Question

What to do: Complete each of the following problems involving ladders and walls by using trigonometric functions and inverse trigonometric functions to determine unknown lengths and angle measures.

Questions:

a) A ladder is placed against the wall of a building such that the bottom of the ladder is 3 ft from the bottom of the wall. If the ladder forms a 14 degree angle with the building, how high up the wall does the ladder reach?


b) A 15 foot ladder is leaned against a wall so that the bottom of the ladder forms a 75 degree angle with the ground. How far up the wall does the ladder reach?


c) You need to paint shutters on a window that is on the second floor of your house. You have a 20 foot ladder that you will use. There is a warning on the ladder that states the angle formed by the ladder and the ground must not be less than 70 degrees, or the ladder may slip and cause serious injury or death. You planned on placing the bottom of the ladder 3 feet from the base of the house.

Will the angle formed between the ground and the ladder be safe? What is the furthest possible distance the ladder can be placed to maintain a safe angle?


How you must answer:

- Sketch:

-We know:

-We want

-Rationale: Which ratio will you use and why?

-Solution


How do you answer:

= you must include an accurate sketch

= a statement identifying the known information - we know

= a statement that tells what you are solving for - we want

= the equation you used to solve the problem

Answer 1

Explanation:

a) x be the height of the wall

`tan \ 14 =(opposite \ side)/(adjacent \ side)\\\\\\0.2493 = (3)/(x)\\\\\\x*0.25 = 3\\\\x = (3)/(0.25)\\\\\\x = 12 ft`

b) Let y be the hieght of the wall

`Sin \ 75 = (opposite \ side)/(hypotenuse)\\\\\\0.9659 = (y)/(15)\\\\\\0.97*15=y\\\\`

y = 14.6 ft

c) Let the angle formed between the bottom of the ladder to the ground be 'x'

`Cos \x =(adjacent \side)/(hypotenuse)\\\\Cos \ x =(3)/(20)\\\\\\Cos \x = 0.15\\\\x = Cos^(-1) \ (0.15)\\\\`

x = 81°

No, the angle formed between the ground and the ladder is not safe as it is more than 70°

Let the farthest possible distance ladder placed from the wall be 'y'

`Cos \ 70 =(adjacent \ side )/(hypotenuse)\\\\0.342 = (y)/(20)\\\\0.3*20=y`

y = 6 ft

6 ft is the furthest possible distance the ladder can be placed to maintain a safe angle.