Find the volume of a frustum of a right circular cone with height 25, lower base radius 30 and top radius 13.
Volume = from 0 to 4 of ∫ pi [f(y)]²dy where f(y) = (-3/4)y + 5
∫pi ((-3/4)y + 5)² dy = pi ∫ ((9/16)y² - (15/2)y + 25) dy
= pi[(3/16)y^3 - (15/4)y² + 25y + C]
evaluated from 0 to 4 gives:
pi[(3/16)(4^3) - (15/4)(4²) + 25(4) - (0-0+0)]
=pi[12 - 60 + 100]
=52pi
no Calculus needed.
extend the sides to form the original cone.
let the height of the added cone be h
by ratios
h/13 = (h+25)/30
30h = 13h + 325
17h = 325
h = 325/17
volume of fulcrum
= (1/3)π(30^2)(25 + 325/17) - (1/3)π(13^2)(325/17)
= (1/3)π[900(25+325/17) - 169(325/17) ]
= (1/3)π (36475)
= 36475π /3 cubic units
Why did the frustum of the cone go to therapy? Because it couldn't handle all the pressure to calculate its own volume! Let's give it a break and I'll do the math for you.
To find the volume of a frustum of a cone, you can use the formula:
V = (1/3) * π * h * (r₁² + r₂² + (r₁ * r₂))
Where:
- V is the volume of the frustum
- h is the height of the frustum
- r₁ is the radius of the lower base
- r₂ is the radius of the top base
Plugging in the given values:
V = (1/3) * 3.14159 * 25 * (30² + 13² + (30 * 13))
Calculating that:
V ≈ 12,134.95 cubic units
So, the volume of the frustum of the right circular cone is approximately 12,134.95 cubic units.
To find the volume of a frustum of a right circular cone, you can use the formula:
V = (1/3) * π * h * (r1² + r2² + r1 * r2),
where:
V is the volume of the frustum,
h is the height of the frustum,
r1 is the radius of the lower base, and
r2 is the radius of the top base.
In this case, the height (h) of the frustum is given as 25, the lower base radius (r1) is 30, and the top radius (r2) is 13. Let's substitute these values into the formula:
V = (1/3) * π * 25 * (30² + 13² + 30 * 13).
Now, let's calculate the value:
V = (1/3) * π * 25 * (900 + 169 + 390)
V = (1/3) * π * 25 * 1459
V ≈ 60449.98 cubic units
Therefore, the volume of the frustum is approximately 60449.98 cubic units.