Question

A 20" ladder needs to have a 5:2 ratio of distance to building and distance to window in order to be safe. How far away does the ladder need to be from the building? How far up the building should the ladder be placed?

Answer 1

Answer (i):
`18.56^(\prime \prime)`

Answer(ii):
`7.42^(\prime \prime)`

Step-by-step explanation:

Let distance of ladder to building
`= s`

and distance to window
`= h`

We Define a parameter
`a` such that from ratio given in the question (i.e
`5:2`);

`s = 5a\ \hspace{0.5cm} \text{and} \hspace{0.5cm} h = 2a`.

Then from the figure attached to the answer and Pythagorean theorem:

`(50.8 \ cm)^2 \ = \ (5a)^2 \ + (2a)^2 \Rightarrow 2580.64 \ cm^2 \ = \ 25 a^2 + 4a^2 = 29 a^2`,

`\Rightarrow 88.99 \ cm^2 = a^2 \ \Rightarrow \ a = \ 9.43 \ cm.`

Therefore:

Distance of ladder to building
`=\ s \ = \ 5a \ = \ 5 (9.43 \ cm ) = 47.15 \ cm \ = 18.56^(\prime \prime)`

Distance to window
`= \ h \ = \ 2a \ = \ 2(9.43 \ cm) = 18.86 \ cm \ = \ 7.42^(\prime \prime)`

Answer 2

Answer: the ladder needs to be 7.43" from the building.

the ladder should be placed 18.68" up the building.

Step-by-step explanation:

A right angle triangle is formed.

The distance from the base of the building to the window represents the opposite side of the right angle triangle.

The distance from the foot of the ladder, h to the base of the building represents the adjacent side of the right angle triangle.

The length of the ladder represents the hypotenuse of the triangle.

The 20" ladder needs to have a 5:2 ratio of distance to building and distance to window in order to be safe. It means that

Opposite side/adjacent side = 5/2

Opposite side/h = 5/2

Opposite side = 5h/2 = 2.5h

We would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side + adjacent side

Therefore,

20² = 2.5h² + h ²

400 = 6.25h² + h² = 7.25h²

h² = 400/7.25

h² = 55.17

h = √55.17

h = 7.43"

The distance from the top of the ladder to the base of the building is

2.5h = 7.43 × 2.5 = 18.58"