Can someone please help me with this optimisation problem? I'm stuck and I don't know what to do.
The cost C ($/h) of running a ferry at a constant speed of v(km/h) is
C=(961/v)+((3v^3)/8469)
The speed which will minimise the cost is...?
I tried differentiating C (to get dC/dv), and equating the equation to 0, but my variable (v) cancels itself out...
Any help would be appreciated! If you could include working out that would also be greatly appreciated. Thanks!
The cost C ($/h) of running a ferry at a constant speed of v(km/h) is
C=(961/v)+((3v^3)/8469)
The speed which will minimise the cost is...?
I tried differentiating C (to get dC/dv), and equating the equation to 0, but my variable (v) cancels itself out...
Any help would be appreciated! If you could include working out that would also be greatly appreciated. Thanks!
-
When v ~ 0 (i.e. very close to 0), 1/v is VERY large...when v >> 1 (i.e. VERY large) v³ is VERY large...this suggests that a minimum DOES exist, so I suspect you took the derivative incorrectly or simplified incorrectly.
dC/dv = -961/v² + 9/8469 * v²
--> find a common denominator then set the numerator to 0
You have v² in the denominator and the other has 1 (if you don't worry about the fraction).
-->
dC/dv = (-961 + 9/8469 * v⁴) / v²
--> now set the numerator to 0
-961 + 9/8469 * v⁴ = 0
--> solve for v⁴
v⁴ = 961 / (9/8469)
--> take the fourth root:
v = ∜(961 / (9/8469)) = (961 / (9/8469))^¼ ~ 30.8374354
This (and -30.8374354) are the ONLY two real solutions to our equation, therefore this MUST be the minimum. You can reason this in two different ways:
1) Reasoning: as I said, around v = 0, you get an infinity, and around v = ∞ you get an infinity, thus a minimum MUST exist and this was the only critical point, so it MUST be a minimum.
2) Cheating: they asked for a minimum and you found only one critical point, so this is probably the minimum.
dC/dv = -961/v² + 9/8469 * v²
--> find a common denominator then set the numerator to 0
You have v² in the denominator and the other has 1 (if you don't worry about the fraction).
-->
dC/dv = (-961 + 9/8469 * v⁴) / v²
--> now set the numerator to 0
-961 + 9/8469 * v⁴ = 0
--> solve for v⁴
v⁴ = 961 / (9/8469)
--> take the fourth root:
v = ∜(961 / (9/8469)) = (961 / (9/8469))^¼ ~ 30.8374354
This (and -30.8374354) are the ONLY two real solutions to our equation, therefore this MUST be the minimum. You can reason this in two different ways:
1) Reasoning: as I said, around v = 0, you get an infinity, and around v = ∞ you get an infinity, thus a minimum MUST exist and this was the only critical point, so it MUST be a minimum.
2) Cheating: they asked for a minimum and you found only one critical point, so this is probably the minimum.