Cone Problem Beginning with a circular piece of paper with a 4- inch radius, as shown in (a), cut out a sector with an arc of length x. Join the two radial edges of the remaining portion of the paper to form a cone with radius r and height h, as shown in (b). What length of arc will produce a cone with a volume greater than 21 in.^3?

To solve this problem, we need to break it down into several steps:

Step 1: Determine the radius and height of the cone.
Since the original circular piece of paper has a radius of 4 inches, the radius of the cone (r) will also be 4 inches. The height (h) of the cone can be calculated using the Pythagorean Theorem. Let's denote the distance from the center of the circular piece to the apex of the cone as d. The radius of the circular piece forms the base of the cone, so the distance from the apex to the base on a perpendicular line is r (4 inches). We can then calculate the height using the formula:

h = √(d^2 - r^2)

Step 2: Calculate the volume of the cone.
The formula for the volume of a cone is:

V = (1/3) * π * r^2 * h

Step 3: Determine the equation for the volume of the cone in terms of x.
To do this, we need to express the height (h) in terms of x. Since the arc length x is given, we can use that to determine the angle θ (in radians) of the sector that is cut out. The formula for the length of an arc is:

Arc Length = radius * angle

So, we have:

x = r * θ

Rearranging the equation, we get:

θ = x / r

Now, we can use this angle to calculate the height (h). Since the angle θ forms part of a circle with a total angle of 2π radians, we can find the remaining angle as:

Remaining angle = 2π - θ

This angle corresponds to the remaining portion of the circle that forms the base of the cone. Since the radius is 4 inches, the circumference of this circle is 2π * 4 = 8π inches. The remaining angle will correspond to a certain fraction of this circumference. The fraction can be calculated as:

Fraction = Remaining angle / Total angle

Therefore:

Fraction = (2π - θ) / (2π)

Now, we can calculate the height (h) as:

h = Fraction * 2r

Step 4: Express the volume in terms of x.
Now that we have the height (h) in terms of x, we can substitute this back into the volume formula:

V = (1/3) * π * r^2 * h

Step 5: Solve for x when the volume is greater than 21 cubic inches.
Set up the inequality:

(1/3) * π * r^2 * h > 21

Substitute the expressions for r and h:

(1/3) * π * (4^2) * (Fraction * 2 * 4) > 21

Simplify:

32π * Fraction > 21

Finally, solve the inequality for x:

x > (21 / (32π * Fraction)) * r

x > (21 / (32π * ((2π - x / r) / (2π)))) * r

After solving this inequality, you will get the range of x values that correspond to a cone volume greater than 21 cubic inches.