A rectangle is inscribed in a circle of radius 2. Find a function that represents the area of the rectangle in terms of x.
I have
(x^2)(y^2)=4 Pythagorean Theorem
(y^2)=4-(x^2)
y=sqrt(4-(x^2))
A=bh or A=xy
A=2x*2sqrt(4-(x^2))
Is this correct or what am I doing wrong?
the diagonal of the rectangle is a diameter of the circle. Since that is 4, if the rectangle has one side of length x, then the other side has length √(16-x^2)
So, the area is x√(16-x^2)
Well, that is not a choice, so if the side is 2x, rather than x, the area is
(2x)√(16-4x^2)
= 4x√(4-x^2)
You are correct.
Thank you
Well, my mathematical skills might need some practice, but let me give it a try.
First, let's draw a diagram of the situation. We have a rectangle inscribed in a circle with radius 2. The radius is basically the hypotenuse of a right triangle formed by the half-length of the rectangle (x) and its height (y).
Using the Pythagorean theorem, we can write the equation:
(x/2)^2 + y^2 = 2^2
Simplifying this, we get:
x^2/4 + y^2 = 4
x^2 + 4y^2 = 16
Now we want to find the area of the rectangle, which is given by A = bh. In this case, the base (b) is 2x, and the height (h) is 2y. So we have:
A = 2x * 2y
Substituting the value of y from the previous equation, we get:
A = 2x * 2 sqrt(4 - x^2)
So it seems that your equation is correct after all! Well done!
Your equation for the area of the rectangle is correct: A = 2x * 2sqrt(4 - x^2). However, there is a mistake in your derivation of the side length of the rectangle.
The Pythagorean theorem states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the circle is inscribed in the rectangle, so the diagonal of the rectangle is equal to the diameter of the circle, which is 2r = 4. Let the width of the rectangle be 2x (since the x-axis divides it into two equal halves), and let the height be y.
By applying the Pythagorean theorem, we can write:
(x^2) + (y^2) = (2r)^2
(x^2) + (y^2) = 16
Solving for y^2, we get:
(y^2) = 16 - (x^2)
Taking the square root of both sides:
y = sqrt(16 - x^2)
Now you can substitute this expression for y into the equation for the area of the rectangle to get the function:
A = 2x * 2sqrt(16 - x^2)
So, the corrected function that represents the area of the rectangle in terms of x is A = 4x * sqrt(16 - x^2).
Your approach is correct, but there is a slight error in your calculation. Let's go through the process step by step to find the correct function representing the area of the rectangle in terms of x.
We know that the rectangle is inscribed in a circle of radius 2, so the lengths of the rectangle's sides will be the diameter of the circle, which is 4. Let's call the length of one side of the rectangle x, and the length of the other side y.
Using the Pythagorean theorem, we can relate x, y, and the radius of the circle:
(x/2)^2 + (y/2)^2 = 2^2
x^2/4 + y^2/4 = 4
x^2 + y^2 = 16
Now let's solve this equation for y^2:
y^2 = 16 - x^2
Taking the square root of both sides, we get:
y = sqrt(16 - x^2)
The area of the rectangle is given by the formula A = xy. Plugging in the value of y, we have:
A = x * sqrt(16 - x^2)
So the correct function representing the area of the rectangle in terms of x is:
A(x) = x * sqrt(16 - x^2)
Therefore, your final expression for the area of the rectangle should be:
A = 2x * sqrt(4 - x^2)