Answer :
To determine the length of \(\overline{ AB }\), we start with the given equation:
[tex]\[ \sin(25^\circ) = \frac{9}{c} \][/tex]
where:
- \( 25^\circ \) is the angle,
- \( 9 \) is the length of the side opposite to the angle,
- \( c \) is the length of the hypotenuse, which is \(\overline{ AB }\).
To find \( c \), we need to solve the equation for \( c \):
[tex]\[ c = \frac{9}{\sin(25^\circ)} \][/tex]
Let's go through the process step-by-step:
1. Calculate \(\sin(25^\circ)\):
[tex]\[ \sin(25^\circ) \approx 0.4226 \][/tex]
2. Substitute \(\sin(25^\circ)\) into the equation:
[tex]\[ c = \frac{9}{0.4226} \][/tex]
3. Perform the division:
[tex]\[ c \approx 21.295814248372487 \][/tex]
4. Round this result to the nearest tenth:
[tex]\[ 21.295814248372487 \approx 21.3 \][/tex]
Therefore, the length of \(\overline{ AB }\) is approximately \(21.3\) inches when rounded to the nearest tenth. Hence, the correct answer is:
[tex]\[ \boxed{21.3 \text{ in.}} \][/tex]
[tex]\[ \sin(25^\circ) = \frac{9}{c} \][/tex]
where:
- \( 25^\circ \) is the angle,
- \( 9 \) is the length of the side opposite to the angle,
- \( c \) is the length of the hypotenuse, which is \(\overline{ AB }\).
To find \( c \), we need to solve the equation for \( c \):
[tex]\[ c = \frac{9}{\sin(25^\circ)} \][/tex]
Let's go through the process step-by-step:
1. Calculate \(\sin(25^\circ)\):
[tex]\[ \sin(25^\circ) \approx 0.4226 \][/tex]
2. Substitute \(\sin(25^\circ)\) into the equation:
[tex]\[ c = \frac{9}{0.4226} \][/tex]
3. Perform the division:
[tex]\[ c \approx 21.295814248372487 \][/tex]
4. Round this result to the nearest tenth:
[tex]\[ 21.295814248372487 \approx 21.3 \][/tex]
Therefore, the length of \(\overline{ AB }\) is approximately \(21.3\) inches when rounded to the nearest tenth. Hence, the correct answer is:
[tex]\[ \boxed{21.3 \text{ in.}} \][/tex]
