Answer :
To determine the width and height of a 72-inch diameter television with an aspect ratio of 5:3, we need to consider the relationship between the diameter and the dimensions based on the aspect ratio.
### Steps to solve the problem:
1. Understanding the Aspect Ratio:
- An aspect ratio of 5:3 means that for every 5 units of width, there are 3 units of height.
- Let's denote the width as 5x and the height as 3x, where x is a scaling factor.
2. Relationship with the Diameter:
- Typically, the diameter of the television is the diagonal of the rectangular screen.
- Using the Pythagorean theorem, we can relate the width, height, and diagonal (diameter):
[tex]\( \text{Diagonal} = \sqrt{(\text{Width})^2 + (\text{Height})^2} \)[/tex].
3. Setting Up the Equation:
- Replace the width and height with their expressions in terms of x:
[tex]\( \text{Diagonal} = \sqrt{(5x)^2 + (3x)^2} \)[/tex].
- Plug in the given diagonal (72 inches):
[tex]\( 72 = \sqrt{25x^2 + 9x^2} \)[/tex].
- Simplify the expression under the square root:
[tex]\( 72 = \sqrt{34x^2} \)[/tex].
4. Solving for x:
- Square both sides to eliminate the square root:
[tex]\( 5184 = 34x^2 \)[/tex].
- Divide by 34:
[tex]\( x^2 = \frac{5184}{34} \)[/tex].
- Solve for x by taking the square root:
[tex]\( x = \sqrt{\frac{5184}{34}} \)[/tex].
5. Calculating Width and Height:
- Width: [tex]\( \text{Width} = 5x \)[/tex].
- Height: [tex]\( \text{Height} = 3x \)[/tex].
After calculating, you will find the dimensions are approximately:
- Width: 61.74 inches (closely matching 61.5 inches)
- Height: 37.04 inches (closely matching 36.9 inches)
Given the choices, the correct dimensions that approximate our calculated result are:
- Width: 61.5 inches
- Height: 36.9 inches
Therefore, the correct answer from the choices is:
- Width 61.5 inches and height 36.9 inches.
### Steps to solve the problem:
1. Understanding the Aspect Ratio:
- An aspect ratio of 5:3 means that for every 5 units of width, there are 3 units of height.
- Let's denote the width as 5x and the height as 3x, where x is a scaling factor.
2. Relationship with the Diameter:
- Typically, the diameter of the television is the diagonal of the rectangular screen.
- Using the Pythagorean theorem, we can relate the width, height, and diagonal (diameter):
[tex]\( \text{Diagonal} = \sqrt{(\text{Width})^2 + (\text{Height})^2} \)[/tex].
3. Setting Up the Equation:
- Replace the width and height with their expressions in terms of x:
[tex]\( \text{Diagonal} = \sqrt{(5x)^2 + (3x)^2} \)[/tex].
- Plug in the given diagonal (72 inches):
[tex]\( 72 = \sqrt{25x^2 + 9x^2} \)[/tex].
- Simplify the expression under the square root:
[tex]\( 72 = \sqrt{34x^2} \)[/tex].
4. Solving for x:
- Square both sides to eliminate the square root:
[tex]\( 5184 = 34x^2 \)[/tex].
- Divide by 34:
[tex]\( x^2 = \frac{5184}{34} \)[/tex].
- Solve for x by taking the square root:
[tex]\( x = \sqrt{\frac{5184}{34}} \)[/tex].
5. Calculating Width and Height:
- Width: [tex]\( \text{Width} = 5x \)[/tex].
- Height: [tex]\( \text{Height} = 3x \)[/tex].
After calculating, you will find the dimensions are approximately:
- Width: 61.74 inches (closely matching 61.5 inches)
- Height: 37.04 inches (closely matching 36.9 inches)
Given the choices, the correct dimensions that approximate our calculated result are:
- Width: 61.5 inches
- Height: 36.9 inches
Therefore, the correct answer from the choices is:
- Width 61.5 inches and height 36.9 inches.