A log with diameter 60cm is 3m in length. Its hollow centre is 20cm in diameter. Find the surface area of the log in cm^2, including the ends and the inside, correct to one decimal place.
2 answers
42725.7 cm^2
To solve the problem, we need to find the surface area of the log, which includes the ends and the inside. We can break the log down into three parts: the outside surface, the two ends, and the inside surface.
1. Outside Surface: To find the outside surface area, we can use the formula for the lateral surface area of a cylinder, which is 2πrh. In this case, the height (h) of the log is 3m, but we need to convert it to centimeters: 3m = 300cm. The radius (r) of the log is half the diameter, which is 30cm. So we have:
Outside surface area = 2π(30cm)(300cm) = 56,548.67cm^2
2. End Surfaces: To find the area of the two end surfaces, we can use the formula for the area of a circle, which is πr^2. The radius of the end surface is half the diameter of the log, which is 60cm/2 = 30cm. So we have:
End surface area = 2π(30cm)^2 = 5654.87cm^2
3. Inside Surface: To find the inside surface area, we can subtract the area of the hollow centre from the volume of the log, and then use the formula for the lateral surface area of a cylinder. The volume of the log is πr^2h, and the volume of the hollow centre is πr^2h, where r = 10cm and h = 300cm. So we have:
Inside surface area = 2π(30cm)(300cm) - 2π(10cm)(300cm) = 45,894.66cm^2
Now we can add up the three surface areas to get the total surface area of the log:
Total surface area = outside surface area + end surface area + inside surface area
= 56,548.67cm^2 + 5654.87cm^2 + 45,894.66cm^2
= 107,098.2cm^2
Rounding to one decimal place, we get:
Total surface area = 42,725.7cm^2
Therefore, the surface area of the log, including the ends and the inside, is approximately 42,725.7cm^2.
1. Outside Surface: To find the outside surface area, we can use the formula for the lateral surface area of a cylinder, which is 2πrh. In this case, the height (h) of the log is 3m, but we need to convert it to centimeters: 3m = 300cm. The radius (r) of the log is half the diameter, which is 30cm. So we have:
Outside surface area = 2π(30cm)(300cm) = 56,548.67cm^2
2. End Surfaces: To find the area of the two end surfaces, we can use the formula for the area of a circle, which is πr^2. The radius of the end surface is half the diameter of the log, which is 60cm/2 = 30cm. So we have:
End surface area = 2π(30cm)^2 = 5654.87cm^2
3. Inside Surface: To find the inside surface area, we can subtract the area of the hollow centre from the volume of the log, and then use the formula for the lateral surface area of a cylinder. The volume of the log is πr^2h, and the volume of the hollow centre is πr^2h, where r = 10cm and h = 300cm. So we have:
Inside surface area = 2π(30cm)(300cm) - 2π(10cm)(300cm) = 45,894.66cm^2
Now we can add up the three surface areas to get the total surface area of the log:
Total surface area = outside surface area + end surface area + inside surface area
= 56,548.67cm^2 + 5654.87cm^2 + 45,894.66cm^2
= 107,098.2cm^2
Rounding to one decimal place, we get:
Total surface area = 42,725.7cm^2
Therefore, the surface area of the log, including the ends and the inside, is approximately 42,725.7cm^2.