The figure below shows the curves y=square root of x, x=9, y=0 and a rectangle with the sides parallel to the axes and its left end at x=a. Find the dimensions of the rectangle having the maximum possible area.
If the lower left corner is at x, the height is sqrt(x) and the width is (9-x)
a = 9x^1/2 - x^3/2
a' = 9/(2x^1/2) - 3/2 x^1/2
= 3/(2x^1/2) * (3 - x)
a' = 0 at x=3
a(3) = sqrt(3) * (9-3) = 6sqrt(3)
To find the dimensions of the rectangle with the maximum possible area, we need to first understand the problem and analyze the given information.
We are given a rectangle with one side parallel to the x-axis, represented by x = a. The other side of the rectangle lies along the curve y = √x. The rectangle's bottom-left corner is at the point (a, 0).
We are tasked with finding the dimensions of this rectangle that maximize its area.
To solve this problem, we need to apply optimization techniques. In this case, we can use calculus to find the maximum area.
Let's proceed step by step:
Step 1: Determine the area function.
The area of a rectangle is given by length × width. In this case, the length is represented by the x-coordinate of the rectangle's top-right corner (which is x = 9). The width is given by the function y = √x.
Therefore, the area function (A) is given by:
A = x × y = x × √x = x^(3/2)
Step 2: Find the derivative of the area function.
To find the maximum area, we need to find the critical points of the area function. This involves finding the derivative of the area function and setting it equal to zero.
Differentiating the area function with respect to x:
dA/dx = (3/2)x^(1/2)
Step 3: Set the derivative equal to zero and solve for x.
Setting dA/dx = 0, we have:
(3/2)x^(1/2) = 0
Simplifying the equation:
3x^(1/2) = 0
Since a square root cannot be zero, there is no critical point in the interval [0, 9].
Step 4: Determine the endpoints of the interval.
In this case, the interval is [0, 9] because the left side of the rectangle is at x = a, which can be any value between 0 and 9.
Step 5: Evaluate the area at the endpoints.
To determine the maximum area, we need to evaluate the area function at the endpoints of the interval [0, 9].
At x = 0:
A(0) = (0)^(3/2) = 0
At x = 9:
A(9) = (9)^(3/2) = 27
Step 6: Compare the values of the area.
Comparing the values, we find that the maximum area occurs at x = 9, where A = 27.
Step 7: Determine the rectangle's dimensions.
The rectangle's left end is at x = a, and the right end is at x = 9. The height of the rectangle is given by the function y = √x.
Therefore, the dimensions of the rectangle with maximum area are:
Length = x-coordinate of the top-right corner - x-coordinate of the bottom-left corner
= 9 - a
Width = y-coordinate of the top-right corner = √9 = 3
Thus, the dimensions of the rectangle with maximum area are (9 - a) × 3.