Math Problem Statement

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.

Solution

To find the width of the alley in terms of aa, hh, and kk, we can use trigonometry and the given angles of the ladders.

Step 1: Define the scenario.

  • Let the width of the alley be ww.
  • One ladder leans against the left wall, extending hh feet up the wall and making a 7575^\circ angle with the ground.
  • The other ladder leans against the right wall, extending kk feet up the opposite wall and making a 4545^\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 7575^\circ).

For the first ladder:

  • The ladder forms a right triangle where the hypotenuse is aa, the angle with the ground is 7575^\circ, and the height it reaches on the wall is hh.

Using the sine function for the vertical height:

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

Solving for hh:

h=asin(75)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)=w1a\cos(75^\circ) = \frac{w_1}{a}

Solving for w1w_1 (the distance from the base of the first ladder to the point where the ladders meet):

w1=acos(75)w_1 = a \cos(75^\circ)

Step 3: Analyze the second ladder (angle 4545^\circ).

For the second ladder:

  • The ladder forms a right triangle where the hypotenuse is aa, the angle with the ground is 4545^\circ, and the height it reaches on the opposite wall is kk.

Using the sine function for the vertical height:

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

Solving for kk:

k=asin(45)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)=w2a\cos(45^\circ) = \frac{w_2}{a}

Solving for w2w_2 (the distance from the base of the second ladder to the point where the ladders meet):

w2=acos(45)w_2 = a \cos(45^\circ)

Step 4: Calculate the total width of the alley.

The total width of the alley, ww, is the sum of the horizontal distances w1w_1 and w2w_2:

w=w1+w2w = w_1 + w_2

Substitute the expressions for w1w_1 and w2w_2:

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

Step 5: Simplify the expression.

Using approximate values for the cosines:

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

Thus, the width of the alley is approximately:

wa(0.2588+0.7071)=a(0.9659)w \approx a(0.2588 + 0.7071) = a(0.9659)

Therefore, the total width of the alley ww is approximately:

w0.9659aw \approx 0.9659a

This is the width of the alley in terms of aa.

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 hh and kk are given?
  4. How do the trigonometric identities change if we use a different angle, say 6060^\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-Angle Triangles
Angle Functions

Formulas

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
Total width of alley: w = a * cos(75°) + a * cos(45°)

Theorems

Trigonometric functions of right-angled triangles

Suitable Grade Level

Grades 10-12