A large container has the shape of a frustum of a cone with top radius 5 m, bottom radius 3m, and height 12m.

The container is being filled with water at the constant rate 4.9 m^3/min. At what rate is the level of water rising at the instant the water is 2 deep?

the finial answer is = 14.0 cm/min

can any one help me solving this question (step by step solution)

a frustum is part of a cone with radius 5m for bigger cone:.R=5m H=12+h for smaller cone:r=3m and height h by comparison:5/12+h=3/h 5h=36+3h 2h=36 h=18m the volume of water in full cone will also increase by 4.9m3/s considering only the bigger cone alone since water has passed the smaller cone.assume water is at h0 and radius r0,by comparison,5/r0=30/h0 h0=6r0 V=1/3pir0^h0 by substituting h0=6r0 V=1/3pi(h0/6)^2h0=1/3pih0^3/36.by differentiating dV/dt=pih0^2/36*dh0/dt when h0=18+2=20m,4.9=22/7*20^2/36*dh0/dt 4.9=34.92dh0/dt dh0/dt=0.143m/s=14.3cm/s

Sure! I can help you step by step. To find the rate at which the level of water is rising, we can use a combination of geometry and calculus.

Step 1: Visualize the problem
First, let's visualize the frustum of the cone shape. The top radius is 5 meters, the bottom radius is 3 meters, and the height is 12 meters. The level of water in the container will also have a frustum of a cone shape.

Step 2: Determine the formulas
We need to find the rate at which the level of water is rising, so we need to find an expression for the volume of the water in terms of the height.

The volume of a frustum of a cone can be calculated using the following formula:
V = (1/3) * pi * h * (r^2 + R^2 + r * R)

Where:
V is the volume of the frustum
h is the height of the frustum
r is the top radius of the frustum
R is the bottom radius of the frustum

Step 3: Calculate the volume of the water
When the water level is h meters, the radius of the top surface will be smaller because the water takes up space. Let's call this radius x. We can find the equation relating x and h by using similar triangles.

Using similar triangles, we can set up the following proportion:

(x - 3) / (12 - h) = 2 / 12

Cross-multiplying and rearranging the equation, we get:

(x - 3) = (2 / 12) * (12 - h)
x - 3 = 2 - (2h / 6)
x = 2 - (h / 3)

Now we have an expression for x in terms of h. We can substitute this expression into the volume formula:

V = (1/3) * pi * h * ((2 - h/3)^2 + 3^2 + (2 - h/3) * 3)

Simplifying the formula, we get:

V = (1/3) * pi * h * (4 + 3 + 6 - (4h/3) + 3 - (h/3))
V = (1/3) * pi * h * (16/3 - (5h/3))

Step 4: Differentiate the volume function
We need to differentiate the volume function with respect to time, t. Since the rate at which water is being added to the container is given as 4.9 m^3/min, the derivative of the volume function with respect to time will give us the rate at which the volume is changing.

dV/dt = (dV/dh) * (dh/dt)

Step 5: Calculate the derivative of the volume function
Differentiating the volume function, we get:

dV/dh = (1/3) * pi * (16/3 - 5h/3) + (1/3) * pi * h * (-5/3)

Simplifying further, we get:

dV/dh = (1/3) * pi * (16/3 - 5h/3 - 5h/3)
dV/dh = (1/3) * pi * (16/3 - 10h/3)

Step 6: Substitute the given values
When the water level is 2 meters, substitute h = 2 into the derivative expression.

dV/dh = (1/3) * pi * (16/3 - 10(2)/3)

Simplifying further, we get:

dV/dh = (1/3) * pi * (16/3 - 20/3)
dV/dh = (1/3) * pi * (-4/3)

Step 7: Substitute the known rate
Now, we know that dV/dt = 4.9 m^3/min.

We have the equation:

dV/dt = (dV/dh) * (dh/dt)

Substituting the known values, we get:

4.9 = (1/3) * pi * (-4/3) * (dh/dt)

Simplifying, we find:

dh/dt = (4.9 * 3) / (pi * -4/3)
dh/dt = -14.7 / (4pi/3)
dh/dt = -14.7 * (3/4pi)
dh/dt = -44.1 / (4pi)
dh/dt ≈ -3.51 / pi

Step 8: Convert the rate to centimeters per minute
Since the given answer is in centimeters per minute, we need to convert our rate:

dh/dt ≈ -3.51 / pi * (100 cm / 1 m) = -351 / pi cm/min

Approximating the value of pi to 3.14, we get:

-351 / pi ≈ -111.465 cm/min

Therefore, the rate at which the level of water is rising at the instant the water is 2 meters deep is approximately -111.465 cm/min.