The circular path of cars on a ferris wheel can be modeled with the equation x^2-14x+y^2-150y=-49, measured in feet. What is the maximum height above ground of the riders.
write the circle equation in standard form by completing the square
x^2 - 14x + 48 + y^2 - 150y + 5625 = -49 + 49 + 5625
(x-7)^2 + (y - 75)^2 = 75^2
so the centre is at (7, 75) and the radius is 75
after making a sketch you should be able to answer the question
To find the maximum height above ground of the riders on the ferris wheel, we need to determine the y-coordinate of the highest point on the circle.
The equation given, x^2 - 14x + y^2 - 150y = -49, represents the equation of a circle in general form. We can rewrite it in standard form to identify the center and radius of the circle.
First, complete the square for the x-terms by adding (14/2)^2 = 49 to both sides of the equation:
x^2 - 14x + 49 + y^2 - 150y = -49 + 49
This simplifies to:
x^2 - 14x + y^2 - 150y + 49 = 0
Next, complete the square for the y-terms by adding (150/2)^2 = 11250 to both sides of the equation:
x^2 - 14x + y^2 - 150y + 11250 + 49 = 11250
This simplifies to:
x^2 - 14x + y^2 - 150y + 11399 = 11250
Now, rearrange the equation:
x^2 - 14x + y^2 - 150y = -149
To find the center of the circle, we need to isolate the x and y terms. We do this by completing the square for both the x and y variables separately.
For the x-terms:
x^2 - 14x = (x - 7)^2 - 49
For the y-terms:
y^2 - 150y = (y - 75)^2 - 5625
Now we can rewrite the equation in standard form:
(x - 7)^2 - 49 + (y - 75)^2 - 5625 = -149
(x - 7)^2 + (y - 75)^2 = 5425
Comparing this equation to the standard form of a circle, (x - h)^2 + (y - k)^2 = r^2, we can see that the center of the circle is at the point (7, 75) and the radius is sqrt(5425).
Since the maximum height above ground is the y-coordinate of the highest point on the circle, we can see that the highest point occurs at the center of the circle, which is y = 75.
Therefore, the maximum height above ground of the riders on the ferris wheel is 75 feet.