Show that a a volume of fruston of a right circular cone of radii r and height
V=π/3h[R^2+rR+r^2]
To show the volume of the frustum of a right circular cone of radii r and R, and height h, we can consider two right circular cones: a large cone with radius R and height H, and a smaller cone with radius r and height h1. The two cones have the same apex, and the difference between their volumes will give us the volume of the frustum.
First, let's find the relationship between the height h1 of the smaller cone and the total height H of the larger cone. Since the radii of the two cones and the height h of the frustum are related, we can use similar triangles to find the relationship between the heights.
We have
h1 / r = (H - h) / R
Let's solve for h1:
h1 = r(H - h) / R ----(1)
Now let's find the volume of the larger cone (VL) and the smaller cone (VS):
Volume of a cone is given by the formula V = (1/3)πr²h.
VL = (1/3)πR²H and
VS = (1/3)πr²h1
We want to find the difference between the two volumes:
V = VL - VS.
V = (1/3)πR²H - (1/3)πr²h1
Factor out the (1/3)π:
V = (1/3)π(R²H - r²h1)
Now, let's substitute the value for h1 from equation (1)
V = (1/3)π[R²H - r²(r(H - h)/R)]
V = (1/3)π[R²H - r³(H - h)/R)]
Now, let's find a common denominator:
V = (1/3)π[(R³R²H - r³R²(H - h))/(R³R)]
Simplify the equation further:
V = (1/3)π[R²H - r³H + r³h]/(R)
Factor out an H in the numerator:
V = (1/3)πH[R² - r³ + r³h/R] / R
Separate the terms in the numerator:
V = (1/3)πH(R²/R) - (1/3)πH(r³/R) + (1/3)πh(r³/R)
Simplify further:
V = (1/3)πHR - (1/3)πHr + (1/3)πhr
The last step is to factor out the (1/3)πh from the expression:
V = (π/3)h(R² + Rr + r²)
This is the formula for the volume of the frustum of a right circular cone of radii r and R, and height h.
V = (π/3)h(R² + rR + r²)