A water resrvoir in form of a cone is mounted on a hemispherical bowl.if the height of that one is six times its radius and the volume of the resevoir is 333 1/3 pi cm cubic,calculate correct to decimal place. a) the total surface area of the resevoir. b) curved surface area of the cone. c) volume of the reservoir
r=6h, so
v = π/3 r^2h = 12πh^3 = 1000/3 π, so h = 3 cm
the slant height s=√(h^2+r^2) = √(3^2+18^2) = 18.25
(a) πr(r+s)
(b) πrs
(c) you just said the volume was 333 1/3 π
But I don't see any mention of the hemisphere, which was apparently not part of the reservoir. Check for typos and make any fixes needed. And I know you have made some typos.
To calculate the required values, we need to break down the problem into smaller steps and use the formulas for the respective geometric shapes involved.
Let's assign variables to the given information:
- Let "r" be the radius of the cone (and also the radius of the hemispherical bowl).
- The height of the cone is given as 6 times its radius, so the height (h) of the cone is 6r.
a) Total surface area of the reservoir:
The total surface area of the reservoir includes the curved surface area of the cone, the curved surface area of the hemispherical bowl, and the base area of the cone.
1. Curved Surface Area of the cone:
The curved surface area of a cone is given by the formula A = πrl, where r is the radius of the base and l is the slant height of the cone.
Since the slant height can be calculated using Pythagoras' theorem as l = sqrt(r^2 + h^2), we can substitute the values of r and h into the formula:
A_cone = π * r * sqrt(r^2 + h^2)
2. Curved Surface Area of the hemispherical bowl:
The curved surface area of a hemisphere is given by the formula A = 2πr^2 (as it only has one curved surface).
So, the curved surface area of the hemispherical bowl is A_hemisphere = 2π * r^2
3. Base Area of the cone:
The base area of a cone is given by the formula A = πr^2.
So, the base area of the cone is A_base = π * r^2
Now, we can add up the three components to find the total surface area:
Total Surface Area = A_cone + A_hemisphere + A_base
b) Curved Surface Area of the cone:
We have already found the formula for the curved surface area of the cone (A_cone) in step 1 above.
c) Volume of the reservoir:
The volume of the reservoir consists of two parts: the volume of the cone and the volume of the hemispherical bowl.
1. Volume of the cone:
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h.
Substituting the values, we get V_cone = (1/3) * π * r^2 * 6r
2. Volume of the hemispherical bowl:
The volume of a hemisphere is given by the formula V = (2/3) * π * r^3.
Since the radius of the hemispherical bowl is the same as the radius of the cone (r), we can substitute the values to find V_hemisphere = (2/3) * π * r^3
Now, we can add up the two volumes to find the volume of the reservoir:
Total Volume = V_cone + V_hemisphere
By substituting the given values into these formulas, you can calculate the required values and round them to the decimal places as necessary.