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

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