This is the Crailsheim water tower in Baden-Württemberg, Germany. Its construction was completed in 1912.
The base of the tower is a graduated cylinder about 19.2 m high with a diameter of 11.5 m at the base and 8.1 m at the top.
a)The spherical tank holds 4.08 102 m3. What is the radius of the sphere?
b)What is the area of the floor at the bottom of the tower?
c)What is the circumference at the top of the graduated cylinder?
d)What is the circumference at the bottom of the graduated cylinder?
e)Calculate the surface area of the walls of the graduated cylinder. (Hint: if you slice the graduated cylinder open and unroll it, you get a trapezoid.)
a) To find the radius of the sphere, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
We are given that the volume of the sphere is 4.08 * 10^2 m^3. Plugging in these values, we can solve for the radius:
4.08 * 10^2 = (4/3) * π * r^3
Dividing both sides by (4/3) * π, we get:
r^3 = (4.08 * 10^2) / [(4/3) * π]
Taking the cube root of both sides, we find:
r = [(4.08 * 10^2) / [(4/3) * π]]^(1/3)
Evaluating this expression will give us the radius of the sphere.
b) The floor at the bottom of the tower is in the shape of a circle. The formula for the area of a circle is:
A = π * r^2
We know the radius of the cylinder at the bottom is 11.5 m. Plugging in this value, we can calculate the area of the floor.
c) The circumference of a circle is given by the formula:
C = 2 * π * r
In this case, we need to calculate the circumference at the top of the graduated cylinder. We know the radius at the top is 4.05 m, so plugging in this value will give us the circumference.
d) Similarly, we can calculate the circumference at the bottom of the graduated cylinder using the same formula, but with the radius of 5.75 m.
e) To calculate the surface area of the walls of the graduated cylinder, we can use the formula for the lateral surface area of a cylinder:
A = 2 * π * r * h
In this case, the height of the cylinder is given as 19.2 m. We need to calculate the surface area using the radius at the top and the radius at the bottom separately. Then we can add these two areas together to get the total surface area.