Find the dimensions of the rectangle with the largest area if the base must be on the x-axis and
its other two corners are on the graph of:
(a) y=16-x²,-4<=x<=4
(b)x²+y²=1
(c)|x|+|y|=1
(d)y=cos(x),-pi/2<=x<=pi/2
a) let the top right vertex be (x,y)
then the area = 2xy
A = 2xy = 2x(16-x^2)
= 32x - 2x^3
dA/dx = 32 - 6x^2
= 0 for a max of A
6x^2 = 32
3x^2 = 16
x^2 = 16/3
x = 4/√3 , then y = 16 - 16/3 = 32/3
so the rectangle has a base of 2(4/√3) = 8/√3
and a height of 32/3
b) follow the same method
c) nice symmetry
http://www.wolframalpha.com/input/?i=%7Cx%7C%2B%7Cy%7C%3D1
let (x, y) be the point of contact
the straight line in first quadrant : y = -x + 1
follow the same steps as a)
d) y = cosx, again, nice symmetry,
make a sketch
Same kind of problem
A = 2xy
= 2x(cosx)
dA/dx = x(-sinx) + cosx
= 0 for a max of A
xsinx = cosx
hard to solve, I will use Wolfram
http://www.wolframalpha.com/input/?i=xsinx+%3D+cosx
x = .8603336 , y = .652185
the rectangle is 1.72067 by .652185
To find the dimensions of the rectangle with the largest area, we need to consider the given constraints for the position of its corners. Let's go through each option:
(a) y = 16 - x², -4 <= x <= 4:
In this case, the base of the rectangle must be on the x-axis, so it will have a length of 2a, where a is the distance from the y-axis to one of the sides of the rectangle. Since the function is symmetric about the y-axis, we can focus on the right side of the graph.
The area of the rectangle will be A = 2a * (16 - a²). To find the maximum area, we can differentiate this expression with respect to a and set it equal to zero:
dA/da = 2(16 - 3a²)
0 = 2(16 - 3a²)
16 = 3a²
a² = 16/3
a = √(16/3)
So one side of the rectangle will be 2a = 2√(16/3), and the other side will be 16 - a² = 16 - (16/3) = 48/3 - 16/3 = 32/3.
(b) x² + y² = 1:
In this case, the rectangle must have two corners on the graph of a circle centered at the origin with a radius of 1. Since a rectangle with corners on a circle is a square, the sides of the square will have equal lengths.
To find the maximum area, we need to find the largest possible square that can fit inside the circle. The side of the square will be equal to the diameter of the circle, which is 2.
So the dimensions of the square (and hence the rectangle) will be 2 units by 2 units.
(c) |x| + |y| = 1:
In this case, the rectangle must have two corners on the graph of a diamond shape centered at the origin with sides of length 1.
To find the maximum area, we need to find the largest possible rectangle that can fit inside the diamond shape. It can be shown that the largest rectangle will be a square with sides of length 1.
So the dimensions of the square (and hence the rectangle) will be 1 unit by 1 unit.
(d) y = cos(x), -π/2 <= x <= π/2:
In this case, the base of the rectangle must be on the x-axis, so it will have a length of 2a, where a is the distance from the y-axis to one of the sides of the rectangle.
The area of the rectangle will be A = 2a * cos(x). Since the function cos(x) is at its maximum value of 1 when x = 0, we can maximize the area of the rectangle by maximizing the value of a.
Since -π/2 <= x <= π/2, we want a to be as large as possible. The distance from the y-axis to the x-axis is 1, so a = 1.
So one side of the rectangle will be 2a = 2, and the other side will be cos(x), which has a maximum value of 1.
Therefore, the dimensions of the rectangle with the largest area in this case will be 2 units by 1 unit.
In summary:
(a) Rectangle dimensions: 2√(16/3) units by 32/3 units
(b) Rectangle dimensions: 2 units by 2 units
(c) Rectangle dimensions: 1 unit by 1 unit
(d) Rectangle dimensions: 2 units by 1 unit