99.3k views
4 votes
A tower under construction in a rural municipality is 24 feet tall. A man of height 6 feet, standing on the same horizontal level of the tower, observes the top of the incomplete tower and finds the angle of elevation to be 30°.

(a) How high must the tower be raised so that the man finds the angle of elevation of the complete tower to be 60° from the same place?
(b) What will be the height of the tower after completing its construction work?​

User Ekreloff
by
8.0k points

2 Answers

1 vote

Explanation:

See image and calcs below

A tower under construction in a rural municipality is 24 feet tall. A man of height-example-1
User Jim Cownie
by
7.4k points
7 votes

Explanation:

Let's first draw a diagram to better visualize the problem:

* T (top of incomplete tower)

/|

/ |

/ |

/ | h (height of incomplete tower)

/ |

/ |

/θ1 |

/___ | M (man's position, height = 6 feet)

d

We can see that we have a right triangle with the tower's height as the opposite side, the distance between the man and the tower as the adjacent side, and the angle of elevation θ1 as 30°. We can use trigonometry to find the height of the incomplete tower:

tan(30°) = h / d

h = d * tan(30°)

We don't know the value of d, but we can use the fact that the man's height plus the height of the incomplete tower equals the distance from the man to the top of the incomplete tower:

d = h / tan(30°) + 6

Now we can use trigonometry again to find the height of the complete tower. Let's call this height H and the new angle of elevation θ2:

* T (top of complete tower)

/|

/ |

/ |

/ | H (height of complete tower)

/ |

/ |

/θ2 |

/___ | M (man's position, height = 6 feet)

d

We have another right triangle, this time with the height of the complete tower as the opposite side, the same distance between the man and the tower as the adjacent side, and the new angle of elevation θ2 as 60°. We can use the tangent function again:

tan(60°) = H / d

H = d * tan(60°)

We can substitute the value of d we found earlier:

H = (h / tan(30°) + 6) * tan(60°)

Simplifying:

H = h * sqrt(3) + 6 * sqrt(3)

(a) To find how high the tower must be raised, we subtract the height of the incomplete tower from the height of the complete tower:

raise = H - h

raise = h * (sqrt(3) - 1) + 6 * sqrt(3)

Substituting the value of h we found earlier:

raise = 24 * (sqrt(3) - 1) + 6 * sqrt(3)

raise ≈ 38.8 feet

(b) The height of the completed tower is simply the height of the incomplete tower plus the raise we found:

height = h + raise

height = 24 + 38.8

height ≈ 62.8 feet

Therefore, the height of the tower after completing its construction work is approximately 62.8 feet.

User Itsundefined
by
8.6k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories