Find the area of a portion of a semicircle of radius 15 m that is outside the area of an inscribed square if the base of the square lies on the diameter of the semicircle.
Please include solution. Thanks.
make a sketch, putting the complete figure on the x-y grid
draw a line from the centre of the semicircle to the vertex of the square , calling that point P(x,y)
so the base of the square is 2x and its height is y
so clearly
y = 2x
also x^2 + y^2 = 15^2
x^2 + (2x)^2 = 225
5x^2 = 225
x^2 = 45
x = 3√5
y = 6√5
area of square = 2xy
= 2(3√5)(6√5) = 180
area of semicircle = (1/2)π(15)^2 = 225π/2
area between square and semicircle
= 225π/2 - 180 or appr 173.43
check my arithmetic
To solve this problem, we can break it down into two parts: finding the area of the semicircle and finding the area of the inscribed square. Then, we can subtract the area of the square from the area of the semicircle to find the area of the portion outside the square.
1. Area of the Semicircle:
The formula for the area of a semicircle is A = (π * r^2) / 2, where r is the radius.
In this case, the radius is given as 15 m.
So, the area of the semicircle is A = (π * 15^2) / 2.
2. Area of the Inscribed Square:
Since the base of the square lies on the diameter of the semicircle, its side length is equal to the radius of the semicircle (15 m). The formula for the area of a square is A = s^2, where s is the side length.
So, the area of the square is A = (15)^2.
3. Area of the Portion Outside the Square:
To find the area of the portion outside the square, subtract the area of the square from the area of the semicircle:
Area outside the square = Area of the semicircle - Area of the square.
Area outside the square = [(π * 15^2) / 2] - (15)^2.
Calculating this expression, we find:
Area outside the square = (π * 15^2) / 2 - 225.
Therefore, the area of the portion of the semicircle that is outside the inscribed square is (π * 15^2) / 2 - 225 square meters.
To solve this problem, we need to find the area of the semicircle and the area of the inscribed square, and then subtract the area of the square from the area of the semicircle.
1. Find the area of the semicircle:
The formula for the area of a semicircle is A = (π * r^2) / 2, where r is the radius.
In this case, the radius is given as 15 m, so we can substitute it into the formula:
A_semicircle = (π * 15^2) / 2
A_semicircle = (π * 225) / 2
A_semicircle = 112.5π m² (rounded to the nearest tenth)
2. Find the area of the inscribed square:
The side length of the square is equal to the diameter of the semicircle, which is twice the radius. So the side length of the square is 2 * 15 m = 30 m.
The formula for the area of a square is A = side^2, so we can substitute the side length to calculate the area:
A_square = 30^2
A_square = 900 m²
3. Find the area of the portion of the semicircle outside the square:
We subtract the area of the square from the area of the semicircle:
A_portion = A_semicircle - A_square
A_portion = 112.5π - 900 m²
A_portion = 112.5π - 900 m² (rounded to the nearest tenth of π)
So the area of the portion of the semicircle outside the inscribed square is approximately 112.5π - 900 square meters.