a rectangle has one corner in quadrant I on the graph of y=25-x^2, another at the origin, a third at the positive y-axis, and the fourth on the positive x-axis
To find the dimensions of the rectangle, we need to determine the coordinates of its corners.
Corner 1: Quadrant I on the graph of y=25-x^2
Since the x-coordinate is positive in quadrant I, we can set x to a positive value. Plugging in x=1 into y=25-x^2 gives us:
y = 25 - (1)^2
y = 24
So, the coordinates of corner 1 are (1, 24).
Corner 2: The origin
The origin is (0, 0).
Corner 3: The positive y-axis
Since this corner lies on the positive y-axis, the x-coordinate is 0. Therefore, the coordinates of this corner are (0, y).
Corner 4: The positive x-axis
Since this corner lies on the positive x-axis, the y-coordinate is 0. Therefore, the coordinates of this corner are (x, 0).
To determine the coordinate of the corner on the positive y-axis, we need to find the y-value when x=0 in the equation y = 25 - x^2:
y = 25 - (0)^2
y = 25
So, the coordinates of corner 3 are (0, 25).
To determine the coordinate of the corner on the positive x-axis, we need to find the x-value when y=0 in the equation y = 25 - x^2:
0 = 25 - x^2
x^2 = 25
x = ±√25
Since the rectangle lies in quadrant I, the x-coordinate will be positive. So, x = √25 = 5.
Therefore, the coordinates of corner 4 are (5, 0).
So, the coordinates of the corners of the rectangle are:
Corner 1: (1, 24)
Corner 2: (0, 0) (origin)
Corner 3: (0, 25)
Corner 4: (5, 0)
To find the coordinates of the rectangle, we need to identify the points where it intersects with the given graph and the axes.
First, let's find the intersection points between the graph of y = 25 - x^2 and the positive x-axis. When y = 0, we have:
0 = 25 - x^2
Solving for x, we get:
x^2 = 25
Taking the square root of both sides, we have:
x = ±√25
So, x = ±5. However, since we need the intersection point on the positive x-axis, our x-coordinate is 5.
Next, let's find the intersection points between the graph and the positive y-axis. When x = 0, we have:
y = 25 - 0^2
y = 25
So, our y-coordinate is 25.
So far, we have the following points on the graph: (0, 25) and (5, 0).
Now, let's determine the point where the rectangle intersects with the origin, which is (0, 0).
Finally, we need to find the last point in quadrant I with the given conditions. Since the rectangle has one corner in quadrant I and another at the origin, the last point must lie on the positive y-axis.
Therefore, the last point is (0, 25), which is the same as the point where the graph intersects with the positive y-axis.
In summary, the coordinates of the four corners of the rectangle are:
(0, 0), (5, 0), (0, 25), and (0, 0).