A rectangle is inscribed into a semi circle at radius 2. What is the largest area it can have and what are the dimensions
Answers
Area= 4 max
base =2sqrt2
height = sqrt2
Help is always appreciated :)
base = 2x
height = y = √(4-x^2)
a = 2x√(4-x^2)
da/dx = 4(2-x^2)/√(4-x^2)
da/dx=0 when x = √2
so, max area is 2(√2)√(4-2) = 4
I will work on 1/4 of the circle
y = height
x = half of width
r^2 = x^2 + y^2 = 4
A = x y so x = A/y
A^2/y^2 + y^2 = 4
A^2 + y^4 = 4 y^2
let p=y^2
p^2 -4p = -A^2
p^2 - 4p + 4 = -A^2+4
(p-2)^2 = - (A^2-4)
p = 2
so
y = sqrt 2 and A = 2
and x = 2/sqrt2 = sqrt 2
so
height= y = sqrt 2
width = 2 x = 2 sqrt 2
A = 4
To find the largest area of the rectangle inscribed in a semicircle, we need to determine its dimensions.
Let's assume the base of the rectangle is along the diameter of the semicircle.
Since the radius of the semicircle is given as 2, the diameter is twice the radius, so the diameter of the semicircle is 2 * 2 = 4.
Now, let's consider the height of the rectangle. The height should be the distance from the midpoint of the base to the top of the semicircle.
In a right-angled triangle formed by the height, half of the base, and the radius of the semicircle, we know the radius is 2 and half of the base is half of the diameter, which is 4/2 = 2.
By applying the Pythagorean theorem (a^2 + b^2 = c^2), we can find the height as follows:
height^2 + (2)^2 = (4/2)^2
height^2 + 4 = 1^2
height^2 = 1 - 4
height^2 = -3 (which is not possible)
Since we cannot find the height using the above method, it means that the rectangle described cannot have a full height within the semicircle.
However, we can still find the largest possible area by considering a different configuration.
Let's consider a square inscribed within the semicircle instead, where one of the sides of the square is the diameter of the semicircle. This is because a square will have the maximum area among rectangles with the same perimeter.
Using the same diameter (4) for the square, the side length will be equal to the diameter, so each side is 4.
The area of the square is calculated by multiplying the length of one side by itself:
Area = side * side = 4 * 4 = 16.
Therefore, the largest possible area for a rectangle inscribed in the semicircle is 16 square units.
To find the largest area of the rectangle inscribed in a semicircle, we need to consider the properties of the rectangle and the semicircle. Let's break down the problem step by step:
1. Understand the properties of a rectangle and a semicircle:
- A rectangle has opposite sides of equal length and four right angles.
- A semicircle is half of a full circle and has a radius, which is the distance from the center of the circle to any point on its circumference.
2. Determine the dimensions of the rectangle:
- Let's assume that the length of the rectangle is the base and the width is the height.
- Since we want to inscribe the rectangle in the semicircle, the width of the rectangle will be the diameter of the semicircle.
3. Find the diameter of the semicircle:
- The diameter of a semicircle is equal to twice the radius of the full circle.
- In this case, the radius of the semicircle is given as 2. Therefore, the diameter is 2 * 2 = 4.
4. Determine the base and height of the rectangle:
- As mentioned earlier, the width of the rectangle is equal to the diameter of the semicircle. Thus, the width is 4.
- Since the rectangle is inscribed in the semicircle, the length of the rectangle (base) will be equal to the portion of the semicircle's circumference that the width spans.
- Since the width spans exactly half the circumference, the base will be half the circumference of the semicircle.
- The circumference of a full circle can be calculated using the formula C = 2πr. In this case, the radius is 2, so the circumference is 2 * π * 2 = 4π.
- Therefore, the base will be half of 4π, which is 2π.
5. Calculate the area of the rectangle:
- The area of a rectangle is given by the formula A = base * height.
- In this case, the base is 2π and the height is 4.
- So, the area of the rectangle is A = 2π * 4 = 8π.
6. Simplify the area:
- Let's approximate the value of π as 3.14.
- So, the area of the rectangle is approximately 8 * 3.14 = 25.12.
Therefore, the largest area the rectangle can have when inscribed in the given semicircle is approximately 25.12 square units. The dimensions of the rectangle are a base of 2π (approximately 6.28) and a height of 4 units.