SOLUTION: The screen size of a television is determined by the length of the diagonal of the rectangular screen. Traditional TVs come in 4 : 3 format, meaning the ratio of the length to the

Algebra ->  Rectangles -> SOLUTION: The screen size of a television is determined by the length of the diagonal of the rectangular screen. Traditional TVs come in 4 : 3 format, meaning the ratio of the length to the       Log On


   

Question 1199156: The screen size of a television is determined by the length of the diagonal of the rectangular screen. Traditional TVs come in 4 : 3 format, meaning the ratio of the length to the width of the rectangular screen is 4 to 3. What is the area of a 37-inch Traditional TV screen? What is the area of a 37-inch LCD TV whose screen is in a 16 : 9 format? Which screen is larger?
Hint given but the textbook:
If x is the length of a 4 : 3 format screen, then (3x/4)is the width.
Let me see.
I know what x represents. I will say let y = length of a 16 : 9 format screen. If this is true, then 9y/16 is the width.
My set up for x goes like this:
37 = x(3x/4)
Solving for x, I get 7.024.
My set up for y goes like this:
37 = y(9y/16)
Solving for y,I get 8.11035.
I see that y > x,meaning the LCD TV screen format is bigger.
However, according to the textbook,I am wrong.
Textbook Answer:
The screen of a 37-inch TV in 4 : 3 format has an area of 657.12 in^2.
The screen of a 37-inch TV in 16 : 9 format has an area of 587.97 in^2.
According to the textbook, the traditional TV has the bigger screen.
Questions:
1. What did I do wrong?
2. What is the correct set up for each TV to find the area?
 
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
4:3 dimensions ratio
say 4n by 3n inches-square

%284n%29%5E2%2B%283n%29%5E2=37%5E2
16n%5E2%2B9n%5E2=37%5E2
25n%5E2=37%5E2
5n=37

n=37%2F5


Length 4%2837%2F5%29
Width 3%2837%2F5%29

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Here are the equations you are trying to use to solve the problem:

(1) 37 = x(3x/4)
(2) 37 = y(9y/16)

Those equations both say (incorrectly!) that the length of the diagonal is the product of the length and width.

That of course is not true; the length of the diagonal is found using the Pythagorean Theorem: (length) squared plus (width) squared equals (diagonal) squared.

NOTE: While your textbook suggests using x and (3/4)x for the dimensions of the traditional TV screen, I would prefer to use 4x and 3x -- why introduce fractions into our calculations when not necessary? I will do that in my response below.

Traditional TV, dimensions 3x and 4x....

The length of the diagonal is 37 inches:

%283x%29%5E2%2B%284x%29%5E2=37%5E2
9x%5E2%2B16x%5E2=1369
25x%5E2=1369
x%5E2=1089%2F25=54.76

The area of the screen is

%283x%29%284x%29=12x%5E2=12%2854.76%29=657.12

That matches the answer in your textbook.

LCD TV, dimensions 9x and 16x....

The length of the diagonal is 37 inches:

%289x%29%5E2%2B%2816x%29%5E2=37%5E2
81x%5E2%2B256x%5E2=37%5E2
337x%5E2=1369
x%5E2=1369%2F337=4.0623 (rounded to 4 decimal places)

The area of the screen is

%289x%29%2816x%29=144x%5E2=584.97 (rounded)

That doesn't quite match what you show as the answer in your textbook. Perhaps you didn't show the textbook answer correctly...?

The bottom line is that the area of the traditional TV screen is greater.