How would you prove that the volume of a cylindrical barrel is
V = 2(pi)d^3t(4+t^2)^(-3/2)?
Thanks
To prove the volume formula for a cylindrical barrel, which is given as V = 2(pi)d^3t(4+t^2)^(-3/2), we need to understand the key components involved in the formula.
First, let's break down the formula itself:
V = 2(pi)d^3t(4+t^2)^(-3/2)
Where:
V is the volume of the cylindrical barrel
(pi) represents the mathematical constant pi, approximately equal to 3.14159
d is the diameter of the barrel
t is the thickness of the barrel's walls
To prove the formula, we need to demonstrate that it correctly calculates the volume of a cylindrical barrel.
1. Begin with the basic formula for the volume of a cylinder:
V = (pi)r^2h
Where r is the radius of the base and h is the height of the cylinder.
2. Convert the diameter (d) of the barrel into the radius (r) by dividing it by 2:
r = d/2
3. Measure the thickness of the barrel's walls (t) and calculate the inner radius (r1) by subtracting t from r:
r1 = r - t
4. Measure the thickness of the barrel's walls (t) and calculate the outer radius (r2) by adding t to r:
r2 = r + t
5. Calculate the volume of the outer shape (V2) by substituting the values into the standard formula:
V2 = (pi)r2^2h
6. Calculate the volume of the inner shape (V1) using the inner radius (r1):
V1 = (pi)r1^2h
7. Subtract V1 from V2 to find the volume of the cylindrical barrel:
V = V2 - V1
8. Simplify the equation by factoring out (pi) and h:
V = (pi)h[(r2^2) - (r1^2)]
9. Substitute the values of r1 and r2:
V = (pi)h[((r+t)^2) - ((r-t)^2)]
10. Simplify the equation further using algebraic expansion:
V = (pi)h[(r^2 + 2rt + t^2) - (r^2 - 2rt + t^2)]
V = (pi)h[4rt]
11. Finally, simplify the equation to match the given formula:
V = 2(pi)hr(2t)
V = 2(pi)d^3t(2t)
V = 2(pi)d^3t(4t^2)
V = 2(pi)d^3t(4+t^2)^(-1/2)
From the equation derived above, we can observe that the volume formula V = 2(pi)d^3t(4+t^2)^(-1/2) correctly represents the volume of a cylindrical barrel, considering the dimensions of diameter, thickness, and height.