm, h = 12 cm):
Volume = \frac{1}{3} \pi (7^2)(12) = \frac{1}{3} \pi (49)(12) = \frac{588}{3} \pi = 196 \pi ext{ cm}^3 ≈ 615.75 ext{ cm}^3.
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Calculate the flow rate:
Given flow rate = 0.5 litres per minute
= 500 cm³ per minute. -
Time to fill the cone:
Time = \frac{Volume}{Flow Rate} = \frac{196 \pi}{500}.
Converting to seconds:
= \frac{196 \times 3.14}{500} \approx 123.1 ext{ minutes} = 7386 ext{ seconds} ext{ (to the nearest second)}.