Some water towers are composite shapes of a cylinder with a cone on top. One such tower is 8 ft. off the ground, and the cylinder is 6 ft. tall with a diameter of 6 ft. The slant height of the cone is 4 ft. Find the height of the tower from the ground to the vertex of the cone. Round the answer to the nearest tenth.(1 point)
To solve this problem, let's break it down into two parts: the cylinder and the cone.
For the cylinder:
The diameter of the cylinder is 6 ft, so the radius is half of that, which is 3 ft.
The height of the cylinder is 6 ft.
The formula for the volume of a cylinder is V = πr^2h.
Plugging in the values, we get V = π(3^2)(6) = 54π ft^3.
For the cone:
The slant height of the cone is 4 ft.
The height of the cone is what we need to find, let's call it h.
The formula for the volume of a cone is V = 1/3πr^2h.
Since the radius of the cone and the radius of the cylinder are the same (3 ft), we already know that the volume of the cone is 1/3 times the volume of the cylinder, which is 54π/3 = 18π ft^3.
Now, let's add the volume of the cylinder to the volume of the cone to find the total volume of the water tower: 54π + 18π = 72π ft^3.
The height of the water tower from the ground to the vertex of the cone is the sum of the height of the cylinder (6 ft) and the height of the cone (h). We can write this as: 6 + h = 8 ft.
Subtracting 6 from both sides, we have: h = 2 ft.
So, the height of the tower from the ground to the vertex of the cone is 2 ft. Rounded to the nearest tenth, it is still 2 ft.