To find the approximate distance between the swing and the slide, we can use the Law of Cosines. The Law of Cosines is helpful in situations where we know two sides of a triangle and the included angle, allowing us to determine the third side.
Here's how we can solve the problem step-by-step:
1. Identify the given values:
- Distance from the observer to the swing, [tex]\( a = 100 \)[/tex] feet.
- Distance from the observer to the slide, [tex]\( b = 60 \)[/tex] feet.
- Included angle between the two distances, [tex]\( \theta = 30^\circ \)[/tex].
2. Convert the angle from degrees to radians:
Because trigonometric functions in some mathematical contexts (such as programming languages or advanced scientific calculators) use radians, we need to convert the angle from degrees to radians. The conversion formula is:
[tex]\[
\text{radians} = \text{degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
For [tex]\( 30^\circ \)[/tex]:
[tex]\[
\theta \text{(in radians)} = 30 \times \left( \frac{\pi}{180} \right) \approx 0.5236 \text{ radians}
\][/tex]
3. Apply the Law of Cosines:
The Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(\theta)
\][/tex]
Substituting the known values:
[tex]\[
c^2 = 100^2 + 60^2 - 2 \cdot 100 \cdot 60 \cdot \cos(30^\circ)
\][/tex]
With [tex]\( \theta \approx 0.5236 \)[/tex]:
4. Calculate [tex]\( \cos(30^\circ) \)[/tex]:
Knowing that [tex]\( \cos(30^\circ) \approx 0.866 \)[/tex]:
[tex]\[
\cos(30^\circ) \approx 0.866
\][/tex]
5. Substitute and Simplify:
[tex]\[
c^2 = 100^2 + 60^2 - 2 \cdot 100 \cdot 60 \cdot 0.866
\][/tex]
[tex]\[
c^2 = 10000 + 3600 - 2 \cdot 100 \cdot 60 \cdot 0.866
\][/tex]
[tex]\[
c^2 = 10000 + 3600 - 10392
\][/tex]
[tex]\[
c^2 = 7208
\][/tex]
[tex]\[
c = \sqrt{7208} \approx 56.64
\][/tex]
6. Result:
The approximate distance between the swing and the slide is about 56.64 feet.
Thus, the distance between the swing and the slide, given the provided angle, is approximately [tex]\( 56.64 \)[/tex] feet.
