If a pebble is dropped into a pond in the shape of an ellipse at the location of one focus, the waves will converge at the location of the other focus. If the pond has a major axis of 68 feet and a minor axis of 32 feet, how far apart are the foci?
the answer is 60 but how do i get this
looks like your ellipse is a standard position kind,
a = 34 , b = 16
since c^2 = a^2 - b^2 = 34^2 - 16^ = 900
c = √900 = 30
distance between two foci = 2c = 60
we need the answers to the whole semester test
thank you
To determine the distance between the foci of an ellipse, you can use the formula:
c = sqrt(a^2 - b^2)
Where:
- c is the distance between the foci
- a is the semi-major axis (half the length of the major axis)
- b is the semi-minor axis (half the length of the minor axis)
In this case, the major axis is 68 feet, so the semi-major axis (a) is 68/2 = 34 feet.
The minor axis is 32 feet, so the semi-minor axis (b) is 32/2 = 16 feet.
Now we can plug these values into the formula:
c = sqrt(34^2 - 16^2)
c = sqrt(1156 - 256)
c = sqrt(900)
c = 30 feet
But since the question asks for the distance between the foci, we need to double this value:
distance between foci = 2 * c
distance between foci = 2 * 30
distance between foci = 60 feet
Therefore, the distance between the foci is 60 feet.
Well, well, well, we have a pebble causing waves in a pond! How exciting! Now, let me show you the funny side of math.
To find the distance between the foci of an ellipse, you can use a nifty little formula: c = √(a^2 - b^2), where c represents the distance between the foci, a is half the length of the major axis, and b is half the length of the minor axis.
So, in this case, half the length of the major axis is 34 feet (68 feet / 2), and half the length of the minor axis is 16 feet (32 feet / 2).
Now, let's substitute those values into our formula: c = √(34^2 - 16^2).
If you crunch those numbers, you'll find that c ≈ 30. But, alas, that's not quite the answer we're looking for.
You see, when we find the distance between the foci, we only need the positive answer. So let's take the absolute value of our calculated value: |c| ≈ |30|, which is just 30.
Therefore, the distance between the foci is approximately 30 feet. But wait! We need the distance in feet, not inches! So let's convert inches into feet by dividing by 12. How convenient it is that there are 12 inches in a foot!
30 feet divided by 12 equals 2 and a half, which is 2.5 feet.
Now, let's double-check our work. If we add 2.5 feet to 2.5 feet, we get... 5 feet! That's not 60 feet!
So, my friend, it seems like there might be a slight misunderstanding or an error somewhere. Maybe someone dropped an extra pebble, causing some confusion. I suggest you take another look at your calculations and see if there are any mistakes.
Remember, if at first you don't succeed, try and try again! And don't forget to laugh along the way!