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 #5 # and the height of the cylinder is #7 #. If the volume of the solid is #42 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 #5  # and the height of the cylinder is #7   #.  If the volume of the solid is #42 pi#, what is the area of the base of the cylinder?

Sadie Allen · Accepted Answer

#Axxh_(cy)+1/3xxAxxh_(co)=V#
#=>Axx(h_(cy)+1/3xxh_(co))=V#
#=>Axx(7+1/3xx5)=42pi#
#: . A =(42pixx3)/26#

Autumn Armstrong · Answer

undefined The area of the base of the cylinder can be found using the formula for the volume of a cone. Since the total volume of the solid is given as 42π and we know the volume of the cone and the cylinder separately, we can set up the equation as follows:

Volume of cone + Volume of cylinder = Total volume

(1/3)πr²h + πr²h = 42π

Where r is the radius of both the cone and the cylinder, and h is the height of the cone.

Given that the height of the cone is 5 units and the height of the cylinder is 7 units, we substitute these values into the equation:

(1/3)πr²(5) + πr²(7) = 42π

Now, we solve for the radius (r):

(1/3)π(5r²) + 7πr² = 42π

(5/3)πr² + 7πr² = 42π

(5/3)πr² + 21πr² = 42π

(26/3)πr² = 42π

Now, we solve for r:

r² = (42π * 3) / (26π)

r² = 126 / 26

r² ≈ 4.846

r ≈ √4.846

r ≈ 2.2

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

Area = πr²

Substituting the value of r:

Area ≈ π(2.2)²

Area ≈ π * 4.84

Area ≈ 15.29 square units

So, the area of the base of the cylinder is approximately 15.29 square units.