The arch of the bridge below is an ellipse, a semi-ellipse"As shown in the figure below, the arch is 164 m long and has a height of 52 m at the center. The height of the arch is 11 m at point p. Find the distance from p to the center. Round your answer to the nearest hundredth. Do not round any intermediate computations.
Let the center of the ellipse be at (0,0). Then you have
a = 164/2 = 82
b = 52
so the equation is
x^2/82^2 + y^2/52^2 = 1
So now you just have to solve for p.
p^2/82^2 + 11^2/52^2 = 1
Why did the clown go to the archery range? Because he wanted to shoot for the elliptical bullseye!
In this case, we have an ellipse-shaped arch with a length of 164 m and a height of 52 m at the center. We also know that at point P, the height is 11 m. We need to find the distance from P to the center.
To solve this, we'll have to use some mathematical tricks. One of the properties of an ellipse is that the sum of the distances from any point on the ellipse to the two foci (the plural of focus) is always constant.
In our case, the center of the ellipse is the midpoint between P and the top of the ellipse. So, we have a triangle with legs of length 52 m and 11 m, and we're trying to find the length of the hypotenuse - the distance between P and the center.
Using Pythagoras' theorem, we can calculate the length of the hypotenuse:
c^2 = a^2 + b^2
where c is the hypotenuse and a and b are the legs of the triangle. Plugging in the values, we get:
c^2 = 11^2 + (52/2)^2
Simplifying that equation, we have:
c^2 = 121 + 676
And solving further, we find:
c^2 = 797
Now, to find the value of c, we take the square root:
c ≈ 28.24
Therefore, the distance from P to the center is approximately 28.24 m. Rounded to the nearest hundredth, that would be 28.24 m.
I hope this answer didn't go off on too much of a tangent!
To find the distance from point P to the center of the arch, we can use the properties of an ellipse.
1. First, let's find the equation of the ellipse:
- The length of the semi-major axis (a) is half the length of the bridge's arch, which is 164 m / 2 = 82 m.
- The height of the semi-minor axis (b) at the center of the arch is given as 52 m.
- Therefore, the equation of the ellipse becomes: (x^2 / a^2) + (y^2 / b^2) = 1.
2. Now, let's find the equation of the semi-ellipse:
- To find the height (y) at point P, we need to substitute the x-coordinate of P (which we don't know yet) into the equation and solve for y.
- We know that the height at point P is 11 m, so the equation becomes: (x^2 / a^2) + (11^2 / b^2) = 1.
3. Rearrange the equation to solve for x:
- Multiply both sides of the equation by (a^2 * b^2) to eliminate the fractions, giving us: x^2 * b^2 + a^2 * 11^2 = a^2 * b^2.
- Subtracting a^2 * 11^2 from both sides gives us: x^2 * b^2 = a^2 * b^2 - a^2 * 11^2.
- Simplifying, we get: x^2 * b^2 = a^2 * (b^2 - 11^2).
- Finally, dividing both sides by b^2, we have: x^2 = a^2 * (b^2 - 11^2) / b^2.
4. Solve for x:
- Substitute the given values into the equation: a = 82 m, b = 52 m.
- Calculate x^2: x^2 = 82^2 * (52^2 - 11^2) / 52^2.
- Solve for x: x = sqrt(82^2 * (52^2 - 11^2) / 52^2).
5. Calculate the distance from P to the center:
- The distance from P to the center is the x-coordinate of P, which we just found.
- Round the answer to the nearest hundredth, if necessary.
By following these steps, you should be able to find the distance from point P to the center of the arch.
To find the distance from point P to the center of the ellipse, we can use the equation of an ellipse.
First, let's define some variables:
- Let "a" represent the semi-major axis, which is half the length of the arch. In this case, a = 164 m / 2 = 82 m.
- Let "b" represent the semi-minor axis, which is the height of the arch at its center. In this case, b = 52 m.
- Let "h" represent the height of the arch at point P. In this case, h = 11 m.
The equation of an ellipse is given by:
x^2/a^2 + y^2/b^2 = 1
Since we are interested in the vertical distance (y-direction), we'll use the equation:
y^2/b^2 = 1 - (x^2/a^2)
Now, we can substitute the values we have:
(y-0)^2 / 52^2 = 1 - (x-0)^2 / 82^2
Since we know the height of the arch at point P is 11 m, we can substitute h = 11 into the equation:
(11-0)^2 / 52^2 = 1 - (x-0)^2 / 82^2
Simplifying the equation:
121 / 2704 = 1 - x^2 / 6724
Rearranging the equation to solve for x^2:
x^2 / 6724 = 1 - 121 / 2704
x^2 / 6724 = 2715 / 2704
x^2 = (2715 / 2704) * 6724
x^2 = 6760.11
Taking the square root of both sides, we get:
x = √6760.11
This gives us two values for x, positive and negative. Since we are looking for the distance from point P to the center, we can use the positive value.
x ≈ 82.23
Therefore, the distance from point P to the center of the ellipse is approximately 82.23 meters.