To solve this problem, we will use trigonometry, specifically the sine and cosine functions. The ladder forms a right-angled triangle with the wall and the ground. Let's denote the following:
- [tex]\( L \)[/tex] as the length of the ladder (10 feet)
- [tex]\( \theta \)[/tex] as the angle between the ladder and the ground (71°)
- [tex]\( d \)[/tex] as the distance from the foot of the ladder to the wall
- [tex]\( h \)[/tex] as the height up the wall where the ladder touches
### a) How far from the wall is the foot of the ladder?
This distance [tex]\( d \)[/tex] can be found using the cosine function, since cosine relates the adjacent side (distance from the wall) to the hypotenuse (length of the ladder):
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{d}{L}
\][/tex]
We can rearrange this to solve for [tex]\( d \)[/tex]:
[tex]\[
d = L \cdot \cos(\theta)
\][/tex]
First, convert the angle from degrees to radians because trigonometric functions in most calculations result accurately when the angles are in radians:
[tex]\[
\theta = 71^\circ = 71 \times \frac{\pi}{180} \approx 1.2392 \, \text{radians}
\][/tex]
Now, calculate [tex]\( d \)[/tex]:
[tex]\[
d = 10 \cdot \cos(1.2392) \approx 10 \cdot 0.3256 \approx 3.256 \, \text{feet}
\][/tex]
So, the foot of the ladder is approximately 3.256 feet from the wall.
### b) How high up the wall does the ladder reach?
This height [tex]\( h \)[/tex] can be found using the sine function, since sine relates the opposite side (height up the wall) to the hypotenuse (length of the ladder):
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\][/tex]
We can rearrange this to solve for [tex]\( h \)[/tex]:
[tex]\[
h = L \cdot \sin(\theta)
\][/tex]
Using the same angle in radians:
[tex]\[
\theta = 71^\circ = 71 \times \frac{\pi}{180} \approx 1.2392 \, \text{radians}
\][/tex]
Now, calculate [tex]\( h \)[/tex]:
[tex]\[
h = 10 \cdot \sin(1.2392) \approx 10 \cdot 0.9455 \approx 9.455 \, \text{feet}
\][/tex]
So, the ladder reaches approximately 9.455 feet up the wall.
### Summary:
a) The foot of the ladder is approximately 3.256 feet from the wall.
b) The height up the wall that the ladder reaches is approximately 9.455 feet.
