a slice of pizza, in the form of a sector of a circle, is to have a perimeter of 24 inches. What should be the radius of the pan to make the slice of pizza largest?

let the arclength of the sector be a

then 2r + a = 24
a = 24-2r

Let the area of the sector be A
then A/(πr^2) = a/(2πr)
A = a(πr^2)/(2πr)
= ar/2
= (24-2r)r/2
A = 12r - r^2
dA/dr = 12 - 2r
= 0 for a max of A
12-2r = 0
r = 6

Oh, the eternal quest for the largest slice of pizza! Sounds like a delicious mathematical challenge.

To find the radius of the pan that will make the slice of pizza largest, let's use our clown logic.

First, let's break it down. The perimeter of a sector of a circle is given by the formula P = 2πr + 2s, where r is the radius and s is the arc length.

Since we want the largest slice of pizza, we want the largest possible arc length. The maximum arc length occurs when it's equivalent to the circumference of the entire circle. So, s = 2πr.

Now we can substitute that into our original formula to find the perimeter: P = 2πr + 2(2πr).

Given that the perimeter is 24 inches, we can solve for r: 2πr + 4πr = 24.

Simplifying that, we get 6πr = 24, which leads us to r = 4 inches.

So, the radius of the pan that will give you the largest slice of pizza is 4 inches. Now you can enjoy a pizza slice big enough to feed not only your hunger but your love of math as well!

To find the radius of the pan that will make the slice of pizza largest, we can make use of some geometrical concepts.

1. Let's assume the slice of pizza is in the form of a sector of a circle, with angle θ in radians.
2. The total circumference of the circle will be C = 2πr, where r is the radius.
3. The arc length of the slice of pizza is proportional to the angle θ and the circumference: arc length = θ * C.
4. In this case, we are given that the arc length (perimeter of the slice) is 24 inches, so arc length = θ * C = 24 inches.
5. Therefore, we have the equation: θ * C = 24 inches.
6. Since C = 2πr, we can substitute this into the equation: θ * 2πr = 24 inches.
7. Now, let's find an expression for the area of the slice of pizza. The area of a sector of a circle is given by A = (θ/2) * r^2.
8. To find the radius that will maximize the area, we need to find the value of r that maximizes the expression A = (θ/2) * r^2.
9. Let's solve the arc length equation for θ: θ = 24 inches / (2πr).
10. Substituting this value of θ into the area equation, we get: A = [24 inches / (2πr)] * r^2.
11. Simplifying further, we have: A = 12 inches / π * r.
12. To find the radius that maximizes the area, we take the derivative: dA/dr = 12 inches / π.
13. Since the radius cannot be negative, we ignore the negative solution and set dA/dr = 0 to find the maximum.
14. Setting dA/dr = 0, we find: 12 inches / π = 0.
15. Solving for r, we get: r = 12 inches / π.
16. Therefore, the radius of the pan that will make the slice of pizza largest is approximately 3.82 inches, when rounded to two decimal places.

To find the radius of the pan that will make the slice of pizza largest, we need to maximize the area of the sector.

The perimeter of a sector is given by the formula: P = 2πr + 2s, where r is the radius and s is the arc length (in this case, the perimeter of the curved part of the slice).

We are given that the perimeter (P) of the pizza slice is 24 inches. So, we have the equation: 24 = 2πr + 2s.

We want to find the radius (r) that will maximize the area of the sector, which is given by the formula: A = (1/2) * r^2 * θ, where θ is the angle of the sector.

To maximize the area (A), we can take the derivative of A with respect to θ and set it equal to zero.

dA/dθ = (1/2) * r^2 * (1 - cos(θ)) = 0

Simplifying the equation, we find:

1 - cos(θ) = 0

cos(θ) = 1

Since the cosine of 0 degrees is 1, we know that the angle θ is 0 degrees. Therefore, the sector of the circle becomes a straight line.

In this case, the maximum area of the sector is achieved when the angle θ is 0 degrees and the sector becomes a straight line. In other words, the largest slice of pizza occurs when the radius (r) is infinity.

Therefore, to make the slice of pizza largest, there is technically no specific radius of the pan.