A cone has a volume of 36pi cubic meters. Its height is four times its radius. Find the surface area of the cone.
To find the surface area of the cone, we need to calculate the lateral surface area and the base surface area separately.
1. Radius of the cone:
Let's assume the radius of the cone is [latex]r[/latex].
2. Height of the cone:
According to the given information, the height [latex]h[/latex] is four times the radius [latex]r[/latex]. So the height can be written as [latex]h = 4r[/latex].
3. Volume of the cone:
The volume [latex]V[/latex] of a cone is given by the formula [latex]V = \frac{1}{3} \pi r^2 h[/latex]. Given that the volume is [latex]36\pi[/latex], we can set up the equation as follows:
[latex]36 \pi = \frac{1}{3} \pi r^2 (4r)[/latex].
4. Solve for the radius:
Simplifying the equation, we get:
[latex]36 \pi = \frac{4}{3} \pi r^3[/latex].
Dividing both sides of the equation by [latex]\frac{4}{3} \pi[/latex], we find:
[latex]r^3 = \frac{9}{4}[/latex].
Taking the cube root of both sides of the equation, we get:
[latex]r = \sqrt[3]{\frac{9}{4}}[/latex].
Thus, the radius of the cone is [latex]r = \sqrt[3]{\frac{9}{4}}[/latex].
5. Lateral Surface Area of the cone:
The lateral surface area [latex]A_l[/latex] of a cone is given by the formula [latex]A_l = \pi r l[/latex], where [latex]l[/latex] represents the slant height of the cone. By using the Pythagorean theorem with the radius [latex]r[/latex] and height [latex]h[/latex], we can find the slant height as follows:
[latex] l = \sqrt{r^2 + h^2}[/latex].
[latex] l = \sqrt{\left(\sqrt[3]{\frac{9}{4}}\right)^2 + (4\sqrt[3]{\frac{9}{4}})^2}[/latex].
Calculate the value of [latex]l[/latex] by simplifying the equation:
[latex]l = \sqrt{\frac{9}{4} + 16\cdot\frac{9}{4}}[/latex].
Thus, the slant height [latex]l[/latex] of the cone is [latex]l = \sqrt{\frac{9}{4} + 16\cdot\frac{9}{4}}[/latex].
Now, we can calculate the lateral surface area [latex]A_l[/latex] using the formula:
[latex]A_l = \pi \cdot \sqrt[3]{\frac{9}{4}} \cdot \sqrt{\frac{9}{4} + 16\cdot\frac{9}{4}} [/latex].
6. Base surface area of the cone:
The base surface area [latex]A_b[/latex] of a cone is given by the formula [latex]A_b = \pi r^2[/latex]. So, we can calculate the base surface area as follows:
[latex]A_b = \pi \cdot \left(\sqrt[3]{\frac{9}{4}}\right)^2[/latex].
7. Total surface area of the cone:
Finally, we can find the total surface area [latex]A_t[/latex] of the cone by adding the lateral surface area [latex]A_l[/latex] and the base surface area [latex]A_b[/latex]:
[latex]A_t = A_l + A_b[/latex].
With these calculations, you can find the surface area of the given cone.
To find the surface area of the cone, we need to know its radius and slant height.
Let's start by finding the radius of the cone. We are given that the height is four times the radius. Let's represent the radius as 'r' and the height as 'h.'
Given: h = 4r
To find the slant height, we can use the Pythagorean theorem. The slant height, represented by 'l', can be found using the formula:
l^2 = r^2 + h^2
Substituting the value of h from the given equation, we have:
l^2 = r^2 + (4r)^2
l^2 = r^2 + 16r^2
l^2 = 17r^2
Now, let's find the radius using the volume of the cone. The formula for the volume of a cone is given by:
V = (1/3) * π * r^2 * h
Substituting the given volume, we have:
36π = (1/3) * π * r^2 * 4r
36 = 4r^3
9 = r^3
r = ∛9
r = 3
Now that we have the radius, we can find the slant height:
l^2 = 17r^2
l^2 = 17 * 3^2
l^2 = 153
l = √153
Finally, we can find the surface area of the cone using the formula:
Surface Area = π * r^2 + π * r * l
Substituting the values we found:
Surface Area = π * 3^2 + π * 3 * √153
Surface Area = 9π + 3π√153
So the surface area of the cone is 9π + 3π√153 square meters.
v = 1/3 pi * r^2 * h
h = 4r
36pi = 1/3 pi * r^2 * 4r = 4/3 pi r^3
27 = r^3
r = 3
If you don't include the area of the base,
a = pi * rs = pi * r*sqrt(r^2 + h^2)
= pi * r*sqrt(r^2 + 16r^2)
= pi * r^2 * sqrt(17)
= 116.6 m^2