1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
erastova [34]
3 years ago
15

The angle of elevation from me to the top of a hill is 51 degrees. The angle of elevation from me to the top of a tree is 57 deg

rees. The tree stands 20 ft off the hill. How high is the top of the tree from the base of the hill?
Mathematics
1 answer:
julia-pushkina [17]3 years ago
6 0

Answer:

Approximately 101\; \rm ft (assuming that the height of the base of the hill is the same as that of the observer.)

Step-by-step explanation:

Refer to the diagram attached.

  • Let \rm O denote the observer.
  • Let \rm A denote the top of the tree.
  • Let \rm R denote the base of the tree.
  • Let \rm B denote the point where line \rm AR (a vertical line) and the horizontal line going through \rm O meets. \angle \rm B\hat{A}R = 90^\circ.

Angles:

  • Angle of elevation of the base of the tree as it appears to the observer: \angle \rm B\hat{O}R = 51^\circ.
  • Angle of elevation of the top of the tree as it appears to the observer: \angle \rm B\hat{O}A = 57^\circ.

Let the length of segment \rm BR (vertical distance between the base of the tree and the base of the hill) be x\; \rm ft.

The question is asking for the length of segment \rm AB. Notice that the length of this segment is \mathrm{AB} = (x + 20)\; \rm ft.

The length of segment \rm OB could be represented in two ways:

  • In right triangle \rm \triangle OBR as the side adjacent to \angle \rm B\hat{O}R = 51^\circ.
  • In right triangle \rm \triangle OBA as the side adjacent to \angle \rm B\hat{O}A = 57^\circ.

For example, in right triangle \rm \triangle OBR, the length of the side opposite to \angle \rm B\hat{O}R = 51^\circ is segment \rm BR. The length of that segment is x\; \rm ft.

\begin{aligned}\tan{\left(\angle\mathrm{B\hat{O}R}\right)} = \frac{\,\rm {BR}\,}{\,\rm {OB}\,} \; \genfrac{}{}{0em}{}{\leftarrow \text{opposite}}{\leftarrow \text{adjacent}}\end{aligned}.

Rearrange to find an expression for the length of \rm OB (in \rm ft) in terms of x:

\begin{aligned}\mathrm{OB} &= \frac{\mathrm{BR}}{\tan{\left(\angle\mathrm{B\hat{O}R}\right)}} \\ &= \frac{x}{\tan\left(51^\circ\right)}\approx 0.810\, x\end{aligned}.

Similarly, in right triangle \rm \triangle OBA:

\begin{aligned}\mathrm{OB} &= \frac{\mathrm{AB}}{\tan{\left(\angle\mathrm{B\hat{O}A}\right)}} \\ &= \frac{x + 20}{\tan\left(57^\circ\right)}\approx 0.649\, (x + 20)\end{aligned}.

Equate the right-hand side of these two equations:

0.810\, x \approx 0.649\, (x + 20).

Solve for x:

x \approx 81\; \rm ft.

Hence, the height of the top of this tree relative to the base of the hill would be (x + 20)\; {\rm ft}\approx 101\; \rm ft.

You might be interested in
The sum of two numbers is 18 and their difference is 4 what are the two numbers?
yarga [219]

Answer:

7 and 11

Step-by-step explanation:

Let the two numbers be x and y

x+y = 18

x-y = 4

Add the two equations together to eliminate y

x+y = 18

x-y = 4

----------------

2x +0y = 22

Divide each side by 2

2x/2 = 22/2

x = 11

Solve for y

x+y = 18

11+y =18

Subtract 11 from each side

11+y-11 =18-11

y = 7

The two numbers are 7 and 11

3 0
2 years ago
Simplify. Assume that a is a positive integer and a>2. (a-2)!/a!
Paul [167]

Answer:

C. 1 / (a(a - 1))

Step-by-step explanation:

View Image

Just know that:

n! = n(n-1)!

   = n(n-1)(n-2)!

   = n(n-1)(n-2)(n-3)!

   = ...

8 0
3 years ago
Read 2 more answers
A 5-pack of key chains costs $5.46. What is the unit price, rounded to the nearest cent?
svlad2 [7]

Answer:

the unit price is $1.10

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
Help me please? 15 points to anyone who can help me!"Find the number of permutations of the first nine letters of the alphabet t
gladu [14]
_{9}P_{6}=\frac{9!}{(9-6)!}=\frac{9!}{3!}=\frac{362,880}{6}=60,480
7 0
3 years ago
What is the solution of x-1/1-x < 0?
kykrilka [37]

Option C is correct option.

Step-by-step explanation:

We need to find the solution of \frac{x-1}{1-x}

The given inequality becomes undefined when x = 1 because the denomiator is: 1-x and if x = 1 then it becomes 1-1 = 0 and anything divided by zero is undefined.

So, all values other than 1 are included in the solution of the given inequality.

So, solution is: x<1 or x>1 but x≠1

So, Option C is correct because all numbers are included in number line except 1. An unfilled circle on 1 shows that it is not included in the solution. Rest of numbers are included.

Option C is correct option.

Keywords: Solving inequalities

Learn more about Solving inequalities at:

  • brainly.com/question/1465430
  • brainly.com/question/6703816
  • brainly.com/question/11788572
  • brainly.com/question/4192226

#learnwithBrainly

8 0
3 years ago
Other questions:
  • What type of number is -12.123 repeating?
    9·1 answer
  • Zuleika was solving 2(x - 3) - 9 = 3x - 9. Zuleika's first step resulted in 2(x - 3) = 3x
    5·1 answer
  • 3y+6=-2x Brainliest to first person to answer
    10·2 answers
  • Suppose I measure the length and width of a rectangle, and get L = 4 +/- 1 m and W = 10 +/- 2m. What is the uncertainty (not fra
    7·1 answer
  • A number cube is rolled 60 times. The results of those
    15·2 answers
  • A. 3<br> B. 1/3<br> C. -3<br> D. -1/3
    15·1 answer
  • Which ordered pair makes both inequalities true?<br> y&gt;-2x + 3<br> y
    6·1 answer
  • What is the name of the shape depicted in the graph below?
    12·2 answers
  • Pls help due today just need 9,10, 11
    11·1 answer
  • NEED HELP ASAP PLUS BRAINLIEST
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!