Question

A case of vintage wine appreciates in value each year, but there is also an annual storage charge. The value of a typical case of investment-grade wine after t years is V(t) = 2000 + 40\sqrt t - 10t dollars (for 0 < t < 25

). Find the storage time that will maximize the value of the wine.

Answer 1

The storage time that will maximize the value of the wine is 4 years.

Step-by-step explanation:

To find the storage time that will maximize the value of the wine, we need to find the maximum value of the function V(t) = 2000 + 40√t - 10t. To do this, we can take the derivative of V(t) with respect to t and set it equal to zero.

dV/dt = 20/√t - 10 = 0

20/√t = 10

√t = 2

t = 4

Therefore, the storage time that will maximize the value of the wine is 4 years.

Continuing from the calculation, after setting the derivative equal to zero and solving for t, we find t. This critical point corresponds to a potential maximum or minimum of the function. To confirm whether it's a maximum, we can check the second derivative or evaluate the function around the critical point. If the second derivative is negative, it indicates a maximum. Alternatively, we can compare the values of V(t) for nearby values of t to verify that \( t = 4 \) indeed yields the maximum value.