a rectangle has its base on the x-axis, and its upper corners in the graph of y=27-x^2. what is the maximal area of this rectangle?
a = 2x(27-x^2)
da/dx = 54-6x^2
max a at x=3
a(3) = 6*18 = 108
To find the maximal area of the rectangle, we need to determine the dimensions that maximize the area. Let's start by understanding the given information:
1. The base of the rectangle lies on the x-axis.
2. The upper corners of the rectangle lie on the graph of y = 27 - x^2.
To determine the dimensions, we need to find the x-coordinates of the upper corners. Since these points lie on the graph, we can set y = 27 - x^2 and solve for x:
27 - x^2 = 0
Rearranging the equation, we get:
x^2 = 27
Taking the square root of both sides, we have:
x = ±√27
Since the base of the rectangle lies on the x-axis, we are only interested in the positive x-coordinate:
x = √27
Now, we have the width of the rectangle. To find the height, we need to substitute the value of x into the equation y = 27 - x^2:
y = 27 - (√27)^2
y = 27 - 27
y = 0
The height of the rectangle is 0 since it lies on the x-axis.
Finally, we can calculate the area of the rectangle:
Area = base × height
Area = √27 × 0
Area = 0
Hence, the maximal area of the rectangle is 0.
To find the maximal area of a rectangle with its base on the x-axis and upper corners on the graph of y = 27 - x^2, we need to maximize the product of the base and the height of the rectangle.
Let's break down the problem into steps:
Step 1: Understand the problem
The problem asks for the maximum area of a rectangle, considering that its base lies on the x-axis and its upper corners lie on the graph of y = 27 - x^2.
Step 2: Understand the rectangle's dimensions
Since the base of the rectangle lies on the x-axis, the length of the base will be the difference between the x-coordinates of the two upper corners. We need to find the x-values at which y = 27 - x^2.
Step 3: Find the x-values of the upper corners
To find the x-values, we set y = 27 - x^2 equal to zero because the y-coordinate of the x-axis is zero. Solving for x in the equation 27 - x^2 = 0, we get x = ±√27. However, since the rectangle is on the base of the x-axis, we only consider the positive value, x = √27.
Step 4: Calculate the area of the rectangle
The height of the rectangle will be the y-coordinate of the upper corners on the graph, which is y = 27 - x^2. So, the height, h, of the rectangle is 27 - (√27)^2 = 27 - 27 = 0.
The area, A, of the rectangle is calculated by multiplying the base, b, and the height, h: A = b * h = √27 * 0 = 0.
Step 5: Understand the result
The maximum area of the rectangle is zero. This means that the rectangle with its base on the x-axis and upper corners on the graph of y = 27 - x^2 is actually just a line segment, rather than a rectangle with positive dimensions. Therefore, it doesn't have an "area" in the typical sense.
In conclusion, the maximal area of the rectangle is 0 (zero).
Note: It's important to carefully read the question and consider the conditions given to avoid situations where a rectangle may not be possible, as in this case.