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User Knabar
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Answer:

15) The length of the line segment DE is 14.908.

16) The measure of the angle W is approximately 31.792°.

17) The length of the ladder is approximately 23.182 feet.

Explanation:

15) We present the procedure to determine the length of segment DE:

(i) Determine the length of the line segment DF by trigonometric relations:


\tan C = (DF)/(CF) (1)

(
C = 61^(\circ),
CF = 24)


DF = CF\cdot \tan C


DF = 24\cdot \tan 61^(\circ)


DF \approx 43.297

(ii) Determine the length of the line segment DE by trigonometric relations:


\tan F = (DE)/(DF) (2)

(
DF \approx 43.297,
F = 19^(\circ))


DE = DF\cdot \tan F


DE = 43.297\cdot \tan 19^(\circ)


DE \approx 14.908

The length of the line segment DE is 14.908.

16) We present the procedure to determine the measure of the angle W:

(i) Determine the length of the line segment XZ by trigonometric relations:


\sin Z = (XY)/(XZ) (3)

(
XY = 15,
Z = 25^(\circ))


XZ = (XY)/(\sin Z)


XZ = (15)/(\sin 25^(\circ))


XZ \approx 35.493

(ii) Calculate the measure of the angle W by trigonometric relations:


\tan W = (XZ)/(WZ) (4)

(
XZ \approx 35.493,
WZ = 22)


W \approx \tan^(-1) \left((22)/(35.493)\right)


W \approx 31.792^(\circ)

The measure of the angle W is approximately 31.792°.

17) The system form by the ladder, the ground and the wall represents a right triangle, whose hypotenuse is the ladder, which is now found by the following trigonometric relation:


\cos \theta = (x)/(l) (5)

Where:


\theta - Angle of the ladder above ground, in sexagesimal degrees.


x - Distance between the foot of the ladder and the base of the wall, in feet.


l - Length of the ladder, in feet.

If we know that
x = 6\,ft and
\theta = 75^(\circ), then the length of the ladder is:


l = (x)/(\cos \theta)


l = (6\,ft)/(\cos 75^(\circ))


l \approx 23.182\,ft

The length of the ladder is approximately 23.182 feet.

User Faytraneozter
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