Computer screens and television screens are usually measured by the diagonal of the screen. For example, a screen of size 60 inches would have a diagonal of 60 inches, a width of 52 inches, and a height of 30 inches.
Yoselin would like to install a new screen on a shelf that is 17 inches tall and 50 inches wide. She would like the screen to be the largest size that will fit on the shelf. She knows the diagonal and width of the four screens listed below.
Screen
A
B
C
D
Diagonal
15”
22”
32”
40”
Width
13”
19”
28”
35”
1: use the Pythagorean theorem to calculate the height of each screen.
2: which screen is the largest that will fit on yoselins shelf?
1: To calculate the height of each screen using the Pythagorean theorem, we can use the formula:
Height^2 = Diagonal^2 - Width^2
For screen A:
Height^2 = 15^2 - 13^2
Height^2 = 225 - 169
Height^2 = 56
Height ≈ 7.48 inches
For screen B:
Height^2 = 22^2 - 19^2
Height^2 = 484 - 361
Height^2 = 123
Height ≈ 11.09 inches
For screen C:
Height^2 = 32^2 - 28^2
Height^2 = 1024 - 784
Height^2 = 240
Height ≈ 15.49 inches
For screen D:
Height^2 = 40^2 - 35^2
Height^2 = 1600 - 1225
Height^2 = 375
Height ≈ 19.36 inches
2: To determine which screen is the largest that will fit on Yoselin's shelf, we need to find the screen with a height less than or equal to 17 inches (the shelf height) and a width less than or equal to 50 inches (the shelf width).
Based on the calculations above, screen A has a height of 7.48 inches and a width of 13 inches, so it can fit on the shelf.
Screen B has a height of 11.09 inches and a width of 19 inches, so it can fit on the shelf as well.
Screen C has a height of 15.49 inches, which is greater than the shelf height, so it cannot fit on the shelf.
Screen D has a height of 19.36 inches, which is also greater than the shelf height, so it cannot fit on the shelf.
Therefore, the largest screen that will fit on Yoselin's shelf is screen B, with a diagonal of 22 inches, a width of 19 inches, and a height of 11.09 inches.
1. To calculate the height of each screen, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (width and height).
Let's calculate the height for each screen using the given width and diagonal:
Screen A:
Width: 13"
Diagonal: 15"
Using the Pythagorean theorem:
Height^2 + Width^2 = Diagonal^2
Height^2 + 13^2 = 15^2
Height^2 + 169 = 225
Height^2 = 225 - 169
Height^2 = 56
Height = √56
Height ≈ 7.48 inches
Screen B:
Width: 19"
Diagonal: 22"
Using the Pythagorean theorem:
Height^2 + Width^2 = Diagonal^2
Height^2 + 19^2 = 22^2
Height^2 + 361 = 484
Height^2 = 484 - 361
Height^2 = 123
Height = √123
Height ≈ 11.09 inches
Screen C:
Width: 28"
Diagonal: 32"
Using the Pythagorean theorem:
Height^2 + Width^2 = Diagonal^2
Height^2 + 28^2 = 32^2
Height^2 + 784 = 1024
Height^2 = 1024 - 784
Height^2 = 240
Height = √240
Height ≈ 15.49 inches
Screen D:
Width: 35"
Diagonal: 40"
Using the Pythagorean theorem:
Height^2 + Width^2 = Diagonal^2
Height^2 + 35^2 = 40^2
Height^2 + 1225 = 1600
Height^2 = 1600 - 1225
Height^2 = 375
Height = √375
Height ≈ 19.37 inches
2. To determine which screen is the largest that will fit on Yoselin's shelf, we can compare the height of each screen to the height of the shelf (17 inches) and the width of each screen to the width of the shelf (50 inches).
Screen A:
Height: 7.48 inches
Width: 13 inches
The height of Screen A (7.48 inches) is less than the height of the shelf (17 inches), but the width of Screen A (13 inches) is less than the width of the shelf (50 inches), so Screen A will fit.
Screen B:
Height: 11.09 inches
Width: 19 inches
Both the height and width of Screen B exceed the dimensions of the shelf, so Screen B will not fit.
Screen C:
Height: 15.49 inches
Width: 28 inches
Both the height and width of Screen C exceed the dimensions of the shelf, so Screen C will not fit.
Screen D:
Height: 19.37 inches
Width: 35 inches
Both the height and width of Screen D exceed the dimensions of the shelf, so Screen D will not fit.
Therefore, the largest screen that will fit on Yoselin's shelf is Screen A.
To calculate the height of each screen, we can use the Pythagorean theorem, which states that in 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.
1. To find the height of a screen, we can use the Pythagorean theorem formula:
Height = sqrt(Diagonal^2 - Width^2)
Now, let's calculate the height of each screen:
Screen A:
Height = sqrt(15^2 - 13^2)
Height = sqrt(225 - 169)
Height = sqrt(56)
Height ≈ 7.48 inches
Screen B:
Height = sqrt(22^2 - 19^2)
Height = sqrt(484 - 361)
Height = sqrt(123)
Height ≈ 11.09 inches
Screen C:
Height = sqrt(32^2 - 28^2)
Height = sqrt(1024 - 784)
Height = sqrt(240)
Height ≈ 15.49 inches
Screen D:
Height = sqrt(40^2 - 35^2)
Height = sqrt(1600 - 1225)
Height = sqrt(375)
Height ≈ 19.36 inches
Now that we have calculated the heights, we can determine which screen is the largest that will fit on Yoselin's shelf.
Yoselin's shelf has a height of 17 inches and a width of 50 inches.
By comparing the heights of the screens, we can see that Screen C has a height of approximately 15.49 inches, which is the largest height that is less than or equal to the shelf's height. Therefore, Screen C is the largest screen that will fit on the shelf.