A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=2–x2. What are the dimensions of such a rectangle with the greatest possible area?
I got Width = 4/1.732 but i don't know why its wrong
I got Height = 4/3 which is right
I assume your equation is y = 2-x^2
The the contact point of the rectangle with the curve be (x,y) in the first quadrant
so the base is 2x, and the height is y
Area = 2xy
= 2x(2-x^2)
= 4x - 2x^3
d(Area)/dx = 4 - 6x^2 = 0 for a max of Area
6x^2 = 4
x^2 = 4/6 = 2/3
x = .8165
so the base is 2x = 1.633
and the height is 2 - (.8165^2 = 1.3333
To find the dimensions of the rectangle with the greatest possible area, we can start by understanding the problem and visualizing the situation.
First, let's consider the rectangle being inscribed within the parabola y = 2 - x^2. The base of the rectangle is on the x-axis, so the rectangle's height will be represented by the y-value of the parabola at the x-coordinate of the rectangle's upper corners. Let's call these x-coordinates a and -a, respectively.
So, the height of the rectangle is given by h = 2 - x^2, where x = a or x = -a.
Next, we need to determine the width of the rectangle. Since the rectangle is bounded by the y-axis and is symmetric about the y-axis, it will have a width of 2a.
The formula for the area of a rectangle is given by A = length * width. In our case, the area of the rectangle is A = (2 - a^2) * 2a.
To find the dimensions of the rectangle that maximize the area, we need to find the maximum value of the area function A.
We can do this by taking the derivative of A with respect to a, setting it equal to zero, and solving for a.
Let's carry out these steps.
1. Find the area function A:
A = (2 - a^2) * 2a
A = 4a - 2a^3
2. Take the derivative of A:
dA/da = 4 - 6a^2
3. Set the derivative equal to zero and solve for a:
4 - 6a^2 = 0
6a^2 = 4
a^2 = 4/6
a = sqrt(4/6) or a = -sqrt(4/6)
Since we are looking for the dimensions of a rectangle, we only consider positive values of a. Therefore, a = sqrt(4/6) ≈ 0.816.
Now, we know that the width of the rectangle is 2a, so the width is approximately 2 * 0.816 ≈ 1.632.
For the height, we substitute the value of a back into the equation for h:
h = 2 - a^2
h ≈ 2 - (0.816)^2
h ≈ 2 - 0.666
h ≈ 1.334
Therefore, the dimensions of the rectangle with the greatest possible area are approximately width = 1.632 and height = 1.334.
Now, let's address the values you calculated.
Width = 4/1.732 is incorrect. It seems you used the square root of 3 (1.732) as the factor for finding the width. However, the correct width is 2a, so substituting a = sqrt(4/6) into 2a gives us approximately 1.632.
Height = 4/3 is correct. You have correctly found the height of the rectangle by substituting the value of a into the equation for h, giving us approximately 1.334.
It's important to carefully follow the steps and use the correct formulas and calculations to find the dimensions of the rectangle with the greatest possible area.