The ceiling in a hallway $20 f t$ wide is in the shape of a semi ellipse and $18 f t$ at the centrre. Find the height of the ceiling $4$ feet from either wall if the height of the side wals is $12 f t$ .

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Solution

The pictorial representation of the problem is shown below

Let $P Q R$ be the height of the ceiling which is $4$ feet from the wall. From the diagram $P Q = 12 f t$ .

We have to find the height $Q R$ .

Since the width is $20 f t$ , take $A, A^{'}$ as vertices with $A$ as $(10, 0)$ and $A^{'}$ as $(- 10, 0)$ .

Take the midpoint of $A A^{'}$ as the centre which is $C (0, 0)$

$A A^{'} = 2 a = 20 \Rightarrow a = 10$ and $b = 18 - 12 = 6$

$∴ \frac{x^{2}}{100} + \frac{y^{2}}{36} = 1$

Let $Q R$ be $y_{1}$ then $R$ is $(6, y_{1})$ .

Since $R$ lies on the ellipse,

$\frac{36}{100} + \frac{y_{1}^{2}}{36} = 1 \Rightarrow y_{1} = 4.8$

$∴ P Q + Q R = 12 + 4.8$

Hence the required height of the ceiling is $16.8 f e e t$

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The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft at the centrre. Find the height of the ceiling 4 feet from either wall if the height of the side wals is 12ft.

The ceiling in a hallway $20 f t$ wide is in the shape of a semi ellipse and $18 f t$ at the centrre. Find the height of the ceiling $4$ feet from either wall if the height of the side wals is $12 f t$ .