Atank in the shape of a right circular cone has height 12 feet and base radius 4 feet. the tank is inverted, with its vertex pointing down and base at the top. the tank contains a liquid with weight density 63 pounds per cubic foot, but is filled to a depth of 8 feet only. set up, but do not evaluate, an integral for the work (in foot-pounds) required to pump all the liquid present to a height one foot over the top of the tank.

V = int(π(y/3)^2, 0, 8)

(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)

Step-by-step explanation:

Set up cartesian axis (x and y) to the system.

Let y axis as the line of the centre of the cone, passing through its vertex and the centre of it's circular base. The x axis could be the 90 degree line to the y axis that passes the vertex. So the origin (0,0) is at the vertex.

I'm this setup , looking it as if we are looking it in 2 dimension, we'll see that there is a signature straight line on the x-y plane, which this line will form the cone as it revolute around the y-axis

Find the equation of the line:

Using height 12 and radius base 4, we can get the slope of the line

m = 12/4 = 3

It passes through origin, so the y-intercept is 0

Hence, y = 3x

Since the volume revolves around y-axis, we use the equation volume of revolution around y-axis

V = int(πx^2,a,b)

(Definite integration of πx^2 with lower boundary a and upper boundary b

Since y=3x

x = y/3

For this que

V = int(π(y/3)^2, 0, 8)

(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)

proofs involving isosceles triangles often require special consideration because an isosceles triangle has several distinct properties that do not apply to normal triangles.(more about triangle types) therefore, when you are trying to prove that two triangles are congruent, and one or both triangles, are isosceles you have a few theorems that you can use to make your life easier.

step-by-step explanation:

an isosceles triangle has two congruent sides and two congruent angles. the congruent angles are called the base angles and the other angle is known as the vertex angle. ∠bac and ∠bca are the base angles of the triangle picture on the left. the vertex angle is ∠abc