Q= 4 million gal, d= 7000 dollar, s=33 cents/gal yr

Given the values:
- $ Q = 4,000,000 $ gallons
- $ d = 7,000 $ dollars
- $ s = 0.33 $ dollars per gallon per year

We can find the optimal number of deliveries $ N $ per year and the length of time $ T $ between deliveries.

### Step 1: Calculate the Optimal Number of Deliveries $ N $

Using the formula derived earlier:
\[ N = \sqrt{\frac{sQ}{d}} \]

Substitute the given values:
\[ N = \sqrt{\frac{0.33 \times 4,000,000}{7,000}} \]

\[ N = \sqrt{\frac{1,320,000}{7,000}} \]

\[ N = \sqrt{188.5714} \]

\[ N \approx 13.73 \]

Since $ N $ must be a whole number (you can't have a fraction of a delivery), we consider the nearest integers, which are 13 and 14. We will need to evaluate the total cost for both to determine which one minimizes the cost.

### Step 2: Calculate the Total Cost for $ N = 13 $ and $ N = 14 $

#### For $ N = 13 $:
\[ \text{Delivery Cost} = 13 \times 7,000 = 91,000 \text{ dollars} \]

\[ T = \frac{1}{13} \]

\[ \text{Yearly Storage Cost} = 0.33 \times 4,000,000 \times \frac{1}{13} \]

\[ \text{Yearly Storage Cost} = 0.33 \times 4,000,000 \times \frac{1}{13} \approx 101,538.46 \text{ dollars} \]

\[ C(13) = 91,000 + 101,538.46 = 192,538.46 \text{ dollars} \]

#### For $ N = 14 $:
\[ \text{Delivery Cost} = 14 \times 7,000 = 98,000 \text{ dollars} \]

\[ T = \frac{1}{14} \]

\[ \text{Yearly Storage Cost} = 0.33 \times 4,000,000 \times \frac{1}{14} \]

\[ \text{Yearly Storage Cost} = 0.33 \times 4,000,000 \times \frac{1}{14} \approx 94,285.71 \text{ dollars} \]

\[ C(14) = 98,000 + 94,285.71 = 192,285.71 \text{ dollars} \]

### Step 3: Compare the Total Costs

\[ C(13) = 192,538.46 \text{ dollars} \]

\[ C(14) = 192,285.71 \text{ dollars} \]

Since $ C(14) < C(13) $, the minimum cost is achieved with $ N = 14 $.

### Conclusion

The optimal number of deliveries per year is $ N = 14 $.

The length of time $ T $ between shipments is:
\[ T = \frac{1}{N} = \frac{1}{14} \approx 0.0714 \text{ years} \approx 26 \text{ days} \]

Would you like more details or have any questions?

Here are 5 questions for you to explore:

1. How would the optimal $ N $ change if the delivery cost $ d $ increased to $10,000?
2. What is the impact on storage costs if the total amount of gasoline $ Q $ changes?
3. How would seasonal variations in gasoline usage affect the optimal delivery schedule?
4. What are the potential consequences of choosing a non-optimal $ N $?
5. Can this model be adjusted for fluctuating storage cost rates over different seasons?

**Tip:** When working with optimization problems, always check the results by substituting the solution back into the original equations to ensure they minimize the cost or other objective functions effectively.

Learn how to calculate the optimal number of deliveries per year for gasoline shipments based on given costs and quantities. This optimization problem involves finding the optimal balance between delivery and storage costs. Suitable for college-level students and professionals.

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