A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=2-x^2
What are the dimensions of such a rectangle with the greatest possible area?
Width=?
Height=?
A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 7.5 . What are the dimensions of such a cylinder which has maximum volume?
To find the dimensions of the rectangle with the greatest possible area, we first need to understand the conditions given in the problem.
The base of the rectangle is on the x-axis, which means the length of the base will be equal to the width of the rectangle. Let's denote the width as "w".
The upper corners of the rectangle are on the parabola y = 2 - x^2. This means that the height of the rectangle will be y-coordinate of the upper corners. Let's denote the height as "h".
Now, let's find the expression for the area of the rectangle. The area of a rectangle is given by the formula: A = length * width.
In this case, the length is equal to the height (since the base of the rectangle is on the x-axis). So, the area can be expressed as: A = h * w.
To maximize the area, we need to maximize the expression A = h * w.
The y-coordinate of the upper corners of the rectangle is given by y = 2 - x^2. We can substitute this expression for y in the area equation:
A = (2 - x^2) * w
To find the greatest possible area, we need to find the values of x and w that maximize this expression. Since w is the width of the rectangle, it must be positive. Therefore, we only need to find the value of x that maximizes the expression.
The graph of the parabola y = 2 - x^2 is symmetric about the y-axis. The maximum value of the y-coordinate occurs at the vertex of the parabola. The vertex is the point where the parabola reaches its highest or lowest point.
The vertex of the parabola can be found using the formula x = -b/2a, where the parabola is in the form of ax^2 + bx + c.
In our case, the parabola is y = -x^2 + 2. Comparing it with the general form of the equation, we have a = -1, b = 0, and c = 2.
Using the formula x = -b/2a, we can find the x-coordinate of the vertex:
x = -0 / 2(-1) = 0
So, the x-coordinate of the vertex is 0.
Now, substitute the x-coordinate (0) into the equation for the y-coordinate of the parabola:
y = 2 - (0)^2 = 2
Therefore, the maximum value of y (which is the height of the rectangle) is 2.
Now that we have the maximum values for x and y, we can find the dimensions of the rectangle:
Width = x-coordinate of the upper corners = x-coordinate of the base = 0
Height = y-coordinate of the upper corners = 2
So, the dimensions of the rectangle with the greatest possible area are:
Width = 0
Height = 2
let the points of contact be (x,y) and (-x,y),
(there is symmetry in your parabola
Area of rectangle
= 2xy
= 2x(2 - x^2)
= 4x - 2x^3
differentiate, set the derivative equal to zero and solve for x
Easy from there.