Question

the elliptical ceiling of a building is 68 ft long and 11 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 \\( \square = 1 \\).

Explanation:

Step1: Identify \(a\), \(b\) for the ellipse

The ellipse is centered at the origin, major axis along the \(x\)-axis. Length of major axis \(2a = 68\), so \(a=\frac{68}{2}=34\). The height is \(b = 11\) (since the vertex on \(y\)-axis is \((0,11)\)).

Step2: Standard form of ellipse

For an ellipse with major axis on \(x\)-axis, standard form is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\). Substituting \(a = 34\) (\(a^{2}=34^{2}=1156\)) and \(b = 11\) (\(b^{2}=11^{2}=121\)), the equation is \(\frac{x^{2}}{1156}+\frac{y^{2}}{121}=1\).

Step3: Calculate \(c\) for focus

We know \(c^{2}=a^{2}-b^{2}\). Substitute \(a^{2}=1156\), \(b^{2}=121\):
\(c^{2}=1156 - 121=1035\). Then \(c=\sqrt{1035}\approx32.17\) (but let's compute exactly: \(1156 - 121 = 1035\), so \(c=\sqrt{1035}\approx32.2\), but let's do the calculation properly. Wait, \(a = 34\), \(b = 11\), so \(c=\sqrt{34^{2}-11^{2}}=\sqrt{(34 - 11)(34 + 11)}=\sqrt{23\times45}=\sqrt{1035}\approx32.17\). But the question is how far along the major axis (x-axis) from the center (origin) to the focus \((c,0)\), so the distance is \(c\approx32.2\) feet. But first, part (a) is the standard form.

Answer:

(for part a):
\(\frac{x^{2}}{1156}+\frac{y^{2}}{121}\)

(For the distance part: The man stood at \(c=\sqrt{34^{2}-11^{2}}=\sqrt{1156 - 121}=\sqrt{1035}\approx32.2\) feet from the center along the major axis. But since the question first asks for the standard form, the answer for (a) is \(\frac{x^{2}}{1156}+\frac{y^{2}}{121}\).)