Question

ANSWER A water tower in New York City has the shape of a cylinder with a cone on top. The cylinder has a diameter of 12 feet and a height of 15 feet. The roof has an inclination angle of 25o . There are 7.48 gallons in a cubic foot. If residents of an apartment building are using the water from the tower at an average rate of 56 gallons per minute, determine how long, to the nearest minute, it will take to drain the entire tower.

Answer 1

To drain the entire water tower, you need to calculate the volume of water in the tower and divide it by the rate of water usage. The volume is the sum of the cylinder and cone volumes. With a usage rate of 56 gallons per minute, it will take approximately 82 minutes to drain the tower.

Step-by-step explanation:

To determine the amount of time it will take to drain the entire water tower, we need to calculate the volume of water in the tower and then divide it by the rate at which the residents are using the water. Here's how:

1. First, let's find the volume of the cylinder. The radius of the cylinder is half its diameter, so it is 6 feet. The formula for the volume of a cylinder is V = πr^2h, where V is the volume, π is a mathematical constant, r is the radius, and h is the height. Plugging in the values, we get: V = π(6^2)(15) = 540π cubic feet.
2. Next, let's find the volume of the cone. The formula for the volume of a cone is V = (1/3)πr^2h, where V is the volume, π is a mathematical constant, r is the radius, and h is the height. The height of the cone can be calculated using trigonometry. It is the product of the sine of the inclination angle and the radius of the cylinder. So, h = sin(25°) x 6 = 2.47 feet. Plugging in the values, we get: V = (1/3)π(6^2)(2.47) = 73.9π cubic feet.
3. Now, let's add the volumes of the cylinder and the cone to find the total volume of water in the tower. V_total = 540π + 73.9π = 613.9π cubic feet.
4. Finally, we can calculate the time it will take to drain the entire tower. The rate at which the residents are using the water is 56 gallons per minute. Since there are 7.48 gallons in a cubic foot, the rate in cubic feet per minute is 56/7.48 = 7.49 cubic feet per minute. Dividing the total volume of the tower by the rate gives us the time it will take to drain it: T = 613.9π / 7.49 ≈ 82.02 minutes. Therefore, it will take approximately 82 minutes to drain the entire water tower.

Answer 2

241 minutes

Step-by-step explanation:

Given:

Height of cylinder, h = 15 ft

Radius, r =
`(d)/(2) = (12)/(2) = 6` (both cylinder and cone have same radius)

Let's find the height of cone, since angle of inclination = 25°C.

`tan25 = (h)/(r)`

`h = r tan25`

`h = 6 tan25 = 2.8`

Height of cone = 2.8 ft

Let's find colume of tower.

Volume = Volume of cone + volume of cylinder.

Formula for volume of cone = ⅓πr²h

Volume of cylinder = πr²h

Therefore,

V = ⅓πr²h + πr²h

V = ⅓π*6²*2.8 + π*6²*15

V = 105.558 + 1696.46

V = 1802.02 ft³

Since volume is 1802.02 ft³, and there are 7.48 gallons in a cubic ft, the total gallon =

1802.02 * 7.48 = 13479.11 gallons

Water is used at an average rate of 56 gallons per minute.

Amount if time to drain the water:

Total gallons / average rate

`= (13479.11)/(56) = 240.698`

≈ 241 minutes