What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 27 - x2
To find the area of the largest rectangle, we need to determine the dimensions that maximize the area. Let's break it down into steps:
Step 1: Understand the problem
The problem asks for the area of the largest rectangle with one side on the x-axis and the opposite vertices on the curve y = 27 - x^2. We want to find the dimensions (length and width) of this rectangle that give us the maximum area.
Step 2: Determine the dimensions
Let's consider the rectangle formed by two vertical sides parallel to the y-axis and two horizontal sides parallel to the x-axis. The length of the rectangle will be the difference between the x-coordinates of the upper vertices on the curve, and the width will be the y-coordinate of the upper vertices.
Step 3: Find the upper vertices
To find the x-coordinate of the upper vertices, we set y = 27 - x^2 equal to zero since the rectangle's upper vertices lie on the curve. Solving for x gives us x^2 = 27, so x = ±√27 = ±3√3. Since we want the lower base on the x-axis, we take x = -3√3.
Step 4: Determine the width
The width of the rectangle will be the y-coordinate of the upper vertices. Substituting x = -3√3 into y = 27 - x^2 gives us y = 27 - (-3√3)^2 = 27 - 27 = 0.
Step 5: Calculate the area
Now that we have the dimensions, we can calculate the area using the formula Area = length * width. The length is given by the difference between the x-coordinates of the upper vertices, so length = 2 * 3√3 = 6√3. Since the width is 0, the area of the largest rectangle is 0.
Therefore, the area of the largest rectangle with a lower base on the x-axis and upper vertices on the curve y = 27 - x^2 is 0.
To find the area of the largest rectangle with the lower base on the x-axis and upper vertices on the curve y = 27 - x^2, we need to find the maximum area.
Step 1: First, let's find the x-values of the upper vertices of the curve by setting y = 27 - x^2 equal to zero.
27 - x^2 = 0
Step 2: Solve for x by moving x^2 to the other side.
x^2 = 27
Step 3: Take the square root of both sides to isolate x.
x = ± √27
However, we are looking for the largest rectangle with the lower base on the x-axis, so we only consider the positive square root, x = √27.
Step 4: Next, we need to find the y-coordinate of the upper vertices. Substitute the value of x into the equation y = 27 - x^2.
y = 27 - (√27)^2
y = 27 - 27
y = 0
Therefore, the upper vertices of the rectangle are (√27, 0).
Step 5: Since the lower base of the rectangle is on the x-axis, the lower vertices are (0, 0).
Step 6: Now we can find the length and width of the rectangle. The length is the difference between the x-coordinates of the upper and lower vertices, which is √27 - 0 = √27. The width is the y-coordinate of the upper vertices, which is 0 - 0 = 0.
Step 7: Finally, we can calculate the area of the rectangle by multiplying the length and width.
Area = length * width
Area = √27 * 0
Area = 0
Therefore, the area of the largest rectangle with the lower base on the x-axis and upper vertices on the curve y = 27 - x^2 is 0.
Did you make a sketch ?
let the vertex in quadrant I be (x,y)
then the vertex in quadratnt II is (-x,y)
the base of our rectange = 2x
and the height is y
Area = xy
= x(27 - x^2)
= -x^3 + 27x
d(area)/dx = 3x^2 - 27
= 0 for a max of area
3x^2 = 27
x^2 = 9
x = ±3
y = 27-9 = 18
largest area = 3(18) = 54 square units