Chapter 1.2, Problem 55E

(a)

To determine

To explain: Thereason for the circumference of the base of the cone is $8\pi -x$ .

Answer to Problem 55E

The circumference of the base of the cone is $8\pi -x$ because piece of arc length is removed from the circumference of original circle $8\pi$ .

Explanation of Solution

Given information:A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.

  

Figure (1)

Calculation:

The formula for the circumference of a circle with radius r is:

  $C=2\pi r$

Substitute 4 for r in the above formula to find the circumference of given circle,

  $\begin{array}{c}C=2\pi \left(4\right)\\ =8\pi \end{array}$

It is given that the circumference of the base of the cone is $8\pi -x$ .

A sector of arc length x is cut out from the circle to form the cone. So, the circumference of the base of the cone is the arc length of sector subtracted from the circumference of the circle.

Therefore, the circumference of the base of the cone is $8\pi -x$ because piece of arc length is removed from the circumference of original circle $8\pi$ .

(b)

To determine

To express: The radius of the cone as a function of x .

Answer to Problem 55E

The radius as a function of x can be written as $r=4-\frac{x}{2\pi }$ .

Explanation of Solution

Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.

  

Figure (1)

Calculation:

It is given that the circumference of the base of the cone is $8\pi -x$ .

The formula for the circumference of a circle with radius r is:

  $C=2\pi r$

Substitute $8\pi -x$ for C in the above formula.

  $\begin{array}{c}8\pi -x=2\pi r\\ r=\frac{8\pi -x}{2\pi }\\ r=4-\frac{x}{2\pi }\end{array}$

Therefore, the radius as a function of x can be written as $r=4-\frac{x}{2\pi }$ .

(c)

To determine

To express: The height h of the cone as a function of x .

Answer to Problem 55E

The height h of the cone as a function x can be written as $h=\frac{\sqrt{16x\pi -x^2}}{2\pi }$ .

Explanation of Solution

Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.

  

Figure (1)

Calculation:

From part (b), the radius as a function of x can be written as $r=4-\frac{x}{2\pi }$ . The slant height of the cone is 4 in.

The formula for the slant height l of a cone is:

  $l^2=h^2+r^2$

Substitute 4 for land $4-\frac{x}{2\pi }$ for r in the above formula.

  $\begin{array}{c}{\left(4\right)}^2=h^2+{\left(4-\frac{x}{2\pi }\right)}^2\\ 16=h^2+16+\frac{x^2}{4{\pi }^2}-\frac{4x}{\pi }\\ h^2=\frac{4x}{\pi }-\frac{x^2}{4{\pi }^2}\\ h^2=\frac{16x\pi -x^2}{4{\pi }^2}\end{array}$

Further simplify.

  $\begin{array}{c}{h}^2=\frac{16x\pi -x^2}{4{\pi }^2}\\ h=\sqrt{\frac{16x\pi -x^2}{4{\pi }^2}}\\ h=\frac{\sqrt{16x\pi -x^2}}{2\pi }\end{array}$

Therefore, the height h of the cone as a function x can be written as $h=\frac{\sqrt{16x\pi -x^2}}{2\pi }$ .

(c)

To determine

To express: The volume V of the cone as a function of x .

Answer to Problem 55E

The volumeV of the cone as a function x can be written as $V=\frac{{\left(8\pi -x\right)}^2\sqrt{16x\pi -x^2}}{24{\pi }^2}$ .

Explanation of Solution

Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.

  

Figure (1)

Calculation:

From part (b), the radius of the cone is $r=4-\frac{x}{2\pi }$ and from part (c) the height h of the cone is $h=\frac{\sqrt{16x\pi -x^2}}{2\pi }$ .

The formula for the volumeV of a cone is:

  $V=\frac{1}{3}\pi r^{2}h$

Substitute $4-\frac{x}{2\pi }$ for r and $\frac{\sqrt{16x\pi -x^2}}{2\pi }$ for h in the above formula.

  $\begin{array}{c}V=\frac{1}{3}\pi {\left(4-\frac{x}{2\pi }\right)}^2\left(\frac{\sqrt{16x\pi -x^2}}{2\pi }\right)\\ =\frac{1}{3}\pi {\left(\frac{8\pi -x}{2\pi }\right)}^2\left(\frac{\sqrt{16x\pi -x^2}}{2\pi }\right)\\ =\frac{{\left(8\pi -x\right)}^2\sqrt{16x\pi -x^2}}{24{\pi }^2}\end{array}$

Therefore, the volume V of the cone as a function x can be written as $V=\frac{{\left(8\pi -x\right)}^2\sqrt{16x\pi -x^2}}{24{\pi }^2}$ .