Sure! To find the area of the rectangular TV screen, we need to follow a series of steps involving the given perimeter and the ratio of the width to the height.
### Step 1: Define Variables and Equations
The perimeter of the rectangle (P) is given by:
[tex]\[ P = 150 \, \text{cm} \][/tex]
The ratio of the width to the height is 16:9. Therefore, let:
- Width (W) = 16x
- Height (H) = 9x
where [tex]\( x \)[/tex] is a constant.
The formula for the perimeter of a rectangle is given by:
[tex]\[ P = 2 \times (W + H) \][/tex]
### Step 2: Substitute the Ratio into the Perimeter Formula
Substitute the expressions for width and height into the perimeter formula:
[tex]\[ 150 = 2 \times (16x + 9x) \][/tex]
[tex]\[ 150 = 2 \times 25x \][/tex]
[tex]\[ 150 = 50x \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
To find the value of [tex]\( x \)[/tex]:
[tex]\[ x = \frac{150}{50} \][/tex]
[tex]\[ x = 3.0 \][/tex]
### Step 4: Calculate Width and Height
Now that we have [tex]\( x \)[/tex], we can calculate the actual width and height:
- Width [tex]\( W \)[/tex]:
[tex]\[ W = 16x = 16 \times 3.0 = 48 \, \text{cm} \][/tex]
- Height [tex]\( H \)[/tex]:
[tex]\[ H = 9x = 9 \times 3.0 = 27 \, \text{cm} \][/tex]
### Step 5: Calculate the Area
Finally, we can calculate the area (A) of the rectangle using the width and height:
[tex]\[ A = W \times H \][/tex]
[tex]\[ A = 48 \, \text{cm} \times 27 \, \text{cm} \][/tex]
[tex]\[ A = 1296 \, \text{cm}^2 \][/tex]
### Conclusion
The area of the rectangular TV screen is:
[tex]\[ 1296 \, \text{cm}^2 \][/tex]
