Question

If the garden is to be 6050 square feet, and the fence along the driveway costs $6 per foot while on the other three sides it costs only $2 per foot, find the dimensions that will minimize the cost. length along the driveway ft length perpendicular to the driveway

Answer 1

Answer:

The dimension that will minimize the cost is 55 and 110(55 × 110)

Explanation:

The garden area is 6050 ft². Along the driveway the costs is $6 per foot while the other 3 area sides cost only $2 per foot.

The garden has four sides so it a rectangle .

a = length along the driveway

b = length perpendicular to the driveway

The area = a × b

ab = 6050

b = 6050/a

minimize cost = 6a + 2a + 2b + 2b

minimize cost = 8a + 4b

cost = 8a + 4b

y = 8a + 4(6050/a)

y = 8a + 24200/a

y = 8a + 24200/a

differentiate the rate of change of the cost with respect to length.

y'(a) = 8 - 24200/a²

(8a² - 24200)/a² = 0

cross multiply

8a² = 24200

a² = 24200/8

a² = 3025

a = √3025

a = 55

inserting the value of a in the area formula

ab = 6050

b = 6050/55

b = 110