A grain storage tank is in the shape of a cylinder covered by half a sphere. The height of the cylinder is 50 feet and its diameter is 80 feet. find the total surface area (including the base) and volume of the tank.

Think of the grain storage tank as two independent, but related, geometric figures for solving this problem. You have a cylinder with only a bottom (no top because it's covered by the half sphere) and a sphere that's cut across the middle (i.e., half sphere).

The surface area formula for a cylinder with no top is pi*r^2 + 2*pi*r*h.

The surface area formula for half a sphere is (1/2)*4*pi*r^2 = 2*pi*r^2.

The volume area formula for a cylinder is pi*r^2*h.

The volume area formula for half a sphere is (1/2)*4/3*pi*r^3 = 3/2*pi*r^3.

Since the diameter is 80 feet, the radius is 1/2 the diameter or 40 feet.

Therefore, the surface area of the grain storage tank is pi*r^2 + 2*pi*r*h + 2*pi*r^2 = 27,632 sq/ft.

The volume of the grain storage tank is pi*r^2*h + 3/2*pi*r^3 = 552,640 cu/ft.

To find the total surface area of the tank, we will consider the surface area of the cylinder and the half sphere.

Surface area of the cylinder:
The formula to find the surface area of a cylinder is:
Surface area = 2πr^2 + 2πrh

Given that the diameter of the cylinder is 80 feet, the radius (r) can be calculated by dividing the diameter by 2:
r = 80 feet / 2 = 40 feet

Using the height (h) of the cylinder as 50 feet, we can substitute the values into the formula to find the surface area of the cylinder:
Surface area of the cylinder = 2π(40 feet)^2 + 2π(40 feet)(50 feet)

Surface area of the cylinder = 2π(1600 square feet) + 2π(2000 square feet)

Surface area of the cylinder = 3200π square feet + 4000π square feet

Simplifying, we get:
Surface area of the cylinder = 7200π square feet

Surface area of the half sphere:
The formula to find the surface area of a sphere is:
Surface area = 4πr^2

Since we have half of a sphere, we will divide the surface area by 2:
Surface area of the half sphere = (4π(40 feet)^2) / 2

Surface area of the half sphere = 2π(1600 square feet)

Surface area of the half sphere = 3200π square feet

Adding the surface area of the cylinder and the half sphere together, we get the total surface area of the tank:
Total surface area = Surface area of the cylinder + Surface area of the half sphere

Total surface area = 7200π square feet + 3200π square feet

Total surface area = 10400π square feet

So, the total surface area of the tank, including the base, is 10400π square feet.

To find the volume of the tank, we will only consider the volume of the cylinder as the top half of the sphere does not affect the volume of the tank.

Volume of the cylinder:
The formula to find the volume of a cylinder is:
Volume = πr^2h

Substituting the values into the formula, we get:
Volume = π(40 feet)^2(50 feet)

Volume = π(1600 square feet)(50 feet)

Volume = 80,000π cubic feet

So, the volume of the tank is 80,000π cubic feet.

To find the total surface area of the tank, we need to calculate the area of the curved surface of the cylinder, the areas of the two bases of the cylinder, and the area of the hemisphere.

1. Area of the curved surface of the cylinder:
The formula to find the area of the curved surface of a cylinder is given by A_cylinder = 2πrh, where r is the radius and h is the height of the cylinder.
Given the diameter of the cylinder is 80 feet, the radius is half of that, so r = 80/2 = 40 feet.
The height of the cylinder is 50 feet, so h = 50 feet.
Therefore, A_cylinder = 2π(40)(50) = 4000π square feet.

2. Area of the bases of the cylinder:
The formula to find the area of a circle is A_circle = πr^2, where r is the radius of the circle.
The radius of the cylinder is 40 feet.
Therefore, A_base = π(40)^2 = 1600π square feet.
Since there are two bases, the total area = 2A_base = 2(1600π) = 3200π square feet.

3. Area of the hemisphere:
The formula to find the area of a hemisphere is A_hemisphere = 2πr^2, where r is the radius of the hemisphere.
The radius of the hemisphere is equal to the radius of the cylinder, which is 40 feet.
Therefore, A_hemisphere = 2π(40)^2 = 3200π square feet.

Now, to calculate the total surface area, sum up the areas of the curved surface of the cylinder, the bases of the cylinder, and the hemisphere:
Total Surface Area = A_cylinder + 2A_base + A_hemisphere
= 4000π + 2(1600π) + 3200π
= 4000π + 3200π + 3200π
= 10400π square feet.

To find the volume of the tank, we need to calculate the volume of the cylinder and the volume of the hemisphere.

1. Volume of the cylinder:
The formula to find the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height of the cylinder.
The radius of the cylinder is 40 feet.
Therefore, V_cylinder = π(40)^2(50) = 80000π cubic feet.

2. Volume of the hemisphere:
The formula to find the volume of a hemisphere is V_hemisphere = (2/3)πr^3, where r is the radius of the hemisphere.
The radius of the hemisphere is 40 feet.
Therefore, V_hemisphere = (2/3)π(40)^3 ≈ 107292π cubic feet.

To find the total volume of the tank, sum up the volume of the cylinder and the hemisphere:
Total Volume = V_cylinder + V_hemisphere
= 80000π + 107292π
= 187292π cubic feet.

Therefore, the total surface area of the tank is 10400π square feet, and the volume of the tank is 187292π cubic feet.