Homework answers

15)
Primary: C = 2x^2 + 8xy
Limiting: 20 = yx^2; y = 20/x^2
Primary (one-variable): C(x) = 2x^2 + 160/x

17)
Primary: P = ab(a-b)
Limiting: a+b = 8; b = 8-a
Primary (one-var.): P(a) = a(8-a)(2a-8); P(a) = -2a^3 + 24a^2 - 64a

18)
Primary: A = (1/2) (l^2) sin (theta)
Limiting: None (l is a constant)

20)
Primary: P = 100 sqrt(x) + 150 sqrt(y)
Limiting: x + y = 40; y = 40 - x
Primary (one-var.): P(x) = 100 sqrt(x) + 150 sqrt (40-x)

22)
Primary: P = xy^2
Limiting: (x/2)^2 + (y/2)^2 = r^2; y^2 = 4r^2 - x^2 (limiting eq. is the pyth. thm.)
Primary (one-var.): P(x) = 4xr^2 - x^3

27)
Primary: A = xy
Limiting: y = (4-x)/(2+x)
Primary (one-var.): A(x) = x(4-x)/(2+x)

28)
Primary: V = hx^2
Limiting: 108 = 4x + h; h = 108 - 4x
Primary (one-var.): V(x) = (108 - 4x) x^2

33)
Primary: S(x) = (x - x1)^2 + (x - x2)^2 + ... + (x-xn)^2
Limiting: None

(The value of x that minimizes S(x) is (x1 + x2 + ... + xn)/n, i.e. the average of x1, x2, ..., xn.)

36)
Primary: C(x) = 30 sqrt(10000 + x^2) + 15 (200 - x)
Limiting: None

44)
Primary: T(x) = (1/3) sqrt (90000 + x^2) + (1/8) (200 - x)
Limiting: None