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

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Question 1208776: The screen size of a television is determined by the length of the diagonal of the rectangular screen. Traditional TVs come in a 4:3 format, meaning the ratio of the length to the width of the screen is 4 to 3.
A. What is the area of a 37-inch traditional TV screen?
B. What is the area of a 37-inch LCD TV whose screen is in a 16 : 9 format?
Hint given in textbook:
If x is the length of a 4 : 3 format screen, then (3/4)(x) is the width.
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Part A

If you were to use that hint, then you'll have this diagram.

The diagonal splits the rectangle into two congruent right triangles.
Use the Pythagorean Theorem a%5E2%2Bb%5E2=c%5E2 to form this equation x%5E2%2B%28+%283%2F4%29x+%29%5E2+=+37%5E2 where a sub-goal is to solve for x.
I'll leave the scratch work for the student to do.
You should find the two solutions to that equation are x = 148/5 and x = -148/5
Ignore the negative x value since we cannot have a negative width.

x = 148/5 = 29.6
(3/4)x = (3/4)*(148/5) = 111/5 = 22.2
For a 37 inch TV to have a 4:3 aspect ratio, the screen needs to be 29.6 inches across horizontally and 22.2 inches tall.
Note that (29.6)^2+(22.2)^2 = 37^2 due to the Pythagorean Theorem.
The area of this TV is therefore 29.6*22.2 = 657.12 square inches.

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Another approach for part A

The use of a fraction in (3/4)x is a bit clunky in my opinion.
We can avoid that fraction by letting 4x be the horizontal portion of the screen and 3x be the vertical height, as shown here,

Note the ratio 4x:3x reduces to 4:3

If you were to solve %283x%29%5E2%2B%284x%29%5E2+=+37%5E2 you would get x = -37/5, x = 37/5 as the two solutions. We ignore the negative solution.

horizontal = 4x = 4*37/5 = 148/5 = 29.6 inches
vertical = 3x = 3*37/5 = 111/5 = 22.2 inches
area = 29.6*22.2 = 657.12 square inches.
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Part B

This is one possible diagram to draw, when having a 16:9 screen

The length of the diagonal is the same as before.
The equation to solve would be %2816x%29%5E2%2B%289x%29%5E2+=+37%5E2
The positive solution from this equation is x = 37*sqrt(337)/337
That would approximate to roughly x = 2.01551843
area = 16x*9x = 16*2.01551843*9*2.01551843 = 584.973294000432
This is roughly a 584.97 square inch screen depending how you round that decimal value.