A rectangle in the first quadrant has one vertex as the origin, and other vertex on the curve y= -x^2 -2x +4. If the rectangle has one side lying on the x-axis, find the largest possible area of the rectangle.

square

Why did the rectangle go to the doctor?

Because it felt a little square!

Now, let's solve your mathematical puzzle, shall we?

To find the largest possible area of the rectangle, we need to maximize the length and width of the rectangle.

Since one side of the rectangle lies on the x-axis, we know that the width of the rectangle will be the x-coordinate of the vertex on the curve y = -x^2 - 2x + 4.

The x-coordinate of the vertex can be found by taking the derivative of the equation and setting it equal to zero:

dy/dx = -2x - 2 = 0

Solving for x, we get x = -1.

So, the width of the rectangle is 1.

To find the length, we need to substitute x = -1 into the equation y = -x^2 - 2x + 4:

y = -(-1)^2 - 2(-1) + 4
= -1 + 2 + 4
= 5.

So, the length of the rectangle is 5.

Now, we can find the area of the rectangle by multiplying the length and width:

Area = length x width
= 5 x 1
= 5.

Hence, the largest possible area of the rectangle is 5 square units.

To find the largest possible area of the rectangle, we need to maximize its dimensions.

Let's assume the rectangle's side lying on the x-axis has length 'x'.

Since the rectangle has one vertex at the origin, the other vertex on the curve y = -x^2 -2x +4 will have coordinates (x, -x^2 -2x +4).

Now, the area of the rectangle is given by the product of its dimensions, which is equal to the length of the side lying on the x-axis (x) multiplied by the height (y).

Therefore, the area (A) of the rectangle is given by:
A = x * (-x^2 -2x +4)

To maximize the area, we need to find the value of x that maximizes the above expression.

To find the value of x, we differentiate the expression for A with respect to x and set it equal to 0 to find the critical points:

dA/dx = -x^2 -2x +4 - 2x = -x^2 - 4x + 4

Setting dA/dx = 0, we have:

-x^2 - 4x + 4 = 0

Simplifying further, we get:

x^2 + 4x - 4 = 0

We can solve this quadratic equation using methods such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of x:

x = (-4 ± √(4^2 - 4*1*(-4))) / (2*1)
x = (-4 ± √(16 + 16)) / 2
x = (-4 ± √32) / 2
x = (-4 ± 4√2) / 2
x = -2 ± 2√2

Therefore, the two possible critical points are x = -2 + 2√2 and x = -2 - 2√2.

Next, we need to determine which critical point maximizes the area. To do this, we substitute these values of x into the expression for A and compare the results.

For x = -2 + 2√2:

A = (-2 + 2√2) * (-(2 - 2√2)^2 - 2(2 - 2√2) + 4)
= (-2 + 2√2) * (-(4 - 8√2 + 8 - 4√2 + 4) - (4 - 4√2) + 4)
= (-2 + 2√2) * (-(-8√2 + 12 - 4√2))
= (-2 + 2√2) * (8√2 - 12 + 4√2)
= (-2 + 2√2) * (12√2 - 12)
= -24 + 24√2 - 24√2 + 24
= -24 + 24
= 0

For x = -2 - 2√2:

A = (-2 - 2√2) * (-(2 + 2√2)^2 - 2(2 + 2√2) + 4)
= (-2 - 2√2) * (-(4 + 8√2 + 8 + 4√2 + 4) - (4 + 4√2) + 4)
= (-2 - 2√2) * (-(-8√2 - 12 - 4√2))
= (-2 - 2√2) * (-12 - 12√2 + 4√2)
= (-2 - 2√2) * (12√2 - 12)
= -24√2 + 24 - 24√2 + 24
= -48√2 + 48

Comparing the values obtained, we see that the area is greater for x = -2 - 2√2.

Therefore, the largest possible area of the rectangle is equal to -48√2 + 48 (approximately 16.97 square units).

To find the largest possible area of the rectangle, we need to maximize the product of its dimensions.

Let's start by defining the rectangle. We know that one of its vertices is the origin (0,0), and the other vertex lies on the curve y = -x^2 - 2x + 4. Since the x-axis is one side of the rectangle, the opposite side will have a length equal to the x-coordinate of the point on the curve.

Let's find the x-coordinate of the vertex on the curve. To do that, we can take the derivative of the curve and set it equal to zero to find the critical points.

The equation of the curve is y = -x^2 - 2x + 4. Taking the derivative with respect to x gives us:
dy/dx = -2x - 2

To find the critical points, we set dy/dx equal to zero and solve for x:
-2x - 2 = 0
-2x = 2
x = -1

So the x-coordinate of the vertex on the curve is -1.

Now let's find the y-coordinate of this point by substituting x = -1 into the equation y = -x^2 - 2x + 4.
y = -(-1)^2 - 2(-1) + 4
y = -1 + 2 + 4
y = 5

Therefore, the vertex on the curve is (-1, 5).

Now we can determine the length of the rectangle's side lying on the x-axis. It is simply the absolute value of the x-coordinate, which is |x| = |-1| = 1.

So the length of the rectangle's side lying on the x-axis is 1, and the width is the y-coordinate of the vertex, which is 5.

Finally, we can calculate the area of the rectangle by multiplying the length and width:
Area = length × width
Area = 1 × 5
Area = 5

Therefore, the largest possible area of the rectangle is 5 square units.

let the vertex which lies on the parabola be (x,y)

So the base of the rectangle is x
and its height is y
Area = xy
but y = -x^2 - 2x + 4
Area = x(-x^2 - 2x + 4)
= -x^3 - 2x^2 + 4x
d(Area)/dx = -3x^2 - 4x + 4 = 0 for a max area
3x^2 + 4x - 4 = 0
(3x-2)(x+2) = 0
x = 2/3 or x is a negative, which is no good

Max area = -(2/3)^3 - 2(2/3)^2 + 4(2/3)
= 40/27