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 #33 # and the height of the cylinder is #5 #. If the volume of the solid is #12 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 #33  # and the height of the cylinder is #5   #.  If the volume of the solid is #12 pi#, what is the area of the base of the cylinder?

Avery Allen · Accepted Answer

Area of base of the cylinder #A_b = (3pi)/4 = color (red)(2.3562)#

Volume of solid #V_s# = vol. of cylinder + vol. of cone

Given #V_s = 12pi, h_(cyl) = 5, h_(co) = 33#

#V_s = 12pi = (pi r^2 h_(cyl) + (1/3) pi r^2 h_(co))#

# 12pi = pi r^2(5 + (1/3)33) = 16 pi r^2#

#r^2 = (12 pi) / (16 pi) = (3/4)#

# Area of base A_b = pi r^2 = pi * (3/4) = 2.3562#

Allison Allen · Answer

undefined Let's denote the radius of both the cone and the cylinder as $ r $. Given that the height of the cone is 33 and the height of the cylinder is 5, we can set up the equation for the volume of the solid:

$$ \text{Volume of cone} + \text{Volume of cylinder} = 12\pi $$

The volume of a cone is given by $ V_{cone} = \frac{1}{3}\pi r^2 h_{cone} $, and the volume of a cylinder is given by $ V_{cylinder} = \pi r^2 h_{cylinder} $.

Substitute the given values:

$$ \frac{1}{3}\pi r^2 \cdot 33 + \pi r^2 \cdot 5 = 12\pi $$

Now, simplify and solve for $ r $:

$$ 11r^2 + 5r^2 = 36 $$

$$ 16r^2 = 36 $$

$$ r^2 = \frac{36}{16} $$

$$ r^2 = \frac{9}{4} $$

$$ r = \frac{3}{2} $$

Now that we have the radius of the base of the cylinder, we can calculate the area of the base of the cylinder using the formula for the area of a circle:

$$ \text{Area of base of cylinder} = \pi r^2 $$

Substitute $ r = \frac{3}{2} $:

$$ \text{Area of base of cylinder} = \pi \left(\frac{3}{2}\right)^2 $$

$$ \text{Area of base of cylinder} = \pi \cdot \frac{9}{4} $$

$$ \text{Area of base of cylinder} = \frac{9\pi}{4} $$

Therefore, the area of the base of the cylinder is $ \frac{9\pi}{4} $.