Sure, let's find the length of \(\( j \)\) given the values \(\( k = 7.2 \)\) inches, \(\( i = 1.3 \)\) inches, and \(\( LJ = 58^\circ \)\). We will use the Law of Sines to solve this.
### Step-by-Step Solution:
1. Understanding the Law of Sines:
The Law of Sines states that in any triangle:
\(\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]\)
where \(\( a \)\), \(\( b \)\), and \(\( c \)\) are the lengths of the sides opposite angles \(\( A \)\), \(\( B \)\), and \(\( C \)\) respectively.
2. Setting Up the Problem:
Given the values:
- \(\( k = 7.2 \)\) inches (one side of the triangle),
- \(\( i = 1.3 \)\) inches (another side of the triangle),
- \(\( LJ = 58^\circ \)\) (angle opposite the side \(\( j \)\)).
We need to find the length of \(\( j \)\).
3. Applying the Law of Sines:
According to the given relationship:
\(\[
\frac{j}{\sin(LJ)} = \frac{k}{\sin(90^\circ)}
\]\)
Because \(\( \sin(90^\circ) = 1 \)\), the equation simplifies to:
\(\[
j = k \cdot \sin(LJ)
\]\)
4. Converting the Angle to Radians:
Since the sine function typically uses radians in most calculators and programming languages, we need to convert \(\( 58^\circ \)\) to radians.
\(\[
\text{Radians} = \frac{58 \times \pi}{180}
\]\)
Calculating this,
\(\[
\text{Radians} \approx 1.01229
\]\)
5. Calculating the Length of \(\( j \)\):
Now, we substitute the values into the equation:
\(\[
j = 7.2 \cdot \sin(1.01229)
\]\)
Evaluating \(\( \sin(1.01229) \)\) (using a calculator or trigonometric table),
\(\[
\sin(1.01229) \approx 0.8480
\]\)
Thus,
\(\[
j \approx 7.2 \cdot 0.8480 \approx 6.1059462923262675
\]\)
6. Rounding the Result:
Finally, we round \(\( j \)\) to the nearest tenth of an inch.
\(\[
j \approx 6.1
\]\)
### Final Answer:
The length of \(\( j \)\), to the nearest tenth of an inch, is \(\( \boxed{6.1} \)\) inches.
