AMBULANCE Some ambulances use weighted averages to find where to park between cities. The weight of a hospital is determined by the average number of emergent patients that need to be taken to the hospital on a daily basis.

The Davis Medical Center usually has a weight 4 times that of Woodland Hospital. Where should the ambulance park? Round your answer to the nearest tenth, if necessary.

To solve this problem, we'll use the concept of weighted averages to determine the optimal parking location for the ambulance.

### Let's define the variables:

- Let \( D \) be the location of Davis Medical Center.
- Let \( W \) be the location of Woodland Hospital.
- Let \( x \) be the distance between \( D \) and \( W \).
- Let the weight of Davis Medical Center be \( 4w \).
- Let the weight of Woodland Hospital be \( w \).

### Weighted Average Formula:
The weighted average location \( P \) for the ambulance can be found using the formula:
\[
P = \frac{( \text{Weight of Davis} \times D ) + ( \text{Weight of Woodland} \times W )}{\text{Total Weight}}
\]

Given:
\[
\text{Weight of Davis} = 4w
\]
\[
\text{Weight of Woodland} = w
\]

The formula for \( P \) becomes:
\[
P = \frac{(4w \times D) + (w \times W)}{4w + w}
\]
\[
P = \frac{4w \times D + w \times W}{5w}
\]

Simplifying:
\[
P = \frac{4D + W}{5}
\]

### Interpretation:
The ambulance should park at a location that is \(\frac{4}{5}\) (or 80%) of the distance from Woodland Hospital toward Davis Medical Center.

### Conclusion:
- If \( D \) and \( W \) are at specific distances, substitute them into the formula.
- Otherwise, the ambulance should park closer to Davis Medical Center, at a point 80% of the distance from Woodland Hospital to Davis Medical Center.

Would you like a detailed example with specific distances, or do you have any questions?

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**Related Questions:**
1. What if the weight of Davis Medical Center was 3 times that of Woodland Hospital?
2. How would the formula change if there were three hospitals?
3. Can we apply the same method if hospitals are located in a 2D plane?
4. What if the ambulance can only park at certain discrete locations between the hospitals?
5. How does the introduction of more hospitals affect the weighted average location?

**Tip:** When dealing with weighted averages, always ensure that the sum of the weights is accurately calculated to avoid errors.

Learn how to determine the optimal parking location for an ambulance using weighted averages. This problem involves calculating distances and weights between hospitals, such as Davis Medical Center and Woodland Hospital, based on emergent patient needs.

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