I need some help on precalculus

Question

Pizzeria Buonapetito makes a triangular-shaped pizza with base width of 30 inches and height 20 inches as shown. Alice wants only a portion of the pizza and does so by making a vertical cut through the pizza and taking the shaded portion. Letting x be the bottom length of Alice’s portion and y be the length of the cut as shown, answer the following questions:

(a) Find a formula for y as a multipart function of x, for 0 ≤ x ≤ 30. Sketch the graph of this function and calculate the range.

(b) Find a formula for the area of Alice’s portion as a multipart function of x, for 0 ≤ x ≤ 30.

(c) If Alice wants her portion to have half the area of the pizza, where should she make the cut?

Answer 1

Note: the solution below assumes that the dimensions in the given diagram are accurate.

Using similar triangles, for 0 ≤ x < 6, y / x = 12 / 6. So, y = 2x

Area of slice = (1/2)(x)(y) = x²

For 6 ≤ x ≤ 18, y / (18-x) = 12/12. So, y = 18 - x.

Area of slice =

(1/2)(6)(12) + (1/2)(12)(12) - (1/2)(18 - x)(y) = 108 - (1/2)(18 - x)² = 108 - 162 + 18x - (1/2)x² =

(-1/2) x² + 18x - 54

Let A(x) = area of slice if the cut is made at x.

Then, A(x) = x², if 0 ≤ x < 6, and A(x) = -(1/2)x² + 18x - 54 if 6 ≤ x ≤ 18

To cut off half of the pizza, -(1/2)x² + 18x - 54 = (1/2)[(1/2)(18)(12)]

-(1/2)x² + 18x - 54 = 54

-(1/2)x² + 18x - 108 = 0

x² - 36x + 216 = 0

x = [36 ± √432] / 2 = 28.4 or 7.6

Since 0 ≤ x ≤ 18, x = 7.6"

Answer 2

(bh-xy)/2=shaded are of #1

" = white " #2

To get "y" on either; a^2+b^2=c^2

a shortest side

c hypotenuse

In these items #1 is for side "b"; so c^2-b^2=a^2.

" " #2 " " "a"; c^2-b^2=a^2. Subtract from the sq/meas.

Draw a line from angle "alpha" (opposite side "a") to the 1/2 pt of side "a". (The easy way).