A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=6–x^2. What are the dimensions of such a rectangle with the greatest possible area?
Find Width=____ & Height=4
just need to find width.. is not 2
let the point of contact be (x,y)
then the base is 2x and the height is y
area = 2xy = 2x(6-x^2) = 12x - 2x^3
d(area)/dx = 12 - 6x^2
= 0 for a max of area
6x^2 = 12
x^2 = 2
x = √2, then y = 6-2 = 4
rectangle has a base of 2√2 and a height of 4
To find the dimensions of the rectangle with the greatest possible area, we can start by determining its height. As given in the question, the height of the rectangle is 4.
Now, let's find the width of the rectangle. Since the base of the rectangle is on the x-axis, it means the top corners of the rectangle lie on the parabola y = 6 - x^2.
To find the width, we need to find the x-values of the top corners of the rectangle.
Setting y = 6 - x^2 = 4 (since the height of the rectangle is 4), we can solve for x:
6 - x^2 = 4
-x^2 = 4 - 6
-x^2 = -2
Dividing both sides by -1:
x^2 = 2
Taking the square root of both sides:
x = ±√2
Since the rectangle is inscribed in the parabola, the width cannot be negative. Thus, the width of the rectangle is √2.
Therefore, the dimensions of the rectangle with the greatest possible area are Width = √2 and Height = 4.
To find the dimensions of the rectangle with the greatest possible area, we need to consider that the base of the rectangle lies on the x-axis and its upper corners lie on the parabola y = 6 - x^2.
Let's denote the width of the rectangle as w and its height as h. Since the base of the rectangle lies on the x-axis, the width of the rectangle is equal to the distance between the x-coordinates of its upper corners. Thus, the width is given by the difference between the x-coordinates of the points where the parabola intersects the x-axis.
To find the x-coordinates of the points where the parabola intersects the x-axis, we set y = 0 in the equation of the parabola:
0 = 6 - x^2
Simplifying the equation, we get:
x^2 = 6
Taking the square root of both sides, we find two possible x-coordinates:
x = √6 and x = -√6
Since the width of the rectangle is the difference between these two x-coordinates, we have:
width (w) = √6 - (-√6) = √6 + √6 = 2√6
Therefore, the width of the rectangle is 2√6.
Since we are given that the height is 4, the dimensions of the rectangle with the greatest possible area are:
Width = 2√6
Height = 4