The cost C of running a ship on a certain voyage is said to be given by c=av+b/v where v is the average speed in km/h and a,b are constant.find the value of v in terms of a and b to give the maximum cost,and also find the cost???
To find the value of v that gives the maximum cost, we need to find the critical points of the cost function and determine whether they correspond to a maximum or minimum. The cost function is given by c = av + b/v.
To find the critical points, we need to take the derivative of the cost function with respect to v and set it equal to zero. Let's do that:
c = av + b/v
Differentiating both sides of the equation with respect to v:
dc/dv = a - b/v^2
Next, we set the derivative equal to zero and solve for v:
a - b/v^2 = 0
b/v^2 = a
v^2 = b/a
v = ±√(b/a)
Since the average speed (v) cannot be negative in this context, we discard the negative square root and obtain:
v = √(b/a)
So, the value of v that gives the maximum cost is √(b/a).
To find the corresponding cost, we substitute this value of v back into the cost function:
c = a(√(b/a)) + b/√(b/a)
Simplifying further:
c = √(ab) + b/√(b/a)
We can rationalize the denominator:
c = √(ab) + b√(a/b)
Hence, the cost of running the ship with the value of v that gives the maximum cost is √(ab) + b√(a/b).
c = av + b/v
cv = av^2 + b
av^2 - cv + b = 0
now just use the quadratic formula to find v.
There will only be a maximum cost if a<0, since then the parabola will open downward.