A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=9-x^2. What are the dimensions of such a rectangle with the greatest possible area?
Well, this is an upside down parabola (sheds water) with x intercepts at x = -3 and x = +3 and vertex at (0,9)
so lets do it just for positive x and double the base at the end.
Area = x y
where y = 9-x^2
A = 9x - x^3
dA/dx = 9 - 3 x^2
that is zero when x^2 = 3
or x = +/- sqrt 3 use + sqrt 3
so y = 9 - 3 = 6
Now we double the base because we only did the right half
base = 2 sqrt 3
height = 6
let the point of contact in the first quadrant be (x,y)
then the base of the rectangle is 2x and its height is y
Area = 2xy
= 2x(9-x^2)
= 18x - 2x^2
d(area)/dx = 18 - 4x
so 18-4x=0
x = ....
take it from here, let me know what you got
change
<< = 18x - 2x^2 >> to
= 18x - 2x^3
then d(area)/dx = 18 - 6x^2
= 0 for a max/min of area
6x^2 = 18
x^2 = 3
x = ±√3
and y = 6
same result as Damon
how do I write a written equation? A jacket cots 28.00 more than twice the cost of a pair of slacks. If the jacket costs 152.00, how much do th slacks cost?
To find the dimensions of the rectangle with the greatest possible area, we need to maximize the area of the rectangle.
Let's consider the rectangle's dimensions:
1. The base of the rectangle lies on the x-axis, so it is a line segment parallel to the x-axis.
2. The upper corners of the rectangle lie on the parabola y = 9 - x^2. These points satisfy the equation of the parabola, and we need to find their x-coordinates.
To find the dimensions of the rectangle, we need to find the x-coordinates where the parabola intersects the x-axis. This occurs when y = 0.
So, we set y = 0 in the equation of the parabola and solve for x:
0 = 9 - x^2
Rearranging the equation, we have:
x^2 = 9
Taking the square root of both sides, we get:
x = ±√9
Since we are considering the rectangle in the coordinate plane, we will ignore negative values of x. So, x = √9 = 3 is the x-coordinate of both upper corners of the rectangle.
Now that we have the x-coordinate, we can find the y-coordinate on the parabola:
y = 9 - (3)^2 = 9 - 9 = 0
Therefore, the coordinates of the two upper corners of the rectangle are (3, 0) and (-3, 0).
The length of the base of the rectangle is the distance between the x-coordinates of the two upper corners, which is equal to the absolute difference between the x-values:
Length of base = |3 - (-3)| = 6
To find the height of the rectangle, we need to find the y-coordinate of the upper corners:
Height = Distance from the x-axis to the parabola
Since the parabola is symmetric about the y-axis, the distance from the x-axis to the parabola at x = 3 is the same as the distance at x = -3.
The distance from the x-axis to the parabola can be found by evaluating the equation of the parabola at x = 3 (or -3):
y = 9 - (3)^2 = 9 - 9 = 0
So, the height of the rectangle is equal to the y-coordinate of the upper corners, which is 0.
Now we can calculate the area of the rectangle:
Area = Length × Height = 6 × 0 = 0.
Therefore, the rectangle with the greatest possible area has dimensions of 6 units for the base and 0 units for the height, resulting in an area of 0 units squared.