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
