Height of building?What should the height of a building be so that a ladder which is 80 m long and is placed 60 m away from the building forms an angle less than 60 degrees with the ground when it...

Height of building?

What should the height of a building be so that a ladder which is 80 m long and is placed 60 m away from the building forms an angle less than 60 degrees with the ground when it leans to touch the top of the building.

 

Asked on by greg1234

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

The inclination of the leaning ladder against the wall is less than 60 degree.

The length of the ladder is 80m. The distance between the wall and the bottom end of the ladder is 60m.

The geometric figure is a triangle with AC = 80m , where A is the top end of ladder of 80m touching the wall. CG = 60m where G is the feet of perpendicular of the wall AG and C is the bottom end of the ladder. Angle GCA < 60 degree.

Now using Pythagoras theorem, AG = sqrt(AC^2-GC^2) = sqrt(80^2-60)^2 = sqrt(2800) = sqrt(400*7) = 20sqrt7 = 52.92 m nearly.  The angle C of inclination = tanC = 20*sqrt7/60. Or C = arcttan {(20sqrt7)/60} = 41.41 deg which is less than 60 degree.

Hope this helps.

 

 

william1941's profile pic

william1941 | College Teacher | (Level 3) Valedictorian

Posted on

From the information in the question we know that the ladder is placed 60 meters away from the building, it is 80m long and the angle that is has to form with the building should be less than 60 degrees.

Now, we assume that the building is a vertical structure.

Therefore a right angled triangle is formed with between the ground, the building and the ladder, with the ladder acting as the hypotenuse. As the angle formed by the ladder and the ground should be less than 60 degrees, let’s take the extreme case of 60 degrees. When this is the case we see that sin 30 = height of the building/ length of the ladder =   sqrt 3 /2

=> height of the building = (sqrt 3/2)*80

=> height of the building = 69.28 m

The angle for any height more than 69.28 m exceeds 60 degree.

Therefore the required height of the building cannot exceed 69.28 m.

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