QuestionThe curved edge of the door mat below is half an ellipse, a "serni-ellipse". As shown in the figure below, the flat edge of the door mat measures 148 cm and the distance from the center (on the flat edge) to the curved edge is 54 cm . The distance from point $p$ to the curved edge is 21 cm . Find the distance from $p$ to the center.
Round your answer to the nearest hundredth. Do not round any intermediate computations. $\longmapsto 148 \mathrm{~cm}$ $\square$ cm

Studdy Solution
Solve for $x$:
Calculate $\frac{21^2}{54^2}$: $$\frac{21^2}{54^2} = \frac{441}{2916}$$
Substitute back into the equation: $$\frac{x^2}{5476} + \frac{441}{2916} = 1$$
Solve for $\frac{x^2}{5476}$: $$\frac{x^2}{5476} = 1 - \frac{441}{2916}$$
Calculate $1 - \frac{441}{2916}$: $$1 - \frac{441}{2916} = \frac{2475}{2916}$$
Thus: $$\frac{x^2}{5476} = \frac{2475}{2916}$$
Solve for $x^2$: $$x^2 = \frac{2475}{2916} \times 5476$$
Calculate $x^2$: $$x^2 = \frac{2475 \times 5476}{2916}$$
Calculate $x$: $$x = \sqrt{\frac{2475 \times 5476}{2916}}$$
Calculate the square root and round to the nearest hundredth: $$x \approx 63.06 \text{ cm}$$
The distance from $p$ to the center is:
$$\boxed{63.06 \text{ cm}}$$