Find the maximum area of a rectangle inscribed in the region bounded by the graph of
y = (3-x)/(2+x)
and the axes. (Round your answer to four decimal places.)
To find the maximum area of a rectangle inscribed in the region bounded by the graph of y = (3-x)/(2+x) and the axes, we need to analyze the critical points of the function.
First, let's find the x-intercepts of the graph:
To find the x-intercept, we set y = 0:
0 = (3-x)/(2+x)
Now, we solve for x:
3-x = 0
x = 3
So, one x-intercept is x = 3.
Next, let's find the y-intercept of the graph:
To find the y-intercept, we set x = 0:
y = (3-0)/(2+0)
y = 3/2
y = 1.5
So, the y-intercept is y = 1.5.
Now, let's determine the critical points. Since we are dealing with a rational function, we need to consider the points where the function is undefined.
The function is undefined when the denominator is equal to zero. So, we set 2+x = 0 and solve for x:
2+x = 0
x = -2
So, the critical point is x = -2.
To find the maximum area, we need to maximize the product of the length and width of the rectangle. In this case, the length of the rectangle will be the distance between the x-intercepts, which is |3 - (-2)| = 5, and the width of the rectangle will be the distance between the y-intercept and the graph, which is 1.5 - 0 = 1.5.
Therefore, the maximum area of the rectangle is 5 * 1.5 = 7.5.
Therefore, the maximum area of the rectangle inscribed in the region bounded by the graph of y = (3-x)/(2+x) and the axes is 7.5 (rounded to four decimal places).
To find the maximum area of a rectangle inscribed in the region bounded by the graph of the given function and the axes, we need to follow these steps:
1. Sketch the graph of the given function to visualize the shape of the region.
2. Recognize that the rectangle will have two vertices on the x-axis and two vertices on the y-axis.
3. Let the length of the rectangle parallel to the x-axis be 2x, where x is a positive number.
4. Determine the equation of the horizontal line passing through (x, 0), which represents the top side of the rectangle.
5. Find the points of intersection between the horizontal line and the curve given by the function.
6. Use these points of intersection to form a function for the height of the rectangle, utilizing the given information.
7. Express the area of the rectangle in terms of x.
8. Find the derivative of the area function with respect to x.
9. Set the derivative equal to zero and solve for x to find critical points.
10. Use the second derivative test to determine whether the critical point is a maximum or minimum.
11. Evaluate the area function at the critical points and endpoints to find the maximum area.
12. Round the answer to four decimal places.
Let's follow these steps to find the maximum area:
1. Sketch the graph of the given function by plotting some points and connecting them to get the overall shape of the curve.
2. Recognize that the rectangle will have two vertices on the x-axis and two vertices on the y-axis.
3. Let the length of the rectangle parallel to the x-axis be 2x, where x is a positive number. This means that the width of the rectangle parallel to the y-axis will be (3 - x)/(2 + x).
4. Determine the equation of the horizontal line passing through (x, 0), which represents the top side of the rectangle. The equation would be y = k, where k is a constant.
5. Find the points of intersection between the horizontal line and the curve given by the function. To do this, we set y = k and solve for x in the equation: (3 - x)/(2 + x) = k
6. Use these points of intersection to form a function for the height of the rectangle, utilizing the given information. Since the height will be the difference between the y-coordinate of the point on the curve and the y-coordinate of the horizontal line, we get: height = (3 - x)/(2 + x) - k
7. Express the area of the rectangle in terms of x. The area can be calculated by multiplying the length (2x) by the height, so the area function is: A = 2x * ((3 - x)/(2 + x) - k)
8. Find the derivative of the area function with respect to x. This can be done by applying the product rule and simplifying the expression.
9. Set the derivative equal to zero and solve for x to find critical points.
10. Use the second derivative test to determine whether the critical point is a maximum or minimum. To do this, find the second derivative of the area function and evaluate it at the critical points.
11. Evaluate the area function at the critical points and endpoints to find the maximum area. Calculate the area using the x-values of the critical points and the endpoints, and compare the results.
12. Round the answer to four decimal places to obtain the maximum area of the rectangle inscribed in the region bounded by the graph of the given function and the axes.
since the center of the rectangle will lie at (-2,-1) if it extends x units right and left of x=-2, then the area of the upper-right quarter is x*5/x = 5
So, the rectangle has a constant area of 5!
If you don't like that answer, obtained by shifting the graph right 2 and up 1 so that it becomes just y=5/x then plug in the real x-y values. If the upper right coordinate of the square is (x,y) then the upper-right area is
(x+2)[((3-x))/((x+2))+1]
= (x+2)(3-x+x+2)/(x+2)
= (x+2)(5)/(x+2)
= 5