1. ## cross sections

Question is
Let f(x, y) = 100 + 10x + 25y – x^2 – 5y^2.
a. Describe the cross section of the surface Z = f(x, y) produced by cutting it with the planes Y = 0, y = 1, y = 2, and y = 3.

b. Describe the cross section of the surface in the planes x = 0, x = 1, x = 2, and x = 3.
c. Describe the surface z = f(x, y)

What I have done for part b:
Z = F(0,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(0) + 25y – (0)^2 – 5y^2
= 100 + 25y - 5y^2

z = F(1,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(1) + 25y – (1)^2 – 5y^2
= 111 + 25y - 5y^2

z = F(2,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(2) + 25y – (2)^2 – 5y^2
= 124 + 25y - 5y^2

z = F(3,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(3) + 25y – (3)^2 – 5y^2
= 139 + 25y - 5y^2

How i described the results:
The cross section of f(x, y) = 100 + 10x + 25y – x2 – 5y2 that correspond to x = 0, x = 1, x = 2, and x = 3 are upper semicircles with centres (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0) respectively.

Is this correct so far?

2. Originally Posted by mellowdano
Question is
Let f(x, y) = 100 + 10x + 25y – x^2 – 5y^2.
a. Describe the cross section of the surface Z = f(x, y) produced by cutting it with the planes Y = 0, y = 1, y = 2, and y = 3.
b. Describe the cross section of the surface in the planes x = 0, x = 1, x = 2, and x = 3.
c. Describe the surface z = f(x, y)

What I have done for part b:
Z = F(0,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(0) + 25y – (0)^2 – 5y^2
= 100 + 25y - 5y^2
Which is a parabola opening downward.

CB

3. For part A i did the following and corrcted my explanation
Z = F(x,0) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(0) – x^2 – 5(0)^2
= -x^2 + 10x + 100z

F(x,1) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(1) – x^2 – 5(1)^2
= - x^2 + 10x + 120

z = F(x,2) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(2) – x^2 – 5(2)^2
= - x^2 + 10x + 130

z = F(x,3) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(3) – x^2 – 5(3)^2
= - x^2 + 10x + 130

The cross section of f(x, y) = 100 + 10x + 25y – x^2 – 5y^2 produced by cutting it with the plane y = 0, y = 1, y = 2, and y = 3 shows that each of these is a parabola that opens downward.

What I have done so far for part C
Describe the surface z = f(x, y)
z = f(0, 0) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(0) + 25(0) – (0)^2 – 5(0)^2
= 100

z = f(1, 1) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(1) + 25(1) – (1)^2 – 5(1)^2
= 141

z = f(2, 2) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(2) + 25(2) – (2)^2 – 5(2)^2
= 194

z = f(3, 3) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(3) + 25(3) – (3)^2 – 5(3)^2
= 259

My textbook has not really explained to me what I had to do for part C so I assumed I did the above correctly. I am not sure how to explain my results.