# Problem 2. A man has a ladder of length 8 m. What is the maximum height he can reach if he props it against the wall at 30 degrees to the horizontal and at 30 degrees to the vertical. (2)

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### 1 Answer

The length of the ladder with the man is 8 m. When it is propped up against the wall a right angled triangle is created with the wall and the ground perpendicular to each other and the ladder as the hypotenuse.

Let the maximum height that the man can reach on the wall using the ladder be H.

When the ladder makes an angle of 30 degrees with the horizontal, H and the length of the ladder are related by the expression sin 30 = H/8

=> 1/2 = H/8

=> H = 4

When the ladder makes an angle of 30 degrees with the vertical, H and the length of the ladder are related by the expression sin(90 - 30) = H/8

=> sin 60 = H/8

=> H = 8*sin 60

=> H ~~ 6.928

The maximum height that the man can reach is 4 m and 6.928 m resp. when the ladder makes an angle of 30 degrees with the horizontal and the vertical.