There is a cylindrical tank lying horizontally on the ground, its diameter is 8 feet, and length is 25 feet, the depth of the water currently in the tank is 2 feet. (1 gallon=231 cubic inches)

How many gallons of water are in the tank?
How many gallons of water will it take to fill the tank?

My teacher asks us to solve this by using 2 ways: non-calculus and calculus. I figured out the non-calculus one by using geometry, but I have no idea about the calculus method. Can anyone possibly help me some?
Thanks.

http://www.arachnoid.com/tank_volume/index.html

Thank you.

But probably this problem is not that hard, this is a Calculus AB level question, so we are suppose to use the disc method, shell method or cross section things like that. But I hardly relate this problem to them...

You can use calculus to solve this problem by finding the volume of the water in the tank using the disk method. First, let's find the equation of the circle that represents the cross-section of the tank. Since the diameter is 8 feet, the radius is 4 feet. Let the ellipse formed by the intersection of the tank and the water be described by the equation:

(x^2 / 4^2) + (y^2 / 2^2) = 1

Now, we will find the volume of the water by revolving the ellipse around the x-axis, which represents the width of the tank.

To apply the disk method, we need to find a representative disk for a given value of x. The thickness of the disk is dx, and the radius of the disk is y.

The volume of a disk can be found by V_disk = π * (y^2) * dx. To find the total volume of the water, we will integrate this equation over the width of the tank, i.e., from x = -12.5 to x = 12.5.

We need to express y in terms of x to integrate. Rearrange the given equation in terms of y:
y^2 = 2^2(1 - x^2 / 4^2)

Now, integrate the equation for the volume of a disk:
V_water = π * ∫[2^2(1 - x^2 / 4^2)] dx from x = -12.5 to x = 12.5

Solve the equation:
V_water = π * (∫ 4 - (x^2 / 16) dx) from -12.5 to 12.5
V_water = π * ([4x - (x^3 / 48)] from -12.5 to 12.5)
V_water = 217.28 cubic feet

Since 1 gallon is equal to 231 cubic inches, we will convert the volume in cubic feet to gallons:
V_water = 217.28 * (700 / 231)
V_water = 651.61 gallons

To find the total volume of the tank, calculate the volume of a full cylinder with a diameter of 8 and length of 25 feet.
V_total = π * 4^2 * 25 = 100π = 314.16 cubic feet

Convert the volume in cubic feet to gallons:
V_total = 314.16 * (700 / 231)
V_total = 950.98 gallons

To find the amount of water needed to fill the tank, subtract the current amount of water from the total volume:
Gallons_needed = V_total - V_water
Gallons_needed = 950.98 - 651.61
Gallons_needed = 299.37 gallons

So, there are approximately 651.61 gallons of water in the tank, and it will take 299.37 gallons to fill the tank.

To solve this problem using calculus, we can use the method of cylindrical shells. This involves dividing the tank into small cylindrical shells and calculating the volume of each shell.

First, let's consider a small horizontal strip of width "dx" at a distance "x" from one end of the tank. The height of this strip will vary depending on the position "x".

We can approximate the volume of this strip as the product of its height, width, and depth. The depth is constant at 2 feet, the width is "dx", and the height can be determined using the Pythagorean theorem:

h = √(r^2 - x^2)

Here, "r" is the radius of the tank, which is half the diameter, so r = 8/2 = 4 feet.

The volume of this small strip is then given by:

dV = h * dx * 2

To find the total volume of the tank, we need to integrate this expression from x = -r to x = r:

V = ∫(-r to r) (h * dx * 2)

Now, we can substitute the expression for "h" into the integral:

V = ∫(-r to r) (√(r^2 - x^2) * dx * 2)

This integration can be quite complex, but fortunately, there is a calculus formula for this type of integral. Using the formula, we can solve the integral to find the volume of the tank.

Once we have the volume of the tank, we can convert it to gallons by multiplying it by the conversion factor of 231 cubic inches per gallon.

Note: The geometry method you mentioned is a simpler approach that can be used for this problem. However, using calculus allows us to generalize the approach to more complex shapes and situations.

To find the number of gallons of water in the tank, we need to calculate the volume of water in the tank. Given that the tank is cylindrical, we can use the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the height.

Using the given information, we know that the diameter of the tank is 8 feet, so the radius (r) is half of that, which is 4 feet. The height (h) of the water in the tank is 2 feet.

To calculate the volume in cubic feet, we substitute the values into the formula:

V = π(4^2)(2) = 32π ft^3.

Now, since 1 gallon equals 231 cubic inches, we need to convert the volume from cubic feet to gallons:

1 ft^3 = 12^3 in^3 = 1728 in^3.

So, 32π ft^3 = (32π)(1728) in^3.

Finally, divide the result by 231 to get the volume in gallons:

Volume in gallons = (32π)(1728) / 231.

Using a calculator or approximation for π, you can solve this to get the answer in gallons.

For the second part of the question, finding how many gallons of water it will take to fill the tank, we need to calculate the total volume of the tank. The tank is 25 feet long, so the height (h) will be the full length of the tank.

Using the formula V = πr^2h again, substitute the radius of 4 feet and the height of 25 feet:

V = π(4^2)(25) = 400π ft^3.

Similar to before, convert cubic feet to gallons by multiplying by 1728 and dividing by 231:

Volume in gallons = (400π)(1728) / 231.

Again, use a calculator or approximation for π to solve this.

So, to summarize:

- The first part uses the formula V = πr^2h to find the volume of water in the tank.
- The second part uses the same formula to find the total volume of the tank.

Note: If your teacher specifically asked for a calculus method, they might want you to use integration to solve the problem. In that case, you would need to set up an integral using the disc or shell method to find the volume.