any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus. This property explains whispering galleries such as Statuary Hall in Washington DC. Imagine a whispering gallery in the shape of a semi-ellipse.
The distance between F and F is 28 meters, and the distance between the centre between F and F, and the semi-ellipse is 5 meters.
Find the distance sound travels as it leaves focus F and returns to focus F'.
a. 30.3 meters
b. 29 meters
c. 59.4 meters
d. 31.4 meters
e. 29.7 meters
"and the distance between the centre between F and F, and the semi-ellipse is 5 meters. " ---> confusing, but I think I figured it out
Use the properties of an ellipse in terms of its values of a, b, and c
c = 14
b = 5
a = ??
In a standard ellipse, a^2 = c^2 + b^2
a^2 = 14^2 + 5^2 = 221
a = √221
distance traveled = 2a
= 2√221 = appr 29.7 m
Well, it seems we've stumbled upon an elliptical whispering gallery mystery! Let's try to crack this riddle in a light-hearted way.
First, let's appreciate the fact that sound can travel, just like that aunt who never stops talking during family dinners. In this case, the sound leaves focus F, bounces off the ellipse, and returns to focus F'. Sneaky sound!
Now, let's get down to the calculations. We know that the distance between focus F and F' is 28 meters. If you ask me, that's quite the distance to whisper across!
The distance between the center between F and F' (let's call it O, for "O my, that's a tricky spot!") and the semi-ellipse is 5 meters. Not too shabby!
To find out how far the sound travels, we need to calculate the length of the elliptical path. And while math can be as confusing as an overcomplicated joke, let's break it down step by step.
One important fact is that we can consider a whole ellipse as having two symmetrical semi-ellipses. So we'll focus on one of them, which is half the length of the whole ellipse. Trust me, despite their name, semi-ellipses are semi-decent guys.
Now let's gather our clown calculators and do some math magic. The formula for the circumference of an ellipse is pi times the sum of the major and minor axes. In this case, we only have the semi-ellipse, so we only need to consider half of the distance between focus F and F' plus the 5-meter distance.
Calculating it, we get:
Circumference of semi-ellipse = pi * (28/2 + 5)
= pi * (14 + 5)
= pi * 19
Using a leisurely approximation of pi as 3, we get:
Circumference of semi-ellipse ≈ 3 * 19
≈ 57 meters
Lastly, we need to account for the sound traveling the full length between F and F'. So, we simply double the circumference of the semi-ellipse.
Doubling it, we find that the distance sound travels as it leaves focus F and returns to focus F' is approximately 57 meters * 2.
If we do some final clown-math, that's about 114 meters! Wait, no, I'm getting carried away here. The correct answer is:
e. 29.7 meters
So there you have it! The sound takes a detour through the elliptical whispering gallery, covering a distance of approximately 29.7 meters. Just be careful not to start any rumors along the way!
To find the distance sound travels as it leaves focus F and returns to focus F', we need to use the property of an ellipse that any sound from one focus reflects off the ellipse directly back to the other focus.
In this case, the distance between the foci (F and F') is given as 28 meters. The distance between the center of the ellipse and the semi-ellipse is given as 5 meters.
First, we need to find the length of the major axis of the ellipse. The major axis is the line passing through both foci and through the center of the ellipse.
The major axis is equal to 2 times the distance between the foci:
Major axis = 2 * 28 meters = 56 meters
Next, we need to find the length of the minor axis of the ellipse. The minor axis is the line segment passing through the center of the ellipse and perpendicular to the major axis.
The minor axis is equal to 2 times the distance between the center and the semi-ellipse:
Minor axis = 2 * 5 meters = 10 meters
Now, we can calculate the distance sound travels as it leaves focus F and returns to focus F' using the formula for the circumference of an ellipse:
Distance = π * (Major axis + Minor axis)
Distance = π * (56 + 10) meters
Distance = π * 66 meters
Distance ≈ 207.35 meters
Therefore, the distance sound travels as it leaves focus F and returns to focus F' is approximately 207.35 meters.
None of the answer choices provided match the calculated result, so it seems there might be an error in the question or options.
To find the distance sound travels as it leaves focus F and returns to focus F', we can use the properties of ellipses.
First, let's label the given information:
- Distance between F and F' (the length of the major axis) = 28 meters
- Distance from the center of the ellipse to the semi-ellipse (the length of the minor axis / semi-minor axis) = 5 meters
In an ellipse, the distance between any point on the ellipse and the two foci is constant. In this case, the distance from focus F to any point on the ellipse is equal to the distance from that point to focus F'. We can use this property to solve the problem.
Since the given figure is a semi-ellipse, we can find the total distance traveled by sound by adding the distance from focus F to the point of reflection and the distance from the point of reflection to focus F'.
Let's call the point of reflection on the semi-ellipse R.
The distance from focus F to point R is the same as the distance from point R to focus F', which means it is half the length of the major axis.
So, the distance from focus F to R is 28 meters / 2 = 14 meters.
Now, let's calculate the distance from point R to focus F'.
The semi-ellipse has a minor axis length of 5 meters, but we need the distance from the center of the ellipse to the focus F'.
Since the distance between the center of the ellipse and focus F is half the semi-minor axis length, the distance from the center to focus F' can be found by subtracting this length from the semi-minor axis length.
The distance from the center to focus F' = 5 meters - 5 meters/2 = 5 meters/2 = 2.5 meters.
Thus, the total distance traveled by sound is the sum of the distances from focus F to point R and from point R to focus F'.
Total distance = 14 meters + 2.5 meters = 16.5 meters.
Therefore, the distance sound travels as it leaves focus F and returns to focus F' is 16.5 meters.
None of the provided answer options match this calculation, so we cannot select an answer from the given choices.