A small truck is to be driven 300 miles on a freeway at a constant speed of x miles per hour. Speed laws on the freeway require that a truck must be driven between 30 miles per hour and 70 miles per hour. assume that fuel cost $3.50 per gallon. in addition, assume that for a given speed of x miles per hour, fuel is consumed at the rate of 1+(x^2/600) gallons per hour. If the salary of the driver is based on $20 per hour, how to find the most economical speed of the total cost of the trip? plz help!

Write the cost function:
C=20t + 3.50(1+(x^2/600)) gives cost as a function of time and speed. But distance, is related to speed x and time, so time= distance/speed or
Cost= 20 *300/x + 3.50(1+(x^2/600)) so now cost is a function of speed.
minimize cost as a function of speed (set first derivative equal to zero).

check my math.

i don't know ask the driver

Your math is correct. To find the most economical speed for the total cost of the trip, you need to minimize the cost function with respect to speed.

The cost function you have correctly written is:

C(x) = 20(300/x) + 3.50(1 + (x^2/600))

To minimize the cost function, you need to find the critical points by setting its first derivative equal to zero.

Let's do that:

C'(x) = -20(300/x^2) + 3.50(x/300) = 0

Simplifying this equation further, we get:

-6000/x^2 + (7x/300) = 0

To solve this equation, let's clear the fractions by multiplying through by 300x^2:

-6000(300) + 7x^3 = 0

-1800000 + 7x^3 = 0

Now, isolate the variable x:

7x^3 = 1800000

Divide both sides by 7:

x^3 = 257142.8571

Take the cubic root of both sides:

x = 63.58 (rounded to two decimal places)

So, the most economical speed for the total cost of the trip is approximately 63.58 miles per hour.