I don't know if this is the place to put it but anyways
Complete each of the following problems in the space provided if you need more space attach additional sheets. You must show all work and/or an explanation in order to receive credit.
1. Older television screens have an aspect ratio of 4:3. That is, the ratio of the width to the height is 4:3. The aspect ratio of many movies is not 4:3, so they are sometimes shown on a television screen by "letterboxing"- darkening strips of equal height at the top and bottom of the screen- as shown below. Suppose a movie has an aspect ratio of 2:1 and is shown on an older television screen with a 27-inch diagonal. What is the height, in inches, of each darkened strip.
Thanks for the help
screen height = h
screen width = w
h = w*3/4 (1)
27 = sqrt(h^2+w^2)
= sqrt((3w/4)^2+w^2)
solve this equation for w
calculate h from (1)
movie width = screen width
movie height = m
m=w/2
darkened strip = (h-m)/2
26.2
To find the height of each darkened strip, we can use the concept of similar triangles. Similar triangles have the same shape but can be different sizes.
Let's assume the height of each darkened strip is represented by 'x' inches.
We can create a proportion using the given aspect ratio of the movie (2:1) and the aspect ratio of the television screen (4:3).
Since the height of the movie's aspect ratio is 2 and the width is 1, we can say:
Height of the movie / Width of the movie = 2/1
Similarly, for the television screen aspect ratio, Height of the screen / Width of the screen = 4/3
Now, we can set up a proportion using these two ratios:
(Height of the movie) / (Width of the movie) = (Height of the screen) / (Width of the screen)
Substituting the given values:
2/1 = (Height of the screen) / (Width of the screen)
2/1 = (Height of the screen) / 3 (since the width of the screen is 3)
Now, we can solve for the Height of the screen:
2 * 3 = Height of the screen
Height of the screen = 6 inches
Since the diagonal of the screen is given as 27 inches (which represents the hypotenuse of a right triangle formed by the width, height, and diagonal), we can use the Pythagorean theorem to find the height and width of the screen.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (height and width):
(diagonal)^2 = (height)^2 + (width)^2
Substituting the given values:
27^2 = 6^2 + (width)^2
729 = 36 + (width)^2
693 = (width)^2
Taking the square root of both sides:
width = √693
width ≈ 26.267 inches
Next, we can find the width of each darkened strip by subtracting the width of the movie (which is equal to the width of the television screen) from the width of the screen:
width of each darkened strip = width of the screen - width of the movie
width of each darkened strip ≈ 26.267 - 1
width of each darkened strip ≈ 25.267 inches
Since the width of each darkened strip is equal to the width of the screen (which is 3 times the height of the darkened strip), we can conclude that:
Width = 3 * Height
Substituting the known values:
25.267 = 3 * Height
Solving for Height:
Height = 25.267 / 3
Height ≈ 8.422 inches
Therefore, the height of each darkened strip is approximately 8.422 inches.