Two ladders of length a lean against opposite walls of an alley with their feet​ touching. One ladder extends h feet up the wall and makes a
75degrees°
angle with the ground. The other ladder extends k feet up the opposite wall and makes a
45degrees°
angle with the ground. Find the width of the alley in terms of​ a, h,​ and/or k. Assume the ground is horizontal and perpendicular to both walls.

To find the width of the alley in terms of \(a\), \(h\), and \(k\), we can use trigonometry and the given angles of the ladders. 

### Step 1: Define the scenario.

- Let the width of the alley be \(w\).
- One ladder leans against the left wall, extending \(h\) feet up the wall and making a \(75^\circ\) angle with the ground.
- The other ladder leans against the right wall, extending \(k\) feet up the opposite wall and making a \(45^\circ\) angle with the ground.
- Both ladders meet at a single point on the ground.

We can break this down into two right-angled triangles, each formed by one ladder.

### Step 2: Analyze the first ladder (angle \(75^\circ\)).

For the first ladder:

- The ladder forms a right triangle where the hypotenuse is \(a\), the angle with the ground is \(75^\circ\), and the height it reaches on the wall is \(h\).
  
Using the sine function for the vertical height:

\[
\sin(75^\circ) = \frac{h}{a}
\]

Solving for \(h\):

\[
h = a \sin(75^\circ)
\]

The horizontal distance from the base of the ladder to the wall is the adjacent side of the triangle, which can be found using the cosine function:

\[
\cos(75^\circ) = \frac{w_1}{a}
\]

Solving for \(w_1\) (the distance from the base of the first ladder to the point where the ladders meet):

\[
w_1 = a \cos(75^\circ)
\]

### Step 3: Analyze the second ladder (angle \(45^\circ\)).

For the second ladder:

- The ladder forms a right triangle where the hypotenuse is \(a\), the angle with the ground is \(45^\circ\), and the height it reaches on the opposite wall is \(k\).

Using the sine function for the vertical height:

\[
\sin(45^\circ) = \frac{k}{a}
\]

Solving for \(k\):

\[
k = a \sin(45^\circ)
\]

The horizontal distance from the base of the second ladder to the wall is the adjacent side of the triangle:

\[
\cos(45^\circ) = \frac{w_2}{a}
\]

Solving for \(w_2\) (the distance from the base of the second ladder to the point where the ladders meet):

\[
w_2 = a \cos(45^\circ)
\]

### Step 4: Calculate the total width of the alley.

The total width of the alley, \(w\), is the sum of the horizontal distances \(w_1\) and \(w_2\):

\[
w = w_1 + w_2
\]

Substitute the expressions for \(w_1\) and \(w_2\):

\[
w = a \cos(75^\circ) + a \cos(45^\circ)
\]

### Step 5: Simplify the expression.

Using approximate values for the cosines:

- \(\cos(75^\circ) \approx 0.2588\)
- \(\cos(45^\circ) \approx 0.7071\)

Thus, the width of the alley is approximately:

\[
w \approx a(0.2588 + 0.7071) = a(0.9659)
\]

Therefore, the total width of the alley \(w\) is approximately:

\[
w \approx 0.9659a
\]

This is the width of the alley in terms of \(a\). 

Would you like more details on the steps or have any other questions?

### 5 Related Questions:
1. How does the width of the alley change if the angle of the second ladder is increased?
2. What happens to the width if both ladders have the same angle with the ground?
3. Can we find the width of the alley if only the heights \(h\) and \(k\) are given?
4. How do the trigonometric identities change if we use a different angle, say \(60^\circ\)?
5. What would be the effect if one ladder is significantly longer than the other?

### Tip:
When working with angles in right triangles, always check whether you need to use sine, cosine, or tangent by identifying which sides of the triangle are known and which are needed.

Two ladders of length a lean against opposite walls of an alley with their feet touching. One ladder extends h feet up the wall and makes a 75 degrees angle with the ground. The other ladder extends k feet up the opposite wall and makes a 45 degrees angle with the ground. Find the width of the alley in terms of a, h, and/or k. Assume the ground is horizontal and perpendicular to both walls.

Discover how to find the width of an alley when two ladders, leaning at angles of 75° and 45°, touch opposite walls. Learn how to apply trigonometric functions to right-angled triangles, using cosine to determine horizontal distances. Suitable for students in Grades 10-12.

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