A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #21 # and the height of the cylinder is #2 #. If the volume of the solid is #75 pi#, what is the area of the base of the cylinder?

Question

A solid consists of a cone on top of a cylinder with a radius equal to that of the cone.  The height of the cone is #21  # and the height of the cylinder is #2   #.  If the volume of the solid is #75 pi#, what is the area of the base of the cylinder?

Noah Atkinson · Accepted Answer

The area of the base is #=26.2u^2#

Let #a=# area of the base

Volume of cone is #V_(co)=1/3*a*h_(co)#

Volume of cylinder is #V_(cy)=a*h_(cy)#

Total amount

#V=V_(co)+V_(cy)#

#V=1/3ah_(co)+ah_(cy)#

#75pi=a(1/3*21+2)#

#75pi=a*9#

#a=75/9pi#

#a=26.2#

Caroline Aucoin · Answer

undefined To find the area of the base of the cylinder, we first need to find the radius of both the cone and the cylinder. Since the radius of the cylinder is equal to that of the cone, we can denote it as "r".

Let's denote the height of the cylinder as "h_cylinder" and the height of the cone as "h_cone".

Given:
- Height of the cone (h_cone) = 21
- Height of the cylinder (h_cylinder) = 2
- Volume of the solid = 75π

We know that the volume of the solid is the sum of the volumes of the cone and the cylinder:

Volume of cone = (1/3)πr²h_cone
Volume of cylinder = πr²h_cylinder

Given that the volume of the solid is 75π, we can set up the equation:

(1/3)πr²(21) + πr²(2) = 75π

Now, we can simplify and solve for "r":

(7/3)πr² + 2πr² = 75π
(7/3)r² + 2r² = 75

Multiplying both sides by 3 to eliminate the fraction:
7r² + 6r² = 225
13r² = 225

Dividing both sides by 13:
r² = 225 / 13
r² ≈ 17.3077

Taking the square root of both sides to solve for "r":
r ≈ √(17.3077)
r ≈ 4.1602

Now that we have found the radius "r", we can find the area of the base of the cylinder using the formula:

Area of the base of the cylinder = πr²

Substituting the value of "r":
Area of the base of the cylinder ≈ π(4.1602)²
Area of the base of the cylinder ≈ π(17.3077)
Area of the base of the cylinder ≈ 68.7928 square units

Therefore, the area of the base of the cylinder is approximately 68.7928 square units.