Two vertices of a rectangle are on the positive x axis. The other two vertices lie on the lines y= 4x and y= -5x + 6. Then the maximum area of the rectangle is?

Draw a diagram. If the base of the rectangle is the segment (a,0) to (b,0) then the upper vertices are (a,4a) and (b,6-5b).

Since it is a rectangle, the height is the same, so 4a = 6-5b.
That is, b = (6-4a)/5

Now, the area is

(b-a)(4a) = ((6-4a)/5-a)(4a) = 12a/5 (2-3a)

This is a parabola with vertex at (1/3, 4/5).

check:
So, b=14/15

and the maximum area is

(14/15 - 1/3)(4/3) = (9/15)(4/3) = 4/5

To find the maximum area of the rectangle, we need to find the points on the positive x-axis and the lines y= 4x and y= -5x + 6 where the rectangle's vertices can be formed.

Let's start by finding the points on the positive x-axis. Since the positive x-axis lies on the y=0 line, the two points on the positive x-axis for the rectangle would be (x, 0) and (x', 0), where x and x' are the x-coordinates of the vertices we want to find.

Next, we need to find the points on the line y=4x and y=-5x+6 where the rectangle's vertices exist. These points can be found by substituting y=4x and y=-5x+6 into the equation of the positive x-axis.

For the point (x, 0):
0 = 4x
x = 0

For the point (x', 0):
0 = -5x' + 6
x' = 6/5

So, the points for the rectangle's vertices are (0, 0), (0, 0), (0, 0), and (6/5, 0), or simply (0, 0) and (6/5, 0).

Now, we can calculate the maximum area of the rectangle by finding the distance between these two points on the positive x-axis and multiplying it by the width of the rectangle, which is the difference between the y-coordinates of the two lines.

The distance between (0, 0) and (6/5, 0) is (6/5 - 0) = 6/5.

The width of the rectangle can be found by subtracting the y-coordinates of the two lines:
y=4x: 4x = 0 => x = 0 => y = 4(0) = 0
y= -5x + 6: -5x + 6 = 0 => -5x = -6 => x = 6/5 => y = -5(6/5) + 6 = 0

Therefore, the width of the rectangle is 0 - 0 = 0.

Lastly, we can calculate the maximum area of the rectangle by multiplying the distance and the width:
Area = Distance × Width = (6/5) × 0 = 0.

Therefore, the maximum area of the rectangle is 0.

To find the maximum area of the rectangle, we need to determine the length of its sides. Let's analyze the information given:

We know that two vertices of the rectangle lie on the positive x-axis. This means that those two points have coordinates (x1, 0) and (x2, 0), where x1 and x2 are the x-coordinates of the vertices.

We also know that the other two vertices lie on the lines y = 4x and y = -5x + 6. This means that the coordinates of those two points are (x, 4x) and (x, -5x + 6), where x is the x-coordinate of the vertices.

To find the length of the sides of the rectangle, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by:

d = √[(x2 - x1)^2 + (y2 - y1)^2]

Let's find the length of the sides of the rectangle:

1. Length of one side:
The two points on the positive x-axis have coordinates (x1, 0) and (x, 4x). Using the distance formula, we have:
d1 = √[(x - x1)^2 + (4x - 0)^2]
= √[(x - x1)^2 + (4x)^2]

2. Length of the other side:
The two points on the positive x-axis have coordinates (x1, 0) and (x, -5x + 6). Using the distance formula, we have:
d2 = √[(x - x1)^2 + (-5x + 6 - 0)^2]
= √[(x - x1)^2 + (-5x + 6)^2]

Now, the area of the rectangle is given by the product of the lengths of its sides. Let's calculate the area:

Area = d1 * d2
= √[(x - x1)^2 + (4x)^2] * √[(x - x1)^2 + (-5x + 6)^2]

To find the maximum area, we need to find the maximum value of this expression. We can achieve that by finding the critical points, which occur when the derivative of the expression is equal to zero. By differentiating the expression and setting it to zero, we can find the values of x that give us the maximum area.

Once we find the critical points, we can substitute them back into the expression for the area to find the maximum area of the rectangle.