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 #42 # and the height of the cylinder is #1 #. If the volume of the solid is #495 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 #42  # and the height of the cylinder is #1   #.  If the volume of the solid is #495 pi#, what is the area of the base of the cylinder?

Mason Adams · Accepted Answer

#33pi#.

Let #r# be the radius of the Cone, so, that of the Cylinder is also #r#. #"Now, the Volume v_1 of the cone="1/3pir^2"(height)"=1/3pir^2(42)#, and, the Vvolume #v_2" of the cylinder="pir^2"(height)"=pir^2(1).# #:." Total Volume of the solid="v_1+v_2=14pir^2+pir^2=15pir^2# As this Volume is #495pi," we have, "15pir^2=495pi# #:."The Area of the base of the cylinder="pir^2=(495pi)/15=33pi#.

Wyatt Anderson · Answer

undefined The volume of the solid is 495 pi, so we have the equation V = V_cylinder + V_cone = πr^2h_cylinder + (1/3)πr^2h_cone. Substituting the given values, we get 495π = πr^2(1) + (1/3)πr^2(42). Simplifying this equation, we find 495 = 1 + 14r^2. Solving for r, we get r = √(494/14). Once we find r, we can calculate the area of the base of the cylinder using the formula A = πr^2. Therefore, A = π(494/14).