Find the area of the largest rectangle with sides parallel to the coordinate axes which can be inscribed in the area bounded by the two parabolas y=26-x^2 and y=x^2+2
Make a sketch and draw in the rectangle.
Let the top-right point be P(x,26-x^2)
and the bottom-right point be Q(x, x^2 + 2)
So the width of the rectangle is 2x
and the height is 26-x^2 - x^2 - 2
= 24 - 2x^2
area = 2x(24-2x^2)
= 48x - 4x^3
d(area)/dx = 48 - 12x^2
= 0 for a max of area
12x^2 = 48
x^2 = 4
x = 2
the largest area = 48(2) - 4(2^3)
= 96 - 32
= 64 units^2
thank you so much!
To find the largest rectangle that can be inscribed in the area bounded by the two parabolas, we need to find the maximum area.
Step 1: Determine the intersection points of the parabolas.
We need to solve the system of equations by setting the two parabolas equal to each other:
26 - x^2 = x^2 + 2
Simplifying the equation, we get:
2x^2 = 24
Dividing both sides by 2, we have:
x^2 = 12
Taking the square root of both sides, we find:
x = ±√12
x = ±2√3
Step 2: Evaluate the y-values of the intersection points.
Substituting the x-values into either of the equations, we get:
For x = 2√3, y = (2√3)^2 + 2 = 12 + 2 = 14
For x = -2√3, y = (-2√3)^2 + 2 = 12 + 2 = 14
So, the two intersection points are (2√3, 14) and (-2√3, 14).
Step 3: Calculate the width of the rectangle.
The width of the rectangle is the distance between the x-values of the intersection points. So, the width is given by:
Width = 2√3 - (-2√3) = 4√3
Step 4: Calculate the height of the rectangle.
To find the height of the rectangle, we need to evaluate the difference in y-values at the intersection points. So, the height is given by:
Height = 14 - 2 = 12
Step 5: Calculate the area of the rectangle.
The area of the rectangle is given by the product of its width and height. So, the area is given by:
Area = Width × Height = 4√3 × 12 = 48√3
Therefore, the area of the largest rectangle that can be inscribed in the area bounded by the two parabolas is 48√3 square units.
To find the area of the largest rectangle that can be inscribed in the area bounded by the two parabolas, we need to determine the coordinates of the rectangle's vertices.
Step 1: Find the x-coordinates of the intersection points of the two parabolas.
Set the equations of the parabolas equal to each other:
26 - x^2 = x^2 + 2
Rearrange this equation to get:
2x^2 = 24
Divide both sides by 2:
x^2 = 12
Take the square root of both sides to find the x-coordinates:
x = ±√12
x = ±2√3
Step 2: Find the corresponding y-coordinates.
Substitute the x-values into either of the parabola equations to find the y-values:
For x = 2√3:
y = (2√3)^2 + 2
y = 12 + 2
y = 14
For x = -2√3:
y = (-2√3)^2 + 2
y = 12 + 2
y = 14
Step 3: Determine the coordinates of the other two vertices of the rectangle.
Since the rectangle has sides parallel to the coordinate axes, the other two vertices will have the same y-coordinates as the found points, with the x-coordinates being the negative of the corresponding found points:
(2√3, 14), (-2√3, 14)
Step 4: Calculate the length and width of the rectangle.
The length of the rectangle is the difference between the x-coordinates of the vertices:
Length = 2√3 - (-2√3) = 4√3
The width of the rectangle is the difference between the y-coordinates of the vertices:
Width = 14 - 14 = 0
Step 5: Calculate the area of the rectangle.
The area of the rectangle is the product of its length and width:
Area = Length * Width = 4√3 * 0 = 0
Therefore, the area of the largest rectangle that can be inscribed in the area bounded by the two parabolas is 0.