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​(a) The standard form of the equation is 

enter your response hereequals=1.

To solve this problem, let's break it down step by step.

### Given:
- The elliptical ceiling has a length of 100 ft and a height of 24 ft.
- The major axis is horizontal (from \((-50, 0)\) to \( (50, 0)\)), so the total length of the major axis is 100 ft, making \(a = 50\).
- The minor axis (the height) is 24 ft, so the semi-minor axis is \(b = 12\).

### Step 1: The Standard Form of an Ellipse
The standard form of the equation of an ellipse with a horizontal major axis is:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Where:
- \(a\) is the semi-major axis (half of the total length along the major axis),
- \(b\) is the semi-minor axis (half of the total height along the minor axis).

### Step 2: Substitute \(a\) and \(b\)
From the given information:
- \(a = 50\), so \(a^2 = 2500\),
- \(b = 12\), so \(b^2 = 144\).

Now substitute these into the standard equation:

\[
\frac{x^2}{2500} + \frac{y^2}{144} = 1
\]

### Step 3: Use the Focus Formula to Find \(c\)
The focus \(c\) is related to \(a\) and \(b\) by the equation:

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

Substitute \(a^2 = 2500\) and \(b^2 = 144\):

\[
c^2 = 2500 - 144 = 2356
\]

So, \(c = \sqrt{2356} \approx 48.54\) ft.

### Step 4: Interpret the Problem
The man stands at the focus, which is \(c\) feet from the center along the major axis. Therefore, the man stands approximately **48.54 feet** along the major axis to hear the conversations clearly.

---

In summary:
- The equation of the ellipse is:

\[
\frac{x^2}{2500} + \frac{y^2}{144} = 1
\]

- The man stands approximately **48.5 feet** along the major axis to hear the conversations.

Would you like further explanation or another example?

The elliptical ceiling of a building is 100 ft long and 24 ft tall. Use the rectangular coordinate system in the figure shown to write the standard form of the equation for the elliptical ceiling. A man discovered that he could hear conversations of colleagues in the entire room if he stood at the focus, (c,0), where c^2 = a^2 - b^2. How far along the major axis did the man stand to hear the conversations? (a) The standard form of the equation is [_] = 1.

Solve the problem of determining the standard form of the equation for an elliptical ceiling and calculate how far along the major axis a person must stand to hear conversations clearly. The problem uses key concepts of ellipses and coordinate geometry, suitable for Grades 10-12.

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