ANSWER IS D.
Process:
See in the attached fig,
BD=length of pole=?
AB=length of vertical(imagine)
BC=length of shadow=20 ft
In traingle ABC,
tan(36)=AB/BC
AB=tan(36)*20
AB=14.5308 ft
Now we find <BDC to use sine law.
For that,
<BDC+<DCB+<CBD=180
<BDC+36+(90+12)=180
<BDC=180-112-36
<BDC=42
Now we use sine law in traingle ABC,
BC/sin (<BDC) =BD/sin (<DCB)
20/sin(42)=BD/sin(36)
BD=(20*sin36)/sin42
BD=17.56=17.6(approx) >>>>>>>>ANSWER D
The length of the pole is [tex]\boxed{17.6{\text{ feet}}}[/tex]. Option (D) is correct.
Further explanation:
The Pythagorean formula can be expressed as,
[tex]\boxed{{H^2} = {P^2} + {B^2}}.[/tex]
Here, H represents the hypotenuse, P represents the perpendicular and B represents the base.
The formula for tan of angle a can be expressed as
[tex]\boxed{\tan a = \frac{P}{B}}[/tex]
Given:
The options are as follows,
(A). [tex]4.5{\text{ feet}}[/tex]
(B). [tex]7.1{\text{ feet}}[/tex]
(C). [tex]11.8{\text{ feet}}[/tex]
(D). [tex]17.6{\text{ feet}}[/tex]
Explanation:
A pole tilts toward the sun at a [tex]{12^ \circ }[/tex] angle from the vertical, and it casts a 20-foot shadow.
The angle of elevation from the tip of the shadow to the top of the pole is [tex]{36^ \circ }.[/tex]
In triangle ABC,
[tex]\begin{aligned}\tan {36^ \circ }&= \frac{{AB}}{{BC}}\\0.727&= \frac{{AB}}{{20}}\\20 \times 0.727&= AB\\\end{aligned}[/tex]
The sum of all angles of triangle is [tex]{180^ \circ }.[/tex]
[tex]\begin{aligned}\angle BDC + \angle DCB + \angle CBD &= {180^ \circ }\\\angle BDC + 36 + \left( {90 + 12} \right) &= {180^ \circ }\\\angle BDC + 148 &= {180^ \circ } \\\angle BDC &= {180^ \circ } - {148^ \circ }\\\angle BDC &= {42^ \circ } \\\end{aligned}[/tex]
Use sine law in triangle ABC.
[tex]\begin{aligned}\frac{{BC}}{{\sin D}} &= \frac{{BD}}{{\sin C}}\\\frac{{20}}{{\sin 42}} &= \frac{{BD}}{{\sin 36}} \\ \frac{{20}}{{\sin 42}} \times \sin 36 &= BD\\\frac{{20}}{{0.669}} \times 0.588 &= BD\\17.6&= BD\\\end{aligned}[/tex]
The length of the pole is [tex]\boxed{17.6{\text{ feet}}}[/tex]. Option (D) is correct.
Learn more:
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Trigonometry
Keywords: Pole, tilts, sun, 12 degree angle, vertical, casts, 200-foot, shadow, top, pole, 36 degree, length, length of pole, nearest tenth, 4.5 feet, 7.1 feet, 11.8 feet, 17.6 feet.
