so its this picture of a rectangle inscribed in a circle. the circle has a radius of 2. the vertexes of the circle are present in every quadrant, but you cant tell exactly whgat point they represent. point Q (x,y) is the vertex of the rectangle in Quadrant I and is on the circle. the equation of the circle is : x^2 + y^2 = 4

Question

so its this picture of a rectangle inscribed in a circle. the circle has a radius of 2. the vertexes of the circle are present in every quadrant, but you cant tell exactly whgat point they represent. point Q (x,y) is the vertex of the rectangle in Quadrant I and is on the circle. the equation of the circle is : x^2 + y^2 = 4

a.) express the area of the rectangle as a function of x.
b.) give the domain of the area
c.) epxress the perimeter as a function of x.
d.) give the domain of Q
PLEASE HELP IM SO LOST. and friday afternoon is my scheduled math homework day. (dont ask im psycho) please this is bother ing me like crazyy

Answer 1

a)

x^2 + y ^2 = 4
area = 4 |x y| (so it works in each quadrant)
y = sqrt(4-x^2)
area = 4 x sqrt(4-x^2)

b)
the area can be 0 (at x = 0 and x = 2)and it has a maximum at x = y = sqrt 2
to show that take derivative and set to zero
0 = -4 x^2/sqrt(4-x^2) +4 sqrt(4-x^2)
0 = -4x^2 + 16 - 4 x^2
8 x^2 = 16
x = sqrt 2
so domain of area is 0 to x = sqrt 2 where the area is 8

c)
L = 2 x + 2 y
= 2 x + 2 sqrt(4-x^2)
= 2 ( x + sqrt (4-x^2))

d)
Q goes from x = 0 to x = 2 in Quadrant 1

Answer 2

What do you mean?

Answer 3

I can help you with your math questions. Let's break down each question and solve them step by step:

a) To express the area of the rectangle as a function of x, we need to find the relationship between x and the sides of the rectangle. Since point Q is on the circle with a radius of 2, the distance from the origin (0,0) to point Q is also 2. So, we have the following relationship:

x^2 + y^2 = 4

Since point Q is in the first quadrant, both x and y are positive. We can rewrite the equation as:

x^2 + y^2 = 4
x^2 + (2 - x)^2 = 4

Simplifying the equation:

x^2 + 4 - 4x + x^2 = 4
2x^2 - 4x = 0

Factoring out 2x:

2x(x - 2) = 0

This gives us two possible solutions: x = 0 and x = 2. However, x cannot be 0 since it represents the length of a side of the rectangle, so we're left with x = 2.

Now, let's find the corresponding y-value for the point Q. Substituting x = 2 into the equation x^2 + y^2 = 4:

2^2 + y^2 = 4
4 + y^2 = 4
y^2 = 0

The equation simplifies to y = 0. So, the coordinates of the vertex Q are (2, 0).

Now, let's find the sides of the rectangle. The length of the rectangle is the diameter of the circle, which is equal to twice the radius (2 * 2 = 4). The width of the rectangle is the chord connecting the points Q and its corresponding point in the second quadrant. The width is equal to the distance between the y-coordinates of the points, which is |0 - (-2)| = 2.

Therefore, the area of the rectangle is A = length * width = 4 * 2 = 8.

So, the area of the rectangle as a function of x is A(x) = 8.

b) The domain of the area refers to the values of x for which the area function is defined. Since x represents the length of a side, it must be a positive number. Therefore, the domain of the area is x > 0.

c) To express the perimeter of the rectangle as a function of x, we need to find the lengths of all four sides. The length is given by the diameter of the circle, which is 4, and the width is 2 (as calculated in part a).

So, the perimeter is P = 2 * (length + width) = 2 * (4 + 2) = 12.

Therefore, the perimeter of the rectangle as a function of x is P(x) = 12.

d) The domain of Q represents the possible values of x for which point Q is on the circle. From part a, we found that x = 2 is the only valid solution. So, the domain of Q is x = 2.

I hope this helps you with your math homework. Let me know if you have any more questions!