The cost of running a ship at a constant speed of v km/h is 160 + 1/100*v^3 dollars per hour.
a)Find the cost of a journey of 1000km at a speed of v km/h.
b)Find the most economical speed for the journey, and the minimum cost.
c)If the ship were to have maximum speed of 16 km/h find what the minimum cost would be.
(a) time = distance/speed, so a trip of 1000km takes 1000/v hours. So, the cost is
c(v) = (1000/v)(160 + 1/100v^3)
(b) Now just find minimum cost where dc/dv = 0
dc/dv = 20(v^3-8000)/v^2
so minimum cost when v=20
(c) Since min occurs at v=20, if top speed is 16, then minimum occurs at 16
a) To find the cost of a journey of 1000 km at a speed of v km/h, we can use the given cost function: 160 + (1/100)*v^3 dollars per hour.
Since the speed is given in km/h and we want to find the cost for 1000 km, we need to convert the speed to km/h.
The time taken for the journey is given by the distance divided by speed, which is 1000/v hours.
Multiplying the time taken by the cost per hour, we can find the total cost for the journey:
Cost = (160 + (1/100)*v^3) * (1000/v) dollars.
b) To find the most economical speed for the journey, we need to minimize the cost function. We can do this by finding the derivative of the cost function with respect to the speed v, setting it equal to zero, and solving for v.
Let's find the derivative first. The derivative of (1/100)*v^3 with respect to v is (3/100)*v^2, and the derivative of 160 with respect to v is 0 since it is a constant.
So, the derivative of the cost function with respect to v is (3/100)*v^2 * (1000/v) - (160/v).
Setting this derivative equal to zero and solving for v will give us the most economical speed for the journey.
(3/100)*v^2 * (1000/v) - (160/v) = 0.
Simplifying this equation, we get:
3v^2 - 1600 = 0.
Solving for v gives us:
v^2 = 1600/3.
v = sqrt(1600/3) ≈ 26.83 km/h.
So, the most economical speed for the journey is approximately 26.83 km/h.
Now, to find the minimum cost, we substitute this speed into the cost function:
Cost = (160 + (1/100)*(26.83)^3) * (1000/26.83) dollars.
c) If the ship were to have a maximum speed of 16 km/h, we can use the same cost function to find the minimum cost.
Substituting v = 16 into the cost function:
Cost = (160 + (1/100)*(16)^3) * (1000/16) dollars.
Simplifying this expression will give us the minimum cost.