A school water tank is in the shape of a frustum of a cone. The height of the tank is 7.2 m and the top and bottom radii are 6m and 12 m respectively.
(a) Calculate the area of the curved surface of the tank, correct to 2 decimal places.
(b) Find the capacity of the tank, in litres, correct to the nearest litre.
(c) On a certain day, the tank was filled with water. If the school has 500 students and each student uses an average of 40 litres of water per day, determine the number of days that the students would use the water.
Think of the frustrum as a large cone with the top missing.
Since the top radius is 1/2 the bottom radius, the missing part has 1/8 the volume of the entire cone, and the area is 1/4 that of the entire cone. The slant height is also 1/2 the slant height.
The slant height s of the entire cone would be
√(12^2 + 7.2^2) = 14
the lateral area would be π r s = π*12*14 = 196π
So, the lateral area of the frustrum would be 3/4 of that, or 126π
See what you can do with the rest, ok?
passing exam
To calculate the area of the curved surface of the tank, we can use the formula for the lateral surface area of a frustum of a cone:
Lateral Surface Area = π(R + r) * l
Where R is the top radius, r is the bottom radius, and l is the slant height.
Given:
Top radius (R) = 12 m
Bottom radius (r) = 6 m
Height (h) = 7.2 m
Firstly, we need to calculate the slant height (l) using the Pythagorean theorem:
l^2 = (R - r)^2 + h^2
l^2 = (12 - 6)^2 + 7.2^2
l^2 = 6^2 + 7.2^2
l^2 = 36 + 51.84
l^2 = 87.84
l ≈ 9.37 m
Next, we can substitute the values into the lateral surface area formula:
Lateral Surface Area = π(12 + 6) * 9.37
Lateral Surface Area ≈ 351.86 m^2
Therefore, the area of the curved surface of the tank is approximately 351.86 square meters.
To find the capacity of the tank, we can use the formula for the volume of a frustum of a cone:
Volume = (1/3) * π * h * (R^2 + Rr + r^2)
Given:
Top radius (R) = 12 m
Bottom radius (r) = 6 m
Height (h) = 7.2 m
Volume = (1/3) * π * 7.2 * (12^2 + 12*6 + 6^2)
Volume ≈ 1209.60 m^3
To convert this to litres, we multiply by 1000:
Capacity = 1209.60 * 1000
Capacity ≈ 1,209,600 litres
Therefore, the capacity of the tank is approximately 1,209,600 litres.
To determine the number of days that the students would use the water, we need to divide the capacity of the tank by the daily water usage per student:
Number of days = Capacity / (Number of students * Water usage per student)
Given:
Number of students = 500
Water usage per student = 40 litres
Number of days = 1,209,600 / (500 * 40)
Number of days ≈ 60.48 days
Since the number of days cannot be in decimal places, we round it to the nearest whole number.
Therefore, the students would use the water for approximately 60 days.
(a) To calculate the area of the curved surface of the tank, we need to find the slant height of the frustum of the cone first.
The slant height (l) can be found using the Pythagorean theorem:
l = √(h^2 + (r2-r1)^2)
where h is the height of the frustum (7.2 m), r1 is the top radius (6 m), and r2 is the bottom radius (12 m).
Plugging in the values, we get:
l = √(7.2^2 + (12-6)^2)
= √(51.84 + 36)
= √87.84
≈ 9.37 m (rounded to two decimal places)
Now, to calculate the curved surface area (A) of the frustum of the cone, we use the formula:
A = π(r1 + r2) l
Plugging in the values, we get:
A = π(6 + 12) * 9.37
= π(18) * 9.37
≈ 167.19 m² (rounded to two decimal places)
Therefore, the area of the curved surface of the tank is approximately 167.19 square meters.
(b) To find the capacity of the tank, we need to calculate its volume. The volume (V) of a frustum of a cone can be calculated using the formula:
V = (1/3) πh (r1^2 + r2^2 + r1r2)
Plugging in the values, we get:
V = (1/3) π * 7.2 * (6^2 + 6*12 + 12^2)
= (1/3) * 3.1416 * 7.2 * 216
≈ 1721.26 m³ (rounded to two decimal places)
To convert cubic meters to liters, we know that 1 cubic meter is equal to 1000 liters.
So, the capacity of the tank is approximately 1,721,260 liters (rounded to the nearest liter).
(c) If the tank is filled with water and the school has 500 students, with each student using an average of 40 liters of water per day, we can calculate the number of days they can use the water by dividing the capacity of the tank by the daily water usage:
Number of days = Capacity of tank / (Number of students * Daily water usage per student)
Number of days = 1,721,260 / (500 * 40)
= 86,063 / 20,000
≈ 4.30 days (rounded to the nearest day)
Therefore, the students would be able to use the water for approximately 4 days.