Q3) 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”
B
22”
19”
C
32”
28”
D
40”
35”
show work
To find the height of each screen, we can 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). The formula is:
c^2 = a^2 + b^2
In this context, 'c' is the diagonal of the screen, 'a' is the width and 'b' is the height we are trying to find. So, we rearrange the formula to solve for 'b':
b^2 = c^2 - a^2
Now, let's calculate the height for each option:
A) Diagonal = 15 inches, Width = 13 inches
b^2 = 15^2 - 13^2 = 225 - 169 = 56
b = √56 ≈ 7.5 inches (rounded to the nearest tenth)
B) Diagonal = 22 inches, Width = 19 inches
b^2 = 22^2 - 19^2 = 484 - 361 = 123
b = √123 ≈ 11.1 inches (rounded to the nearest tenth)
C) Diagonal = 32 inches, Width = 28 inches
b^2 = 32^2 - 28^2 = 1024 - 784 = 240
b = √240 ≈ 15.5 inches (rounded to the nearest tenth)
D) Diagonal = 40 inches, Width = 35 inches
b^2 = 40^2 - 35^2 = 1600 - 1225 = 375
b = √375 ≈ 19.4 inches (rounded to the nearest tenth)
Filling out the table with the calculated heights:
Screen Diagonal Width Height
A 15” 13” 7.5”
B 22” 19” 11.1”
C 32” 28” 15.5”
D 40” 35” 19.4”
Now we want to find the biggest TV that will fit on a 17 x 50-inch shelf. The diagonal of the TV can't be the only measurement we consider – both the width and the height of the TV must not exceed the shelf dimensions.
A) Height 7.5" and Width 13", fits
B) Height 11.1" and Width 19", fits
C) Height 15.5" and Width 28", fits
D) Height 19.4" and Width 35", does not fit
From the above calculations, option D will not fit in height, as it exceeds 17 inches. Among A, B, and C, which have a height equal to or less than 17 inches, only option C has a width that fits within the 50-inch shelf width. Hence, a TV with a diagonal of 32 inches is the largest TV that will fit on the shelf.
The answer is C, a 32-inch TV.