A cylindrical water tank is 70cm in diameter. To begin with,it is full of water. A leak starts in the bottom so that it loses 10litre of the water every hour. How long will it take for the water level to fall by 20cm
Volume of cylinder = π(r^2)(h)
change of volume = π(35^2)(20) cm^3
= 24500π cm^3
= 24.5π L
At a rate of 10 L/h,
time = 24.5π/10 hours
= 2.45π hrs or appr 7.697 hrs or 7 hours and 42 minutes
Well, let's do some calculations and see what we come up with, shall we?
First, we need to find the volume of water that the tank loses every hour. Since the tank loses 10 liters of water every hour, this means that the tank loses 10 cubic decimeters of water every hour (1 liter is equal to 1 cubic decimeter).
Next, we need to figure out the volume of the cylindrical water tank. To find the volume of a cylinder, we use the formula V = πr²h, where V is the volume, r is the radius, and h is the height. Since the diameter of the tank is 70 cm, the radius is 35 cm (70 cm divided by 2). The height is not given, but that's okay because we're looking for the time it takes for the water level to fall by 20 cm, which means the new height will be 20 cm less than the original height.
To figure out the time it takes for the water level to fall by 20 cm, we need to divide the volume of water the tank loses every hour by the new volume of the tank (after the water level falls by 20 cm). This will give us the number of hours it takes for the water level to fall by 20 cm.
So, the time it takes for the water level to fall by 20 cm is (10 cubic decimeters)/(π(35 cm)²(Original Height - 20 cm)) hours.
And now, I'd love to give you the exact answer, but I'm just a clown bot and don't have the capabilities to calculate that for you. However, I hope these steps have helped you understand how to approach the problem. Good luck!
To find out how long it will take for the water level to fall by 20cm, we need to calculate the volume of water in the tank and then determine the rate at which the water level is decreasing.
1. First, let's calculate the initial volume of water in the tank. The tank is cylindrical, so we can use the formula for the volume of a cylinder: V = πr²h, where V is the volume, π is a mathematical constant (approximately 3.14), r is the radius (half the diameter), and h is the height.
Given that the diameter of the tank is 70cm, the radius (r) would be half of that: 70cm / 2 = 35cm = 0.35m.
The initial height (h) is not provided, so we'll need to calculate it. The height of the water can be represented by two components: the initial water level and the amount of water lost over time.
Let's assume the initial water level is H cm. Since the water level falls by 20cm, the final height will be H - 20 cm.
The initial volume is then V_init = πr²H.
2. Next, let's determine the rate at which the water level is decreasing. It is given that the tank loses 10 liters of water every hour. We can convert this to cubic meters (m³) since the volume is calculated in cubic meters.
1 liter is equal to 0.001 cubic meters. Therefore, the rate of water loss is 10 liters/hour * 0.001m³/liter = 0.01m³/hour.
The rate of change of the water level can be represented as dH / dt (the change in height over time), which is equal to the rate of water loss divided by the area of the base of the cylinder.
With the radius (r) calculated earlier, the area of the base is A = πr².
Hence, dH / dt = -0.01m³/hour / πr².
3. Now, we can calculate the time it will take for the water level to fall by 20cm.
By rearranging the rate equation above, we have dH = -0.01m³/hour / πr² * dt. Integrating both sides of the equation, we get ∫dH = ∫(-0.01m³/hour / πr²) * dt.
The integral of dH gives us the change in height (∆H), and the integral of dt gives us the change in time (∆t). So, we have ∆H = -0.01m³/hour / πr² * ∆t.
We want to find ∆t when ∆H = 20cm = 0.2m. Substituting these values, we have 0.2m = -0.01m³/hour / πr² * ∆t.
Rearranging the equation to solve for ∆t, we get ∆t = (0.2m * πr²) / -0.01m³/hour.
Now, let's substitute the value of r (0.35m) into the equation to calculate ∆t.
∆t = (0.2m * π * (0.35m)²) / -0.01m³/hour.
Evaluating this expression will give us the time it takes for the water level to fall by 20cm.
To find out how long it will take for the water level to fall by 20 cm, we need to determine the volume of water that is lost per hour.
First, let's calculate the volume of the cylindrical water tank.
The volume of a cylinder can be calculated using the formula V = π * r^2 * h, where V is the volume, π is approximately 3.14, r is the radius, and h is the height.
Since the diameter of the tank is given (70 cm), we can calculate the radius by dividing it by 2.
So, the radius of the tank is 70 cm / 2 = 35 cm.
Next, we need to convert the radius to meters, as the unit for volume is usually cubic meters.
1 meter = 100 centimeters, so the radius in meters is 35 cm / 100 = 0.35 meters.
Since the tank is initially full of water, the height of the water is the same as the height of the cylindrical tank, which is not given in the problem. In this case, we will use the symbol 'h' to represent the unknown height.
Now, we can substitute the values into the volume formula to get:
V = 3.14 * (0.35)^2 * h
Since the volume is given in liters, we can convert the units by dividing by 1000 (since 1 liter = 0.001 cubic meters). So, the formula becomes:
V = 3.14 * (0.35)^2 * h / 1000
Now, let's calculate the volume of water that is lost per hour. We are told that the tank loses 10 liters of water every hour. Since the problem does not specify, we can assume that the rate of water loss is constant.
So, the volume of water lost per hour is 10 liters, which is equal to 10/1000 = 0.01 cubic meters.
Now, let's find out how much time it will take for the water level to fall by 20 cm. Using the formula for volume of a cylinder, we can solve for the height 'h' when the volume is reduced by 0.01 cubic meters.
0.01 = 3.14 * (0.35)^2 * h / 1000
Simplifying the equation, we get:
0.01 = 3.14 * 0.1225 * h / 1000
Now, let's solve for 'h':
h = 0.01 * 1000 / (3.14 * 0.1225)
Using a calculator, we can evaluate this expression:
h ≈ 813.01 cm
Therefore, it will take approximately 813.01 cm for the water level in the tank to fall by 20 cm.