A rectangle is bounded by the x-axis, the y-axis, and the line y = (4-x)/4. Find the maximum area of such a rectangle.
let the point on the line be (x,y)
line is y = (4-x)/4 = 1 - (1/4)x
area of rectangle:
A = xy
= x( 1 - (1/4)x)
= x - (1/4)x^2
this is a downwards parabola. We need the vertex
the x of the vertex is -b/2a = -1/(-1/2) = 2
then y = 1- (1/4)(2) = 1/2
max area = xy = 2(1/2) = 1
check my arithmetic
Thank you so much!
To find the maximum area of the rectangle bounded by the x-axis, y-axis, and the line y = (4-x)/4, we need to determine the dimensions that will give us the largest possible area.
Let's break down the problem into steps:
Step 1: Understand the problem.
The problem asks for the maximum area of a rectangle with specific boundaries.
Step 2: Define the variables.
Let's assume the length of the rectangle is x units and the height of the rectangle is y units.
Step 3: Formulate the area equation.
The area of a rectangle is calculated by multiplying its length (x) by its height (y): A = x * y.
Step 4: Determine the constraints.
We are given that the rectangle is bounded by the x-axis, y-axis, and the line y = (4 - x)/4.
Since the rectangle is bounded by the x-axis and y = (4 - x)/4, the height of the rectangle (y) can be represented as y = (4 - x)/4 within the given range of x.
Also, since the rectangle is bounded by the y-axis, the length (x) should be non-negative.
Step 5: Rewrite the area equation with the constraints.
Substitute the value of y from the given equation into the area equation:
A = x * ((4 - x)/4)
Simplify the equation:
A = (4x - x^2)/4
A = (4x - x^2) * (1/4)
A = x - (x^2)/4
Step 6: Find the critical points.
To find the maximum area, we need to find the critical points of the area equation by taking its derivative and setting it equal to zero.
dA/dx = 1 - (1/2)x
Setting dA/dx equal to zero:
1 - (1/2)x = 0
x = 2
Step 7: Check the endpoints.
Since the length (x) should be non-negative, we need to check the area at the endpoints of the given range.
When x = 0, A = 0.
When x = 4, A = 0.
Step 8: Determine the maximum area.
Comparing the critical point and the endpoints, we find that the only critical point is x = 2, which corresponds to the maximum area of the rectangle. Therefore, the maximum area is achieved when x = 2.
Substituting x = 2 into the area equation:
A = 2 - (2^2)/4
A = 2 - 1
A = 1
Therefore, the maximum area of the rectangle is 1 square unit.