1)Forty solid hemisphere each of diameter 2cm , are melted to form a solid cone with a base diameter 6cm .Find the height of the cone?
2)A pyramid has a square base of side 12cm and a height of 15cm .Calculate its volume and total surface area?
1) 8.88
To find the height of the cone formed from melting the hemispheres, we can follow the steps below:
Step 1: Find the volume of each hemisphere
The volume of a hemisphere is given by the formula: V = (2/3) * π * r³, where r is the radius of the hemisphere. In this case, the radius (r) is half of the diameter, so r = 2/2 = 1 cm.
V_hemisphere = (2/3) * π * (1 cm)³
Step 2: Find the total volume of all the hemispheres
Since there are 40 hemispheres, we multiply the volume of one hemisphere by 40.
Total_volume_hemispheres = 40 * V_hemisphere
Step 3: Find the volume of the cone by equating it to the volume of the melted hemispheres
The volume of a cone is given by the formula: V_cone = (1/3) * π * r² * h, where r is the radius of the base and h is the height of the cone.
V_cone = (1/3) * π * (3 cm)² * h
Since the diameter of the base is 6 cm, the radius (r) is 6/2 = 3 cm.
We equate the volume of the cone to the total volume of the hemispheres:
(1/3) * π * (3 cm)² * h = Total_volume_hemispheres
Simplify the equation and solve for h.
Now let's move on to the second question.
To calculate the volume and total surface area of a pyramid with a square base, follow these steps:
Step 1: Calculate the volume
The volume of a pyramid is given by the formula: V = (1/3) * A_base * h, where A_base is the area of the base and h is the height of the pyramid.
In this case, the base is a square with side length 12 cm, so the area of the base (A_base) is 12 cm * 12 cm.
V = (1/3) * (12 cm * 12 cm) * 15 cm
Simplify the expression to calculate the volume.
Step 2: Calculate the total surface area
The total surface area of a pyramid is given by the formula: A_total = A_base + A_side, where A_base is the area of the base and A_side is the combined area of all the sides.
In this case, A_side can be calculated by finding the area of each triangular face and summing them up. Each triangular face has a base equal to a side of the square base (12 cm) and a height equal to the height of the pyramid (15 cm).
Calculate the area of one triangular face and multiply it by 4 to get the total area of the sides.
A_side = (1/2) * (12 cm) * (15 cm) * 4
Then, calculate the total surface area by adding the area of the base and the area of the sides.
A_total = A_base + A_side.
Now you have the formulas and steps to calculate the height of the cone and the volume and total surface area of the pyramid. Just plug in the values and solve the equations.
V = 40 (4/3) pi r^3
r = 1
V = (160/3) pi
V = (1/3)pi r^2 h
(160/3) pi = (1/3) pi 3^2 h
160 = 9 h
h = 160/9
2)
(1/3) 144 (15) = V
distance up slope of side :
15^2 + 6^2 = L^2
L = sqrt (225+36) = sqrt(261)
area of one side = (1/2)(12)sqrt 261
area of 4 sides = 24 sqrt 261
It is not clear if we should add the bottom area
If so then 144+24 sqrt 261