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 #9 # and the height of the cylinder is #6 #. If the volume of the solid is #130 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 #9  # and the height of the cylinder is #6   #.  If the volume of the solid is #130 pi#, what is the area of the base of the cylinder?

Avery Allen · Accepted Answer

The answer is #=45.4#

Let area of base #=a#

The volume of the cone #=1/3*a*h#

The volume of the cylinder #=a*H#

Total volume #V=1/3ah+aH=a(h/3+H)#

#V=130pi#

height of cone #h=9#

height of cylinder #H=6#

So,

#a(9/3+6)=130pi#

#9a=130pi#

#a=130/9pi=45.4#

Sadie Allen · Answer

undefined To find the area of the base of the cylinder in the given solid, we first need to find the volume of the cone and the volume of the cylinder separately.

The volume of a cone is given by the formula $ V_{\text{cone}} = \frac{1}{3} \pi r^2 h $, where $ r $ is the radius and $ h $ is the height.

Given that the height of the cone is 9 and its radius is equal to that of the cylinder, we have $ r_{\text{cone}} = r_{\text{cylinder}} $ and $ h_{\text{cone}} = 9 $.

Substitute these values into the volume formula for the cone: $ V_{\text{cone}} = \frac{1}{3} \pi (r_{\text{cylinder}})^2 \times 9 $.

Now, let's find the volume of the cylinder. The volume of a cylinder is given by the formula $ V_{\text{cylinder}} = \pi r^2 h $, where $ r $ is the radius and $ h $ is the height.

Given that the height of the cylinder is 6 and its radius is equal to that of the cone, we have $ r_{\text{cylinder}} = r_{\text{cone}} $ and $ h_{\text{cylinder}} = 6 $.

Substitute these values into the volume formula for the cylinder: $ V_{\text{cylinder}} = \pi (r_{\text{cylinder}})^2 \times 6 $.

We are given that the total volume of the solid (cone on top of the cylinder) is 130 pi. Therefore, the sum of the volumes of the cone and the cylinder is $ V_{\text{cone}} + V_{\text{cylinder}} = 130 \pi $.

Substitute the volume formulas for the cone and cylinder into the equation: $ \frac{1}{3} \pi (r_{\text{cylinder}})^2 \times 9 + \pi (r_{\text{cylinder}})^2 \times 6 = 130 \pi $.

Simplify and solve for $ r_{\text{cylinder}} $:
$$ 3(r_{\text{cylinder}})^2 + 2(r_{\text{cylinder}})^2 = 390 $$
$$ 5(r_{\text{cylinder}})^2 = 390 $$
$$ (r_{\text{cylinder}})^2 = \frac{390}{5} $$
$$ (r_{\text{cylinder}})^2 = 78 $$
$$ r_{\text{cylinder}} = \sqrt{78} $$

Now that we have the radius of the cylinder, we can find the area of its base. The area of the base of a cylinder is given by the formula $ A_{\text{base}} = \pi (r_{\text{cylinder}})^2 $.

Substitute the value of $ r_{\text{cylinder}} $ into the formula: $ A_{\text{base}} = \pi (\sqrt{78})^2 $.

Calculate the area:
$$ A_{\text{base}} = \pi \times 78 $$
$$ A_{\text{base}} = 78\pi $$

Therefore, the area of the base of the cylinder is $ 78\pi $ square units.