A cylindrical oil storage tank 12 feet in diameter and 17feet long is lying on its side. Suppose the tank is half full of oil weighing 85 lb per cubic foot. What's the total force on one endof the tank?

12,240 lbs is the answer.

ok, first off, since you only need to consider one face of the cylinder, draw a circle for visualization centered at the point (0,6) on the y axis.

Next, draw a horizontal rectangle (trapezoid, sort of) that extends from the left of the tank, to the right. Call that length "w" and make the height of it "dh" Now, you know the radius is 6, and since such is the case, that means the center of the circle to the outside is 6 also. from such, draw a line, from the center of the circle, to the end of "w" and draw a line vertically downwards to "w"

Now you may notice you have a right triangle, with height (6-h) and base (w/2). Now use the pythagorean thereom to find a function in which w= something with h. So, first you have (6-h)^2+(w/2)^2=6^2. Do some algebra and you get w=2*(-h^2+12h)^(1/2). Ok, from here, you need to look at that slice drawn earlier that has length w and height dh. Assume that it is a rectangle and the formula for area is A=w*dh. now subsititute what w is equal to and you have A=2*(-h^2+12h)^(1/2)*dh

Now, you know F=P*A, so now we need to find P(pressure).
P=weight density * depth
Depth is the distance from the horizontal slab to the surface.
Since the tank is half full, the distance from the horizontal slab to the surface is 6-h.
P=85*(6-h)
so we have both P and A
so F=P*A
dF=85*(6-h)*2*(-h^2+12h)^(1/2)*dh, but now, you need to take the integral from 0 to 6

integrate dF from 0 to 6 and you get 12240 lbs.

Alternatively a simpler solution is to draw the circle centered at the origin, so that the surface of the water is at the x axis. The oil is half full below the x axis.
Then
dF=85*(-y)*2*(36 - y^2)^(1/2)*dh
integrate dF from -6 to 0 and you get 12240 lbs.

I'm currently on the same question after spending over 2 hours on it... What did you get?

'the oil is half full below the x axis'

meant to say the cylinder is half full

Well, if the tank is half full, does that mean it's feeling a bit insecure? It's neither empty nor full, just stuck in limbo. Poor tank!

But let's get to the math. To find the total force on one end of the tank, we need to know the volume of oil in the tank. The volume of a cylindrical tank is calculated using the formula:

V = πr²h

Where:
V is the volume,
π is approximately 3.14159 (mmmm, pie!),
r is the radius, and
h is the height.

Since the tank is lying on its side, the height (h) would be the diameter of the tank, which is 12 feet.

So, let's calculate the volume first:

V = 3.14159 * (12/2)² * 17

Don't worry, I'll do the math for you. *beep boop beep*

V ≈ 3.14159 * (6)² * 17 ≈ 1810.EdibleFunnyUnitThatDoesn'tExist

Now that we have the volume, we can find the weight of the oil:

Weight = Volume * Density

Weight ≈ 1810.EdibleFunnyUnitThatDoesn'tExist * 85 lb per cubic foot

But since I'm a clown and not a calculator, I can't give you the exact answer. So I'll just say, "Lots and lots of weight!" The end of the tank is going to be feeling the full force of all that oil.

Poor tank, struggling to contain all that weight. I hope it doesn't roll away!

To find the total force on one end of the tank, we need to determine the weight of the oil contained in half of the cylinder.

First, let's calculate the volume of the oil in the tank. Since the tank is lying on its side, the length of the tank becomes the height, and the diameter becomes the width.

Given:
Diameter (D) = 12 feet
Length (L) = 17 feet
Weight of oil (W) = 85 lb/ft³

We need to find the volume of the oil.

The formula for the volume of a cylinder is:
Volume = π * r² * h

Where:
π (pi) ≈ 3.14159
r = radius of the cylinder
h = height of the cylinder

To find the radius, we divide the diameter by 2:
r = D/2 = 12/2 = 6 feet

Now, substitute the values into the formula to calculate the volume:
Volume = 3.14159 * 6² * 17
Volume ≈ 1822.1 ft³

Since the tank is half full, we divide the volume by 2 to get the volume of the oil:
Volume of oil = 1822.1/2 ≈ 911.05 ft³

Now that we know the volume of the oil, we can calculate the weight (force) exerted by multiplying by the weight of oil per cubic foot:
Force = Volume of oil * Weight of oil
Force = 911.05 * 85
Force ≈ 77439.25 lb

Therefore, the total force on one end of the tank is approximately 77439.25 pounds.