Certainly! Let's solve the problem step-by-step:
### Given Data
- Length of the ladder ([tex]\(L\)[/tex]) = 32 feet
- Angle formed with the ground ([tex]\(\theta\)[/tex]) = 29.37 degrees
### Objective
To find the distance from the base of the building to the base of the ladder, which we'll call [tex]\(d\)[/tex].
### Step-by-Step Solution
1. Convert the angle from degrees to radians:
Since many trigonometric functions use angles in radians, we first convert the given angle to radians.
[tex]\[
\theta_{\text{radians}} \approx 0.5126 \text{ radians}
\][/tex]
2. Use the Cosine of the Angle:
The cosine function relates the adjacent side (base of the ladder, [tex]\(d\)[/tex]) to the hypotenuse (the ladder length, [tex]\(L\)[/tex]):
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
[tex]\[
\cos(29.37^\circ) = \frac{d}{32}
\][/tex]
3. Calculate the Adjacent Side:
To isolate [tex]\(d\)[/tex], multiply both sides of the equation by the hypotenuse (32 feet):
[tex]\[
d = 32 \times \cos(29.37^\circ)
\][/tex]
[tex]\[
d \approx 32 \times 0.871 \approx 27.89 \text{ feet}
\][/tex]
4. Round to the Nearest Hundredth:
The calculated value already appears as [tex]\(27.89\)[/tex], which remains the same after rounding.
### Conclusion
The distance from the base of the ladder to the building, rounded to the nearest hundredth, is:
[tex]\[
\boxed{27.89} \text{ feet}
\][/tex]
So, the correct answer to the multiple-choice question is 27.89 feet.
