Answer :
To determine the length and width of the TV with a given diagonal of 30 inches, we utilize the properties of a [tex]$30-60-90$[/tex] right triangle.
### Step-by-Step Solution:
1. Identify the properties of a [tex]$30-60-90$[/tex] triangle:
In a [tex]$30-60-90$[/tex] triangle, the sides are in a specific ratio:
- The shortest side (opposite the [tex]$30^\circ$[/tex] angle) is of length \( x \).
- The side opposite the [tex]$60^\circ$[/tex] angle (the longer leg) is \( x\sqrt{3} \).
- The hypotenuse (the side opposite the right angle, which in this case is the diagonal of the TV) is \( 2x \).
2. Given the diagonal (hypotenuse) is 30 inches:
We use the fact that the hypotenuse in a [tex]$30-60-90$[/tex] triangle is \( 2x \).
[tex]\[ 2x = 30 \][/tex]
3. Solve for \( x \):
[tex]\[ x = \frac{30}{2} = 15 \][/tex]
4. Determine the lengths of the sides:
- The shorter leg (opposite the [tex]$30^\circ$[/tex] angle) is \( x = 15 \) inches.
- The longer leg (opposite the [tex]$60^\circ$[/tex] angle) is \( x\sqrt{3} = 15\sqrt{3} \) inches.
5. Result:
The lengths corresponding to the sides of the triangle are therefore 15 inches and 15\(\sqrt{3}\) inches respectively.
Combining these steps, the length and width of the TV are 15 inches and \( 15\sqrt{3} \) inches, respectively.
Therefore, the correct answer is:
[tex]\[ \boxed{ \text{B. } 15 \text{ inches by } 15 \sqrt{3} \text{ inches}} \][/tex]
### Step-by-Step Solution:
1. Identify the properties of a [tex]$30-60-90$[/tex] triangle:
In a [tex]$30-60-90$[/tex] triangle, the sides are in a specific ratio:
- The shortest side (opposite the [tex]$30^\circ$[/tex] angle) is of length \( x \).
- The side opposite the [tex]$60^\circ$[/tex] angle (the longer leg) is \( x\sqrt{3} \).
- The hypotenuse (the side opposite the right angle, which in this case is the diagonal of the TV) is \( 2x \).
2. Given the diagonal (hypotenuse) is 30 inches:
We use the fact that the hypotenuse in a [tex]$30-60-90$[/tex] triangle is \( 2x \).
[tex]\[ 2x = 30 \][/tex]
3. Solve for \( x \):
[tex]\[ x = \frac{30}{2} = 15 \][/tex]
4. Determine the lengths of the sides:
- The shorter leg (opposite the [tex]$30^\circ$[/tex] angle) is \( x = 15 \) inches.
- The longer leg (opposite the [tex]$60^\circ$[/tex] angle) is \( x\sqrt{3} = 15\sqrt{3} \) inches.
5. Result:
The lengths corresponding to the sides of the triangle are therefore 15 inches and 15\(\sqrt{3}\) inches respectively.
Combining these steps, the length and width of the TV are 15 inches and \( 15\sqrt{3} \) inches, respectively.
Therefore, the correct answer is:
[tex]\[ \boxed{ \text{B. } 15 \text{ inches by } 15 \sqrt{3} \text{ inches}} \][/tex]
