Water is flowing at the rate of 50 m^3/min from a conical reservoir (vertex down) of base radius 45 meters and height 6 meters. What is the water level when the reservoir has 5000 m^3 of water left in it?

Question

Water is flowing at the rate of 50 m^3/min from a conical reservoir (vertex down) of base radius 45 meters and height 6 meters. What is the water level when the reservoir has 5000 m^3 of water left in it?

Answer 1

Since the ratio of radius/height is constant at any water depth (45/6 when full), r = 15/2 h. You want h when v=5000.

v = 1/3 πr^2 h, so
π/3 (15/2 h)^2 h = 5000
π/3 * 225/4 h^3 = 5000
h^3 = 12*5000/225 = 800/3
h = 6.43m

Answer 2

To find the water level when there are 5000 m^3 of water left in the conical reservoir, we can use the concept of similar triangles and volume ratios.

1. The volume of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14), r is the radius of the base, and h is the height.

2. We can set up a ratio of the remaining volume to the original volume to determine the corresponding ratio of the heights. Given that the original volume is 50 m^3/min and there are 5000 m^3 remaining, the ratio is 5000/50 = 100/1.

3. Since the ratios of the volumes and heights are equal, we can set up the following proportion using the ratio of the volumes:
(1/3) * π * 45^2 * h = (1/3) * π * 45^2 * 6

4. Cancel out the common terms, simplify, and solve for the height h:
h = (6 * (5000/50)) / 45^2

Now, let's calculate the water level when there are 5000 m^3 of water left in the reservoir.

Answer 3

To find the water level when the reservoir has 5000 m³ of water left, we can use the concept of similar triangles.

First, let's denote the height of the water level in the reservoir as h. We need to find this value.

We can set up the following proportion:

(45 / h) = ((45 - 6) / (h + 6))

Here, the left side of the equation represents the ratio of the base radius of the reservoir to the height of the water level (before the water has been consumed), while the right side represents the ratio after the water has been consumed.

Now, let's solve this equation step by step:

1. Cross-multiply to get rid of the denominators:

45 × (h + 6) = (45 - 6) × h

2. Expand and simplify both sides:

45h + 270 = 45h - 6h

3. Simplify further:

270 = 39h

4. Solve for h:

h = 270 / 39

h ≈ 6.92

So, the water level when the reservoir has 5000 m³ of water left is approximately 6.92 meters.