Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y=4−x^2.
Width =
Height =
Why did it take 5 years to answer this question?
A=x(12-x^2),
A'=12-3x^2
Finding critical value,u find x=2
Then A(2)=16
Thus (2,16) is a critical point and
A''(2)=-8<0 therefore (2,16) is relative maximum point,
Solving for y from y=12-x^2 we get 8
Therefore;length (x)=2 & width(y)=8
Area= 16 units squared
To find the dimensions of the rectangle of largest area, we need to find the coordinates of the two vertices that lie on the parabola y=4−x^2. Let's call these vertices (x, y).
Since the base of the rectangle is on the x-axis, the y-coordinate of both vertices will be zero. Therefore, we need to find the x-coordinate of the vertices.
Setting y=0 in the equation of the parabola, we get:
4 - x^2 = 0
Solving for x:
x^2 = 4
x = ±√4
x = ±2
So, the two x-coordinates of the vertices are -2 and 2.
Now, we can calculate the width and height of the rectangle based on these coordinates.
The width of the rectangle is the difference between the x-coordinates of the two vertices:
Width = |2 - (-2)| = 4
The height of the rectangle is the y-coordinate of the vertices:
Height = 0 (Since the vertices lie on the x-axis)
Therefore,
Width = 4
Height = 0
To find the dimensions of the rectangle of the largest area, we need to determine the width and height of the rectangle.
Let's denote the width of the rectangle as "w" and the height as "h". Since the base of the rectangle lies on the x-axis, the width will be the distance between the x-intercepts of the parabola.
To get the x-intercepts, we set y = 0, which gives us:
4 - x^2 = 0
Solving this equation, we find two x-values:
x = 2 and x = -2
Therefore, the width of the rectangle is:
w = distance between x = -2 and x = 2
w = 2 - (-2) = 4
Now, let's find the height of the rectangle. The height will be the distance between the vertex of the parabola and the highest point on the parabola.
The vertex of the parabola y = 4 - x^2 is at (0, 4). To find the highest point, we need to find the y-coordinate when x = 0.
Substituting x = 0 in the equation, we get:
y = 4 - (0)^2
y = 4
Therefore, the height of the rectangle is:
h = 4 - 0 = 4
Hence, the dimensions of the rectangle of largest area are:
Width = 4
Height = 4