Please show work/explain steps taken to solve this: 1) A right cone is 16 cm hig
ID: 2982459 • Letter: P
Question
Please show work/explain steps taken to solve this:1) A right cone is 16 cm high and has a base radius of 12 cm. A cut is made through the cone 4 cm from the vertex and parallel to the base. The discarded top is a right cone with base diameter 6 cm and slant height 5 cm. The part that remains is a frustum with a slant height 15 cm. A hole with radius 3 cm is drilled through the frustum, from the center of one base to the center of the other. The drilled frustum is then dipped in a vat of paint.
a) Sketch the original cone, the un-drilled frustum, the discarded cone, and the drilled frustum. Label all relevant measurements.
b) Calculate the exact area of the painted surface of the frustum. Explain the steps in your calculation procedure. Please show work/explain steps taken to solve this:
1) A right cone is 16 cm high and has a base radius of 12 cm. A cut is made through the cone 4 cm from the vertex and parallel to the base. The discarded top is a right cone with base diameter 6 cm and slant height 5 cm. The part that remains is a frustum with a slant height 15 cm. A hole with radius 3 cm is drilled through the frustum, from the center of one base to the center of the other. The drilled frustum is then dipped in a vat of paint.
a) Sketch the original cone, the un-drilled frustum, the discarded cone, and the drilled frustum. Label all relevant measurements.
b) Calculate the exact area of the painted surface of the frustum. Explain the steps in your calculation procedure.
Explanation / Answer
I'll call the original cone the "big cone" and the cut-off cone the "small cone". The slant height of the big cone is 20 cm (you can get it via pythagorean theorem). The entire surface area of the big cone is (pi)(12)(12+20)=384pi. The entire surface area of the small cone is (pi)(3)(3+5)=24pi. But the surface area of the base of the small cone is not part of the entire surface area of the big cone, so the area of the small cone not including its base is 24pi - 9pi = 15pi. Subtract this from the area of the big cone to get 384pi - 15pi = 369pi. But now to complete the frustum you have to add the area of its top, which is 9pi, so 369pi + 9pi = 378pi.
Now for the drilled hole. Each of its ends has an area of 9pi (radius of 3 cm). Subtract the area of both ends from the entire area of the frustum: 378pi - (2)(9pi) = 360pi. Then find the surface area of the inside of the hole, which is the same as the surface area of the outside of a cylinder of equal diameter, (pi)(diameter)(height) = (pi)(6)(12) = 72pi. This area has to be added to the area of the frustum since it wasn't there before the hole was drilled: 360pi + 72pi = 432pi.
:
surface area of cone = (pi)(r)(r+s), where r = radius of base and s = slant height