A water reservoir in the form of a cone mounted on a hemispere is built such that the plane face of the hemisphere is built fits exactly to the base of the cone and the height of the cone is 6 times the radius of its base.
(a) Illustrate this information in a diagram,
(b) If the volume of the reservoir is 3331/3 pi m3 , calculate, correct to the nearest whole number, the:
(i) volume of the hemisphere
(ii) total surface area of the reservoir. (Take pi = 22/7)
Is this physics or geometry?
If the radius of the hemisphere (and cone base) is r, then the volume of the hemisphere + cone is
2/3 π r^3 + 1/3 π r^2 (6r) = 8/3 π r^3
The area is thus 2πr^2 + 2π√37 r^2
You can use your given volume to find r.
(a) Here is a diagram illustrating the information given:
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.-' '-.
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/ \
/ \
/_______________________________\
/| |\
/ | | \
\_| Cone Hemispehre |/
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|______________________________|
```
(b) To calculate the volume and surface area of the reservoir, we need to determine the radius and height of the cone.
Let's assume the radius of the cone is "r". Since the height of the cone is 6 times the radius, the height of the cone is 6r.
The volume of the cone can be calculated using the formula for the volume of a cone: V_cone = (1/3) * π * r^2 * h.
Given that the volume of the reservoir is (3331/3) * π m^3, we can set up the following equation:
(1/3) * π * r^2 * 6r = (3331/3) * π
Simplifying this equation, we get:
2 * r^3 = 3331/3
Taking the cube root on both sides, we find:
r = (∛(3331/3) / ∛2)
Now, let's calculate:
(i) Volume of the hemisphere:
The volume of a hemisphere is given by the formula: V_hemisphere = (2/3) * π * r^3
Using the value of r we just found, we can calculate the volume of the hemisphere:
V_hemisphere = (2/3) * π * (∛(3331/3) / ∛2)^3
(ii) Total surface area of the reservoir:
The surface area of the cone is given by: A_cone = π * r * (r + √(r^2 + h^2))
The surface area of the hemisphere is given by: A_hemisphere = 2 * π * r^2
The total surface area of the reservoir is: A_reservoir = A_cone + A_hemisphere
Substituting the values we found, we can calculate the total surface area of the reservoir.
(a) To illustrate this information, we need to draw a diagram representing the water reservoir.
Start by drawing a cone - a shape with a circular base and a pointed top. The base of the cone should be a circle, and the height of the cone should be 6 times the radius of the circle. Label the radius of the base as 'r' and the height as '6r'.
Next, draw a hemisphere - a shape that looks like half of a sphere. The flat, circular face of the hemisphere should fit exactly onto the top of the cone. Label the radius of the hemisphere as 'r' as well.
Join the cone and the hemisphere at their bases so that they are connected and form a continuous shape.
(b) Now, let's calculate the volume of the hemisphere and the total surface area of the reservoir.
(i) Volume of the hemisphere:
The volume of a hemisphere is given by the formula (2/3) * pi * r^3.
In this case, the radius of the hemisphere is 'r'. Therefore, the volume of the hemisphere is:
Volume = (2/3) * (22/7) * r^3
(ii) Total surface area of the reservoir:
The total surface area of the reservoir is the sum of the area of the curved surface of the cone and the curved surface area of the hemisphere, as well as the area of the circular base of the cone.
The curved surface area of the cone is given by the formula pi * r * l, where 'l' is the slant height of the cone. In this case, the slant height is given by pythagoras theorem as sqrt(r^2 + (6r)^2) = sqrt(r^2 + 36r^2) = sqrt(37r^2) = r * sqrt(37).
So, the curved surface area of the cone is pi * r * r * sqrt(37).
The curved surface area of the hemisphere is half of the surface area of a sphere, which is 2 * (22/7) * r^2.
The area of the circular base of the cone is pi * r^2.
Therefore, the total surface area of the reservoir is:
Surface Area = pi * r * r * sqrt(37) + 2 * (22/7) * r^2 + pi * r^2
Now, to solve for the values of volume, surface area, and the radius 'r', we need to solve the following equation for 'r':
(2/3) * (22/7) * r^3 = 3331/3 * (22/7)
Simplifying the equation:
r^3 = (3331/3 * 22/7) * (7/2) * (3/22) =~ 3331
To find the value of 'r', we can cube root both sides of the equation:
r ≈ cube root of 3331, approximately 14.85
Finally, substitute the value of 'r' into the formulas for the volume and surface area to get the solutions.