A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 20 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
My base length is correct: (10/1+pi/4)
But I need the total height, thanks so much!!
It is showing up as incorrect :(
To find the dimensions of the window that admits the greatest possible amount of light, we can use the concept of optimization.
Let's assume that the width of the rectangle is 'w' ft. Since the window's perimeter is given as 20 ft, we can set up the equation:
Perimeter of rectangle + Circumference of semicircle = 20 ft
2w + (πw)/2 = 20
Simplifying the above equation, we get:
2w + (πw)/2 = 20
Multiplying through by 2 to clear the fraction, we have:
4w + πw = 40
Now, we can rearrange the equation to solve for 'w':
(4 + π)w = 40
w = 40 / (4 + π)
w ≈ 5.966 ft (rounded to 3 decimal places)
Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle is half of this width:
r = w / 2
r ≈ 2.983 ft (rounded to 3 decimal places)
Now, to find the total height of the window, we need to sum the height of the rectangle and the diameter of the semicircle:
Total height = height of rectangle + diameter of semicircle
Total height = w + 2r
Substituting the values we calculated earlier:
Total height ≈ 5.966 + 2 * 2.983
Total height ≈ 5.966 + 5.966
Total height ≈ 11.932 ft (rounded to 3 decimal places)
Therefore, the dimensions of the window that admit the greatest possible amount of light are approximately:
Width ≈ 5.966 ft
Total Height ≈ 11.932 ft
To find the dimensions of the window, we can start by assigning variables to the width and height of the rectangle.
Let's say the width of the rectangle is 'w'. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle is 'w/2', and the height of the rectangle is 'h'.
Now, let's calculate the perimeter of the window using the given information. The perimeter of the window is the sum of the lengths of all its sides.
Perimeter = 2 * (width of rectangle) + circumference of the semicircle
Given that the perimeter is 20 ft, we have:
20 = 2w + πw/2
To solve this equation, let's simplify it:
20 = 2w + πw/2
40 = 4w + πw
40 = (4 + π)w
Dividing both sides by (4 + π), we find:
w = 40 / (4 + π)
Now, let's find the total height of the window.
The height of the window is equal to the height of the rectangle plus the radius of the semicircle (since the diameter of the semicircle is equal to the width of the rectangle).
So, the total height (h_total) can be calculated as:
h_total = h + w/2
Since we know the width 'w', we can substitute it into the equation above to find the total height.
h_total = h + (40 / (4 + π))/2
h_total = h + 20 / (4 + π)
Therefore, the total height of the window is h + 20 / (4 + π).
do you mean
base = 2 r = 10/(1+pi/4) ??
if that is correct then
2r + 2h + pi r = 20
total height = h + r
10/(1+pi/4) + 2 h + pi(5/(1+pi/4) = 20
10 + 2h(1+pi/4) +5 pi = 20
2h(1+pi/4) = 10-5pi
h = (10-5pi)/(2+pi/2)
then to get total height add r
r+h = 5/(1+pi/4) + (5-2.5pi)/(1+pi/4)
= (10-2.5 pi) /(1+pi/4)