Use the Pythagorean Theorem to find a relationship between h, a, and r.
C
c
Use the Pythagorean Theorem to find a relationship between h, a, and r.
D
d
Use the Pythagorean Theorem to find a relationship between h, a, and r.
A
a
52π/3≈ 54.45 ft^3
B
b
23 648π/3≈ 24 764 cm^3
C
c
1984π/3≈ 2077.64 m^3
D
d
166 212π ≈ 522 170 inches^3
Practice makes perfect
a
Let's look into the formula for the volume of a spherical cap.
V=π h/6(3 a^2 + h^2 )
We want to find the volume of a spherical cap that we know has a base with a radius a of 4 feet. We also know that the sphere has a radius of 5 feet. To find the volume, we also need to know the spherical cap's height, h. Let's consider a diagram of a sphere and a spherical cap.
Notice that the segments x and h together make a radius of the sphere.
h+ x= r ⇔ x= r- h
Let's write this into the diagram.
We have a right triangle in the diagram. Let's write the measures of its sides into into the Pythagorean Theorem.
a^2+b^2=c^2 ⇒ a^2+( r- h)^2= r^2
We can find h by using this equation.
Since 5-h is the side of a triangle it must be non-negative, which is why we only kept the principal root when solving the equation. Now that we know that h= 2 we can find the volume of the spherical cap. Let's do it!
The volume of the spherical cap is 52π3≈ 54.45 ft^3.
b
We have a sphere with a radius of 34 cm. This sphere has a spherical cap for which we know that the radius of the base a is 30 cm. To find the volume of the spherical cap we also need to find the height of the cap, h. Let's use the formula we found in Part A to find the cap's height.
The spherical cap has a volume of 23 648π3≈ 24 764 cm^3.
c
A sphere with a radius of 13 meters has a spherical cap with a height of 8 meters. To find the volume we need to find the radius of the base, a. We can use the expression we found in Part A.
a^2+( r- h)^2= r^2Let's substitute the corresponding values and solve for a.
Since a is a distance it must be non-negative, which is why we only kept the principal root when solving the equation. Now we have enough information about the spherical cap to find its volume.
The spherical cap's volume is 1984π 3≈ 2077.64 m^3.
d
For a sphere with a radius of 75 inches we want to know the volume of a spherical cap with a height of 54 inches. Let's begin by finding a, the radius of the base.
