Sure, let's solve this problem step by step:
### Step 1: Understand the Problem
Gabriel leans an 18-foot ladder against a wall, and it forms an angle of 73 degrees with the ground. We need to determine how high up the wall the ladder reaches, rounding the answer to the nearest foot if necessary.
### Step 2: Identify the Variables
- Length of the ladder (hypotenuse of the right triangle): 18 feet
- Angle between the ladder and the ground: 73 degrees
### Step 3: Use Trigonometry
To find the height up the wall (opposite side of the right triangle), we can use the sine function in trigonometry. The sine of an angle in a right triangle is equal to the opposite side divided by the hypotenuse.
\(\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]\)
### Step 4: Set Up the Equation
Let \(\( h \)\) be the height up the wall:
\(\[ \sin(73^\circ) = \frac{h}{18} \]\)
### Step 5: Solve for \(\( h \)\)
\(\[ h = 18 \times \sin(73^\circ) \]\)
### Step 6: Calculate the Sine of 73 Degrees
First, convert the angle in degrees to radians because trigonometric functions in calculators or most programming languages use radians.
\(\[ 73^\circ = 1.2740903539558606 \text{ radians} \]\)
Using this, we find:
\(\[ \sin(1.2740903539558606) \approx 0.970 \]\)
### Step 7: Multiply by the Hypotenuse
\(\[ h \approx 18 \times 0.970 \approx 17.213485607334636 \text{ feet} \]\)
### Step 8: Round to the Nearest Foot
\(\[ h \approx 17 \text{ feet} \]\)
### Conclusion
The ladder reaches approximately 17 feet up the wall.
