NEED HELP NOT AI - a truck is to travel 1000 kilometers at a constant speed of v km/h. When travelling at v km/h, the truck consumes fuel at the rate of (6+v^2/50) litres per hour. The truck company pays 50c/litre for fuel and pays each of the two drivers $20 per hour whilst the truck is travelling.
A) Let the total cost for the trip be C dollars. Show that C= 10v +43000/v
B)The truck must take no longer than 12 hours to complete the trip, and speed limits require that c is less than or equal to 100. at what speed should the truck travel to minimise the cost C?
I only need help with part B).
To minimize the cost C, we need to find the minimum of the function C(v) = 10v + 43000/v subject to the constraints that the trip must be completed in 12 hours or less, and the speed must be less than or equal to 100.
To find the minimum of C(v), we take the derivative with respect to v and set it equal to 0:
dC/dv = 10 - 43000/v^2
Setting dC/dv = 0:
10 - 43000/v^2 = 0
43000/v^2 = 10
v^2 = 4300
v = √4300
v ≈ 65.57
So, the truck should travel at a speed of approximately 65.57 km/h to minimize the cost C.