a water feature in the shape of an ellipse is built in front of city hall. the architect plans to place the fountains at the 2 foci. if the pool of the water is 20 ft long and 12 ft wide, how far from the center should the fountains be located?

The given water feature is shaped like an ellipse, and the fountains are located at the two foci of this ellipse. The key is to use the relationship between the dimensions of the ellipse and the foci.

### Formula:
For an ellipse, the equation for the relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance of the foci from the center \(c\) is:

\[
c^2 = a^2 - b^2
\]

Where:
- \(a\) is the semi-major axis (half of the longer dimension),
- \(b\) is the semi-minor axis (half of the shorter dimension),
- \(c\) is the distance from the center to each focus.

### Step 1: Find \(a\) and \(b\)

- The length of the pool is 20 ft, so the semi-major axis \(a\) is:
  \[
  a = \frac{20}{2} = 10 \, \text{ft}
  \]
  
- The width of the pool is 12 ft, so the semi-minor axis \(b\) is:
  \[
  b = \frac{12}{2} = 6 \, \text{ft}
  \]

### Step 2: Find \(c\)
Now, use the formula \(c^2 = a^2 - b^2\):

\[
c^2 = 10^2 - 6^2 = 100 - 36 = 64
\]

So,
\[
c = \sqrt{64} = 8 \, \text{ft}
\]

### Conclusion:
The fountains should be located 8 feet from the center of the pool.

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Would you like further details or have any questions?

Here are some related questions:
1. How do you calculate the area of an ellipse?
2. What happens to the foci if the ellipse becomes a circle?
3. Can the same formula be applied to ellipses with different orientations?
4. How would you find the equation of this ellipse in standard form?
5. How does the eccentricity of an ellipse relate to the distance of the foci?

**Tip:** The sum of the distances from any point on the ellipse to the two foci is constant!

A water feature in the shape of an ellipse is built in front of city hall. The architect plans to place the fountains at the 2 foci. If the pool of the water is 20 ft long and 12 ft wide, how far from the center should the fountains be located?

Learn how to calculate the distance of the foci from the center in an elliptical water feature. This detailed solution explains the geometry behind ellipses and how to apply the formula c^2 = a^2 - b^2 using a 20 ft long and 12 ft wide ellipse. Perfect for students in Grades 9-10.

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