Three students have ordered a 14 inch diameter pizza. They slice it with 2 parallel cuts, giving them 3 pieces with equal area. The two cuts are equidistant from the center of the pizza. Find the horizontal distance between one cut and the center of the pizza.

I figured the area of each piece would be 1/3(pi)r^2 and my teacher said to think about anti derivatives but I'm not sure what information that would give you

To find the horizontal distance between one cut and the center of the pizza, let's start by finding the radius of the pizza.

Given that the pizza has a diameter of 14 inches, we can find the radius by dividing the diameter by 2:

Radius (r) = Diameter / 2 = 14 inches / 2 = 7 inches.

Since the two cuts are equidistant from the center of the pizza, let's denote the horizontal distance between one cut and the center as "x" inches.

Now, let's consider the two outer pieces of the pizza after the cuts. Each of these pieces has an equal area, which is 1/3 times the area of the whole pizza.

So, the area of one piece is given by:

Area of one piece = (1/3) * π * r^2,

where r is the radius of the pizza.

Substituting the value of the radius, we get:

Area of one piece = (1/3) * π * (7 inches)^2 = (1/3) * π * 49 square inches = (49/3) * π square inches.

Since the two outer pieces have equal areas, their combined area would be (2/3) times the area of the whole pizza.

Therefore, the combined area of the two outer pieces is:

Combined area = (2/3) * (π * r^2) = (2/3) * (π * (7 inches)^2) = (2/3) * (π * 49) square inches = (98/3) * π square inches.

Now, we can set up an equation to find the value of x, representing the horizontal distance between one cut and the center:

Area of one piece = Combined area.

Therefore, we have:

(49/3) * π = (98/3) * π.

To simplify the equation, we can cancel out the common factors:

49 = 98.

As 49 is not equal to 98, we find that there is no horizontal distance (x) between one cut and the center of the pizza that would result in three equal area pieces.

Hence, it is not possible to make three pieces with equal areas by dividing the pizza with two parallel cuts equidistant from the center.

To find the horizontal distance between one cut and the center of the pizza, we can start by calculating the radius of the pizza.

Given that the diameter of the pizza is 14 inches, the radius (r) can be found by dividing the diameter by 2:

r = 14 inches / 2 = 7 inches

Now, let's think about the area of each piece. Since the three pieces have equal area, each piece should have an area of 1/3 of the total pizza area.

The total area of the pizza can be calculated using the formula for the area of a circle:

A = π * r^2

Substituting the radius we found earlier:

A = π * 7 inches^2

To find the area of each piece, we divide the total area by 3:

Area of each piece = (π * 7 inches^2) / 3

Now we have the area of each piece, but we need to relate it to the horizontal distance between one cut and the center of the pizza.

By slicing the pizza with two parallel cuts, the pieces are divided into three parts of equal area. Since the cuts are equidistant from the center of the pizza, each cut passes through the middle of the pizza.

Imagine a line connecting the center of the pizza with one of the cuts, forming a right-angled triangle. The horizontal distance between the cut and the center of the pizza is one of the legs of this triangle.

To relate the area of each piece to the horizontal distance between the cut and the center, we can use calculus and integration. Here's how:

Let x be the horizontal distance from the cut to the center of the pizza.

The area of each slice, A_slice, is given by integrating the area of a slice from x to r, which is from the cut to the edge of the pizza:

A_slice = ∫[x to r] (1/2)*(r^2 - y^2) dy

Where y represents the vertical distance from the center (which is always less than or equal to r).

Since we know that the area of each slice is equal to (π * 7^2) / 3, we can set up the integration equation:

(π * 7^2) / 3 = ∫[x to r] (1/2)*(r^2 - y^2) dy

By evaluating this integral and solving for x, we can find the horizontal distance between one cut and the center of the pizza.

This is where your teacher mentioned using antiderivatives. Integrating the expression (1/2)*(r^2 - y^2) with respect to y will give you the antiderivative, which you can then evaluate within the integration limits of x and r.

However, it's worth noting that this approach involves advanced math techniques and can be quite complex. It may not be necessary for solving the problem unless specifically required.

Alternatively, you could consider approximations or numerical methods to find the value of x. This would involve dividing the interval between x and r into small increments, calculating the slice area for each increment, and summing them until you reach the desired area.

See the first related question below. Just change the numbers as needed