Applied Min/Max Help Calculus

Question

You have $9000 dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $5 per foot and for the sides it costs $3 per foot. Find the dimensions, x and y, of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

b. You have T dollars to buy fence to enclose a rectangular plot of land (see figure at right). The fence for the top and bottom costs $5 per foot and for the sides it costs $3 per foot. Find the dimensions of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

c. You have T dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $A per foot and for the sides it costs $B per foot. Find the dimensions of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

Answer 1

Note that (a) is a special case of (b), and that (b) is a special case of (c). We will solve (c) first, then solve(b), the solve (a).

c) The total cost T is simply T = 2*A*x + 2*B*y

The Area G of the plot is G = x*y, and we wish to maximize A.

Solve for y as follows:

2*A*x + 2*B*y = T

2*B*y = T - 2*A*x

y = ( T - 2*A*x ) / (2*B)

Write G as a function of x alone:

G(x) = x*y = x*( T - 2*A*x ) / (2*B)

dG/dx = 1 * ( T - 2*A*x ) / (2*B) + x*( - 2*A ) / (2*B) = ( T - 2*A*x - 2*A*x ) / (2*B) = (T - 4*A*x) / (2*B)

= 0 when

T - 4*A*x = 0

x = T / (4*A)

y = (T - 2*A*x ) / (2*B) = ( T - 2*A*T/(4*A) ) / (2*B) = (T - T/2 ) / (2*B) = T / (4*B)

x,y define a maximum G because the second derivative of G is negative:

d²G/dx² = -4A / 2B < 0 as long as A and B > 0.

b) Here, A=5, B=3.

x = T / (4*A) = T / (4*5) = T / 20

y = T / (4*B) = T / (4*3) = T / 12

T = 2Ax + 2By = 2(5)(T / 20) + 2(3)(T / 12) = T / 2 + T / 2

Cost to build side x: T / 2

Cost to build side y: T/ 2

a) Here, A=5, B=3, and T=9000

x = T/20 = 9000 / 20 = 450

y = T/12 = 9000 / 12 = 750

T = 2Ax + 2By = 2(5)(450) + 2(3)(750) = 4500 + 4500

Cost to build side x: $4500

Cost to build side y: $4500