What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y equals 27 minus x^2?
if the base has width 2x, then the area is
a = 2xy = 2x(27-x^2) = 54x - 2x^3
da/dx = 54 - 6x^2 = 6(9-x^2)
da/dx=0 when x=3
So, the rectangle is 6 by 18 with area=108
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: Understanding the Problem
The problem asks for the largest rectangle, so we need to find the dimensions of the rectangle that maximize its area. The rectangle's lower base is on the x-axis, so its height will be determined by the curve y = 27 - x^2.
Step 2: Finding the Dimensions of the Rectangle
Let's consider a rectangle with one of its vertices at (x, y). The dimensions of this rectangle are:
- Base: x (on the x-axis by definition)
- Height: y (determined by the curve y = 27 - x^2)
Step 3: Finding the Area of the Rectangle
The area of a rectangle is calculated by multiplying its base by its height. In this case, the area (A) can be expressed as A = x * y.
Step 4: Expressing the Area in terms of x
Substituting y = 27 - x^2 into the area formula, we get A = x * (27 - x^2), which simplifies to A = 27x - x^3.
Step 5: Finding the Maximum Area
To find the maximum area, we differentiate the area function A with respect to x and set it equal to zero:
dA/dx = 27 - 3x^2 = 0
Solving this equation, we find that x = ±√9. Since the rectangle has its lower base on the x-axis, we consider only the positive value x = √9 = 3.
Step 6: Calculating the Area
Substituting x = 3 back into the area function A = 27x - x^3, we get:
A = 27(3) - (3)^3 = 81
The maximum area of the rectangle is 81 square units.
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 81 square units.