A ladder 28 feet long leans against the side of a building, and the angle between the ladder and the building is 22°.

Question

A ladder 28 feet long leans against the side of a building, and the angle between the ladder and the building is 22°.

(a) Approximate the distance from the bottom of the ladder to the building. (Round your answer to two decimal places.)

 ft

(b) If the distance from the bottom of the ladder to the building is increased by 2.0 feet, approximately how far does the top of the ladder move down the building? (Round your answer to two decimal places.)

  ft

Answer 1

height of top of ladder on the building is 10.49 feet

28 times sin22

distance of bottom of the ladder to the building is 28cos22 = 25.96 feet

add 2 feet to get 27.96 feet

28cosA = .9986 where A is the new smaller angle

A = 3.032 degrees

sinA = 0.0529

28sinA = 1.48 feet for the new lower height of the top of the ladder.

moving the ladder an extra 2 feet lowered the top of the ladder from 10.49 to 1.48 or lowered it 9 feet

Answer 2

To label: Sides: a shortest

b mid

c hypotenuse=28

angles: α=22ˆ opposite side a.

β b

γ=90° c

a/sinα=b/sinβ=c (sin of a rt angle=1)

a/.374=b/sinβ=28

a=28*.374=10.4889'

a^2+b^2=c^2

10.49^2+b^2=28^2

784-110.02=674^.5=25.96'=b.

a=12.5

b=

c=28

α=

β=

γ=90

12.5/sinα=b/sinβ=28

12.5/28=sinα; .446asin=26.51°=α

c^2-a^2=b^2

784-156.25=627.75^.5=25.055'

Answer 3

(a) distance d from the bottom of the ladder to the building d = lsin22° = 28sin22° = 10.49 feet

(b) Distance Δs top of ladder move down: Δ s = lcos22° - (l² - (d + 2)²)^1/2 = 28cos22ˆ - (28² - 12.49²)^1/2 =

0.9 feet

Answer 4

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FOLLOW the steps I have shown you!

Answer 5

To answer the first question we just need to remember what all of the trig functions actually mean. We have the mnemonic soh (sine = opposite/hypotenuse) cah (cosine = adjacent/hypotenuse) toa (tangent = opposite/adjacent) to remember exactly what they correspond to. In this case, we have the angle between the ladder and building and the length of the ladder and we want to know the height of the building (let's call it h). In this case, we have the angle, we want to know the side adjacent to it, and we know the length of the hypotenuse (it is the length of the ladder, 28 ft). It seems like in this case, cosine would be useful since it relates the adjacent and hypotenuse. If we set everything up correctly, we get that cos 22 = h/28, which implies h = 28 cos 22 ≈ 25.96.

In this next question, it would seem like a good idea to know the original distance the ladder is away from the building (d). In this case, we would be looking for the length of the side opposite the 22-degree angle, so it would be useful to use sine here since it relates to the opposite and hypotenuse of the triangle. Specifically, we know that sin 22 = d/28 which implies that d = 28 sin 22. If we move the ladder 2 ft further from the building, the new distance it would be away would be (28 sin 22) + 2 ft. In order to figure out the new height (and thus figure out how much it changes), we can use the Pythagorean theorem here. We know that a²+b² = c². We can either let a or b be the new height of the ladder on the building. Just for the sake of argument, we will let it be b and thus let a be the distance of the ladder from the building, and let c be the hypotenuse, which is the length of the ladder, 28ft. If we sub everything in, we get that ((28 sin 22) +2)² + b² = 28². Rearranging and simplifying slightly, we get that b² = 784 - ((28sin22)+2)²≈ 784 - 156 = 628. If we square root both sides we get that b is approximately equal to 25.06 feet. The original height was 25.96 feet, so the ladder moved down the building approximately (25.96-25.06) = 0.90 feet.

Answer 6

Check this video out for an explanation!