A steamship sails upriver from A to B against a current of three km/h. If the amount of fuel consumed per hour varies as the cube of the speed through the water, how fast should the captain order the ship to steam, if the fuel consumption for the journey from A to B is to be a minimum.
I gave this as a final exam problem in HS calculus many years ago.
Fuel=K1*(speedrelative to water)^3 *time
But time= distanceAtoB/(speedrelativewater-3)
fuel=K1*(speedrelativewater^3)distancAB/(speedrelativewater-3)
let s be speed relative to water
dfuel/ds= 3s^2*distance/(s-3) - s^3*distance(s-3)^-2
set to zero...
3s^2/(s-3)=s^3/s-3)^3
3=s/(s-3)
3s-9=s
2s=9
s=4.5
the minu
what does K1 stand for??
its the constant term; every time two things vary directly there's always a constant K that you have to consider.
Hi, Kavya! :D LOL
To minimize the fuel consumption for the journey from A to B, we need to find the speed at which the rate of fuel consumption is at a minimum.
Let's break down the problem and find an equation that describes the relationship between the fuel consumption rate and the speed of the ship.
Let the speed of the ship in still water be V km/h, and let the speed of the current be 3 km/h. When the ship is moving against the current, the effective speed is reduced by the speed of the current. Hence, the speed of the ship against the current is (V - 3) km/h.
According to the information provided, the rate of fuel consumption varies as the cube of the speed through the water. We can express the fuel consumption rate as C(V), where C(V) represents the rate of fuel consumption while moving at speed V km/h.
Since the fuel consumption varies with the cube of the speed, we can write:
C(V) = k * V^3
Where k is a constant of proportionality.
To find the minimum fuel consumption rate, we need to find the minimum value of C(V). To do that, we can take the derivative of C(V) with respect to V and set it equal to zero.
dC/dV = 3k * V^2
Setting dC/dV equal to zero, we have:
3k * V^2 = 0
Solving for V, we find that V = 0.
However, we need to keep in mind that the ship must move to reach point B. We can't have the speed equal to zero.
Therefore, the minimum fuel consumption rate occurs at the lowest possible positive speed, or when V = 0 + 3.
So the captain should order the ship to steam at a speed of (0 + 3) km/h, or simply 3 km/h, to minimize the fuel consumption for the journey from A to B.