Question

Televisions and monitors come in two common aspect ratios, 4:3 and 16:9 (sometimes

16:10 is also used). A 42 inch TV suggests that the main diagonal of the TV is 42 inches.

Determine the dimensions of the screen of a 42 inch TV with a 4:3 aspect ratio.

Answer 1

To find the dimensions of a 42 inch TV with a 4:3 aspect ratio, set up a proportion with the aspect ratio and use the Pythagorean theorem. The dimensions are calculated to be 33.6 inches wide and 25.2 inches high.

Step-by-step explanation:

To determine the dimensions of a 42 inch TV with a 4:3 aspect ratio, we first note that the aspect ratio describes the relative dimensions of the width to the height. For a 4:3 aspect ratio, the width is 4 units for every 3 units of height. The diagonal, the TV screen size of 42 inches, forms a right triangle with the width and the height.

We can solve for the width (W) and height (H) using the Pythagorean theorem: W^2 + H^2 = Diagonal^2. Let's set up a proportion using the aspect ratio with W as 4x and H as 3x, where x is a scaling factor we need to determine.

Using the Pythagorean theorem:

(4x)^2 + (3x)^2 = 42^2

16x^2 + 9x^2 = 1764

25x^2 = 1764

x^2 = 1764 / 25

x = sqrt(1764 / 25)

x = sqrt(70.56)

x = 8.4

Now plug the value of x into W and H:

W = 4x = 4(8.4) = 33.6 inches

H = 3x = 3(8.4) = 25.2 inches

Therefore, the dimensions of a 42 inch TV with a 4:3 aspect ratio are 33.6 inches wide by 25.2 inches high.

Answer 2

The dimension of the TV = 11.2 inches × 8.4 inches

Step-by-step explanation:

Let x be the length of the TV

Let y be the width of the TV

x : y = 4:3

`(x)/(y) = (4)/(3) \\cross-multiplying\\3x = 4y \\x = (4)/(3) y- - - - - (1)`

From the figure attached:

using Pythagoras theorem

x² + y² = 42²

where:

`x = (4)/(3) y\\((4)/(3)y)^2 + y^2 = 42^2\\(16)/(9)y^2 + y^2 = 1764\\ (16y^2)/(9) + (y^2)/(1) = 1764\\(16y^2 +9y^2)/(9)= 1764\\ cross-multiplying\\16y^2 + 9y^2 = 1764\\25y^2 = 1764\\y^2 = (1764)/(24)\\y^2 = 70.56\\y = √(70.56) \\y = 8.4\ inches\\Finding\ x:\\x = (4)/(3) y\\x = (4)/(3) * 8.4\\ x = 1.33 * 8.4\\x = 11.2\ inches\\\therefore\ the\ dimensions\ are\ 11.2\ inches * 8.4\ inches`