Question

(8 points) A fence 6 feet tall runs parallel to a tall building at a distance of 6 feet from the building. We want to find the the length of the shortest ladder that will reach
from the ground over the fence to the wall of the building.
Here are some hints for finding a solution:
Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and
just touching the top of the fence
A. If the ladder makes an angle 0.61 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the
ground to the top of the fence.
B. The distance along the ladder from the top of the fence to the wall is
C. Using these hints write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just
reaches the wall
L( x ) =
D. Use this function to find the length of the shortest ladder which will clear the fence.
The length of the shortest ladder is ___feet

Answer 1

A.10.47 ft   B.7.32 ft    C.L(x) =14.58 /cos 34.95°  D.10.47 ft

Step-by-step explanation:

Finding the distance along the ladder from the ground to the top of the fence given the angle the ladder makes with the fence and height of the fence

Applying the formula for tangent of an angle where

tan α = length of opposite side/length of adjacent  side

α= 0.61 rad -----change to degree by multiplying 0.61 by 57.2958

α= 0.61 × 57.2958 =34.95°

Let x to be distance of ladder from ground to the fence and height of fence=6 ft

tan 34.95°=O/A

0.69890780818=6/x

x=6/0.69890780818 =8.58 ft

To get distance of ladder from ground to the fence apply Pythagorean relationship where sum of squares of the sides equals square of the hypotenuse

8.5²+6²=h²

109.699 =h²

√109.699= h

10.47 ft =h

Hence ⇒ the distance along the ladder from the

ground to the top of the fence is 10.47 ft

B. Finding the distance along the ladder from the top of the fence to the wall is;

Apply the cosine of an angle formula, where cos α = adjacent side length/hypotenuse

Adjacent side length = 8.58+6 = 14.58 ft

Hypotenuse =?

α=34.95°

cos 34.95°=14.58/h

h=14.58/cos 34.95°

h=length of ladder from ground to the wall= 17.79 ft  hence the distance along the ladder from the top of the fence to the wall is'

17.79 ft - 10.47 ft = 7.32 ft

C.

The function for total length of the ladder which touches the ground at an angle x, touches the top of the fence and just

reaches the wall will be;

L(x) =14.58 /cos 34.95°

D.

The length of the shortest ladder which will clear the fence will be;

(14.58 /cos 34.95°)- 7.32 ft where 7.32 ft is the length of the ladder from top of the fence to the wall

=17.79 ft - 7.32 ft = 10.47 ft

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Sine, Cosine and Tangent of an angle :https://brainly.com/question/12465995

Keywords : fence,building, angel, ladder, wall, building , parallel

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