A company manufactures a hallow cylinder component that is closed at the end. The r=4" & h=8". To make each cylinder 42 pi cubic inches of material is needed. 1. Write a function to express the volume of the material needed. 2. Write an equation than can be solved to find the required thickness of the component.
you need to specify the thickness of the material, and assuming that the end cap is the same thickness t, and the inner radius is 4, then the volume is
πt(4+t)^2 + π((4+t)^2-4^2)*8 = πt^3 + 16πt^2 + 80πt
= πt(t^2 + 16t + 80)
so to find t, just solve
πt^3 + 16πt^2 + 80πt = 42π
t^3 + 16t^2 + 80t - 42 = 0
t = 0.48 in
adjust things if the outer radius was 4", rather than the inner radius
Thank you!
1. To express the volume of the material needed, we need to determine the volume of the hollow cylinder component. The volume of a hollow cylinder can be calculated by subtracting the volume of the smaller hollow cylinder (with radius r and height h) from the volume of the bigger hollow cylinder (with radius R and height H) using the formula:
Volume = π * ((R^2 * H) - (r^2 * h))
In this case, the radius of the smaller hollow cylinder is 4 inches (r = 4) and the height is 8 inches (h = 8). The volume required for each cylinder is given as 42π cubic inches. Therefore, the function to express the volume of the material needed can be written as:
Volume_needed(R, H) = π * ((R^2 * H) - (4^2 * 8))
2. To find the required thickness of the component, we need to solve an equation derived from the volume expression. The thickness of the component can be obtained by subtracting the radius of the smaller hollow cylinder from the radius of the bigger hollow cylinder. Let's denote the required thickness as T.
The radius of the bigger hollow cylinder (R) can be calculated using the following equation:
(R^2 * H) - (4^2 * 8) = 42π
Simplifying the equation will give:
R^2 * H = 42π + 4^2 * 8
Now, we need to solve for R and H. Since we already know the value of r and h, we can substitute them in the equation to find the values of R and H.
Once we determine the value for R, we can calculate the required thickness (T) by subtracting r from R:
T = R - r
This equation can be used to find the required thickness of the component.