a rectangle with one side on the x axis and one side on the line x=2 has it upper left vertex on the graph of y=x^2, for what values of x does the area of the rectangle attain its maximum value?
height = y = x^2
width = 2-x
area = a = x^2(2-x)
a = 2 x^2 -x^3
da/dx = 0 for max/min = 4 x -3x^2
0 = x (4 - 3 x)
x = 4/3
Ah, the classic tale of geometry and parabolas. Let me entertain you with an answer that will tickle your mathematical funny bone.
Well, to find the maximum area of the rectangle, we need to optimize it. First, let's imagine the rectangle placed on the coordinate plane. We know that one side of the rectangle is on the x-axis, and the other side is on the line x = 2. So, the width of the rectangle is 2 - x, right?
Now, to find the length of the rectangle, we need to look at the graph of y = x^2. The upper left vertex of the rectangle is on this parabolic graph. As we move along the x-axis, the y-coordinate of the parabola increases, indicating the height of the rectangle.
To maximize the area, we need the rectangle to be as tall as possible. Since the vertex of the parabola is a minimum or maximum point, we want to find the x-coordinate of this vertex. Using some fancy calculus, we can determine that the x-coordinate of the vertex is x = 0.
So, the maximum area of the rectangle can be found when x = 0. At this value, the width of the rectangle is 2 - 0 = 2, and the height is the y-coordinate at x = 0 on the parabola, which is y = 0^2 = 0.
Hence, the maximum area is achieved when x = 0, resulting in a rectangle with a width of 2 and a height of 0. It may not be the most exciting rectangle to look at, but hey, it's all about optimization. Keep those mathematical laughs coming!
To find the area of the rectangle, we need to determine the length of the base on the x-axis and the height on the graph of y=x^2.
Let's start by finding the equation of the line x=2. Since the line is vertical and passes through the point (2, 0), its equation is simply x=2.
Now, let's find the equation of the graph y=x^2. This is a standard parabolic function and its equation is y=x^2.
To find the length of the base of the rectangle, we need to determine the x-coordinate of the point where the line x=2 intersects with the graph of y=x^2.
Setting both equations equal to each other, we have:
x^2 = 2
To solve this equation, we take the square root of both sides:
x = ±√2
Since we are looking for the positive x-values, the intersection point on the x-axis is x = √2.
Next, we need to find the corresponding y-coordinate on the graph of y=x^2. Substituting x = √2 into the equation, we have:
y = (√2)^2
y = 2
Therefore, the upper left vertex of the rectangle is (√2, 2).
Now, we can find the area of the rectangle by multiplying the base length and height:
Area = (2 - √2) * 2
= 4 - 2√2
To determine when the area attains its maximum value, we usually take the derivative of the area equation with respect to x, set it equal to zero, and solve for x. However, in this case, the area is already in simplified form, and its derivative is not necessary.
Therefore, the area of the rectangle attains its maximum value when x = √2.
So, the answer is x = √2.
To find the values of x for which the area of the rectangle attains its maximum value, we need to analyze the problem step by step.
First, let's visualize the situation described in the problem:
1. The rectangle has one side on the x-axis, which means that one of its sides will have a length of x. Let's call this side "width."
2. The other side of the rectangle lies on the vertical line x=2. Since the line is vertical, the height of the rectangle will be the difference between the y-coordinates of the upper left vertex and the lower right vertex.
3. The upper left vertex of the rectangle lies on the graph of y=x^2. So, the y-coordinate of this vertex is given by y=x^2. Let's call this coordinate "y1".
4. To find the y-coordinate of the lower right vertex, we need to substitute x=2 into the equation y=x^2. Thus, the y-coordinate of the lower right vertex is y=2^2=4. Let's call this coordinate "y2".
Now, let's calculate the area of the rectangle:
Area = width * height
Given that the width is x and the height is y2 - y1, the area becomes:
Area = x * (4 - x^2)
Now, to find the maximum area, we should take the derivative of the area function and set it equal to zero. Let's do that:
d(Area)/dx = 4 - 3x^2
Set the derivative equal to zero:
4 - 3x^2 = 0
Now, solve for x:
3x^2 = 4
x^2 = 4/3
x = ±√(4/3)
Therefore, the area of the rectangle attains its maximum value at x = ±√(4/3).
Note: Since the rectangle has one side on the x-axis, negative values of x will yield the same area as positive values (as the area is always positive). Thus, we can conclude that the area attains its maximum value when x = √(4/3).