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.
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...
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