rectangle is inscribed between the x-axis and the parabola y=16-x^2 wiht one side along the x-axis as shown below
I assume you want the size of the rectangle with the largest area?
Se the related questions below.
To find the dimensions of the rectangle, we need to determine the x-coordinate where the rectangle is located on the x-axis.
Since the rectangle is inscribed between the x-axis and the parabola, the top side of the rectangle will touch the parabola at two points. To find these points, we set y = 0 and solve for x:
0 = 16 - x^2
Rearranging the equation:
x^2 = 16
Taking the square root of both sides:
x = ±√16
x = ±4
This means the rectangle is located between x = -4 and x = 4 on the x-axis.
Next, let's find the height of the rectangle. Since the top side of the rectangle is formed by the parabola, the height will be the difference in y-coordinate between the points on the parabola where it intersects the x-axis (x = -4 and x = 4).
Substituting x = -4 into the equation of the parabola:
y = 16 - (-4)^2
y = 16 - 16
y = 0
Substituting x = 4 into the equation of the parabola:
y = 16 - 4^2
y = 16 - 16
y = 0
Therefore, the height of the rectangle is 0.
To find the width of the rectangle, we need to find the difference in x-coordinate between the two points on the x-axis where the rectangle is located (x = -4 and x = 4).
The width of the rectangle is 4 - (-4), which is equal to 8.
Thus, the dimensions of the rectangle are 8 for the width and 0 for the height.
To find the dimensions of the rectangle inscribed between the x-axis and the parabola y = 16 - x^2, we can start by visualizing the situation.
The rectangle is inscribed between the x-axis and the parabola, which means that two of its opposite vertices lie on the parabola, and the other two lie on the x-axis.
Let's assume the coordinates of the vertices of the rectangle are A(a, 0), B(b, 0) on the x-axis, and C(a, 16 - a^2), D(b, 16 - b^2) on the parabola.
Since the rectangle has sides parallel to the coordinate axes, we can deduce that the lengths of AB and CD are equal to the difference in x-coordinates, which is b - a.
Now, to find the coordinates of points C and D, we need to substitute the x-values into the equation of the parabola y = 16 - x^2.
For point C:
x-coordinate = a, y-coordinate = 16 - a^2.
So, C(a, 16 - a^2).
Similarly, for point D:
x-coordinate = b, y-coordinate = 16 - b^2.
So, D(b, 16 - b^2).
Now, let's find the x-values where the rectangle touches the parabola. Since the rectangle touches the parabola at points C and D, we need to find the values of a and b.
At these points, the y-coordinate of the parabola is zero because the rectangle lies on the x-axis. So, we have:
16 - a^2 = 0 => a^2 = 16 => a = ±4,
16 - b^2 = 0 => b^2 = 16 => b = ±4.
Now, we can calculate the width of the rectangle, which is the difference between the x-coordinates b - a:
b - a = (4) - (-4) = 8.
So, the width of the rectangle is 8 units.