Answer:
[tex]1,\! 200[/tex] square inches.
Step-by-step explanation:
It is given that the aspect ratio of this TV (a rectangle) is [tex]12:9[/tex]. In other words, there exists an unknown [tex]x[/tex] for which the width would be [tex]12\, x[/tex] inches, and the height would be [tex]9\, x[/tex] inches. To find the value of this unknown [tex]x[/tex], try to express the only known length measure in this question- the diagonal- in terms of [tex]x[/tex].
Refer to the diagram attached. If the width and height of this rectangle are considered as the two sides of a right triangle adjacent to the right angle, the diagonal would be the hypotenuse. By the Pythagorean Theorem, the length of the diagonal would be:
[tex]\sqrt{(12\. x)^{2} + (9\, x)^{2}} = 15\, x[/tex].
In other words:
[tex]15\, x = 50[/tex].
[tex]\displaystyle x = \frac{50}{15} = \frac{10}{3}[/tex].
The area of this rectangle would be the product of width and height:
[tex]\begin{aligned}(\text{area}) &= (\text{width})\, (\text{height}) \\ &= (12\, x)\, (9\, x) \\ &- 108\, x^{2} \\ &= (108) \left(\frac{10}{3}\right)^{2} \\ &= 1200\end{aligned}[/tex].
To find the area of the TV screen with a 12:9 aspect ratio and 50-inch diagonal, we use the aspect ratio to relate width and height, apply the Pythagorean theorem, and then calculate the area, resulting in 1200 sq. inches.
Finding the area of a TV screen that has an aspect ratio of 12:9, given that its diagonal measures 50 inches. To solve this, we can utilize the aspect ratio to express the width and height of the screen in terms of 'x' where the width is 12x and the height is 9x.
Using the Pythagorean theorem for the diagonal (d), we get:
d^2 = width^2 + height^2
50^2 = (12x)^2 + (9x)^2
2500 = 144x^2 + 81x^2
2500 = 225x^2
x^2 = 2500 / 225
x = sqrt(2500 / 225)
x = 50 / 15
Now, we compute the width and height:
Width = 12x = 12 * (50 / 15) = 40 inches
Height = 9x = 9 * (50 / 15) = 30 inches
Finally, the area (A) of the TV screen is:
A = width * height
A = 40 * 30
A = 1200 sq. inches
Therefore, the area of the TV screen is 1200 sq. inches (Option b).