Screen size: Television screens are usually measured by the diagonal of the screen. The length and width of the rectangular screen become the legs of our right triangle with a diagonal that is our hypotenuse.
Find the height of each screen in the table below, round to tenth. Remember which is the legs and which is the diagonal or c. We want the TV to fit on a 17 x 50-inch shelf. What is the biggest TV (longest diagonal) that will fit on the shelf? Choose A, B, C, D
Screen
Diagonal
Width
Height
A
15”
13”
7.5”
B
22”
19”
11.1”
C
32”
28”
15.5”
D
40”
35”
19.4”
show work
To find the height of each screen, we need to use the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this context, the hypotenuse is the screen's diagonal, and the other two sides are the width and height of the screen.
The formula of the Pythagorean theorem is: c² = a² + b², where:
c = diagonal
a = width
b = height
Let's rearrange this formula to solve for the height (b):
b² = c² - a²
b = √(c² - a²)
Now let's calculate the height of each screen:
A. Diagonal: 15", Width: 13"
Height (b) = √(15² - 13²)
b = √(225 - 169)
b = √(56)
b ≈ 7.5"
So, for screen A, the height has been provided correctly in the table as 7.5".
B. Diagonal: 22", Width: 19"
Height (b) = √(22² - 19²)
b = √(484 - 361)
b = √(123)
b ≈ 11.1"
For screen B, the height has also been provided correctly in the table as 11.1".
C. Diagonal: 32", Width: 28"
Height (b) = √(32² - 28²)
b = √(1024 - 784)
b = √(240)
b ≈ 15.5"
Screen C's height given in the table is 15.5", which is also correct.
D. Diagonal: 40", Width: 35"
Height (b) = √(40² - 35²)
b = √(1600 - 1225)
b = √(375)
b ≈ 19.4"
Lastly, for screen D, the calculated height (19.4") matches the value in the table.
To decide which TV is the biggest that will fit on a 17 x 50-inch shelf, we need to ensure both the width and height of the TV are less than or equal to the shelf's dimensions.
Looking at the table:
- Screen A will fit since its width and height (13" and 7.5") are both smaller than the shelf dimensions (17" and 50").
- Screen B will also fit, as its width and height (19" and 11.1") are smaller than the shelf.
- Screen C exceeds the height of the shelf with a height of 15.5" (shelf height is only 17"), so it will not fit.
- Screen D's width is 35", which will fit on the shelf, but the height is 19.4", which exceeds the shelf's height.
Therefore, the biggest TV that will fit is screen B, with a diagonal of 22 inches. It is both shorter in height and less in width than the shelf dimensions (17" height and 50" width). So the correct choice is B.