Begin with a circular piece of paper with a4-in. radius as shows in (a). cut out a sector with an arc length of x.Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b). a) Explain why the circumference of the bas eofthe cone is 8(π)−x. b) Express the radius r as a function of x. c) Express the height h as a function of x. d) Express the volume V of the cone as afunction of x.

Question

Begin with a circular piece of paper with a4-in. radius as shows in (a). cut out a sector with an arc length of x.Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b).   a) Explain why the circumference of the bas eofthe cone is 8(π)−x. b) Express the radius r as a function of x. c) Express the height h as a function of x. d) Express the volume V of the cone as afunction of x.

2k1enyvp · Accepted Answer

a) The circumference of a circle with radius 4 is 8π. Since the circumference of the base of the cone is equal to the arc length of the remainder of the circle, the conference is 8π−x  b) since r is the radius of the base of the cone, the circuference of that base is equal to 2π⋅r.  From answer a) that circuference is also equal to 8π−x so:  8π−x=2πr  r=8π−x2π=4−x2π  c) since h, r and 4 make up a right triangle, we can use the pythagorean theorem:  r2+h2=16  next we substitute the answer from b) for r and solve for h:  h=16−(4−x2π)2  d) the volume of a cone is:  V=13πr2h  which means:  V=π3(4−x2π)216−(4−x2π)2

Begin with a circular piece of paper with a4-in. radius as shows in (a).

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