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 degrees. The tree stands 20 ft off the hill. How high is the top of the tree from the base of the hill?
Answers
Answer:
Approximately [tex]101\; \rm ft[/tex] (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 [tex]\rm O[/tex] denote the observer.Let [tex]\rm A[/tex] denote the top of the tree.Let [tex]\rm R[/tex] denote the base of the tree.Let [tex]\rm B[/tex] denote the point where line [tex]\rm AR[/tex] (a vertical line) and the horizontal line going through [tex]\rm O[/tex] meets. [tex]\angle \rm B\hat{A}R = 90^\circ[/tex].Angles:
Angle of elevation of the base of the tree as it appears to the observer: [tex]\angle \rm B\hat{O}R = 51^\circ[/tex].Angle of elevation of the top of the tree as it appears to the observer: [tex]\angle \rm B\hat{O}A = 57^\circ[/tex].
Let the length of segment [tex]\rm BR[/tex] (vertical distance between the base of the tree and the base of the hill) be [tex]x\; \rm ft[/tex].
The question is asking for the length of segment [tex]\rm AB[/tex]. Notice that the length of this segment is [tex]\mathrm{AB} = (x + 20)\; \rm ft[/tex].
The length of segment [tex]\rm OB[/tex] could be represented in two ways:
In right triangle [tex]\rm \triangle OBR[/tex] as the side adjacent to [tex]\angle \rm B\hat{O}R = 51^\circ[/tex].In right triangle [tex]\rm \triangle OBA[/tex] as the side adjacent to [tex]\angle \rm B\hat{O}A = 57^\circ[/tex].For example, in right triangle [tex]\rm \triangle OBR[/tex], the length of the side opposite to [tex]\angle \rm B\hat{O}R = 51^\circ[/tex] is segment [tex]\rm BR[/tex]. The length of that segment is [tex]x\; \rm ft[/tex].
[tex]\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}[/tex].
Rearrange to find an expression for the length of [tex]\rm OB[/tex] (in [tex]\rm ft[/tex]) in terms of [tex]x[/tex]:
[tex]\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}[/tex].
Similarly, in right triangle [tex]\rm \triangle OBA[/tex]:
[tex]\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}[/tex].
Equate the right-hand side of these two equations:
[tex]0.810\, x \approx 0.649\, (x + 20)[/tex].
Solve for [tex]x[/tex]:
[tex]x \approx 81\; \rm ft[/tex].
Hence, the height of the top of this tree relative to the base of the hill would be [tex](x + 20)\; {\rm ft}\approx 101\; \rm ft[/tex].
Related Questions
Zachary earns $8.00 per hour. He get paid time and a half for any hours over 40 that he works in a week. If Zachary works 9 hours per day from Monday through Friday, what are his gross earnings for the week?
Answers
Answer:
27 bc he warms hard and should get a raise
Step-by-step explanation:
The two graduates with the greatest GPA from each of 14 randomly selected law schools were recruited to investigate a new method of preparation for a law exam. For each school, one student was randomly assigned to prepare using the standard method and the other student was assigned to prepare using the new method. At the end of the preparation, all 28 students were given the same exam, and their scores were recorded. Why can we conclude cause and effect from this type of study
Answers
Incomplete question. The missing question read;
Which of the following best describes why a matched-pairs design is an appropriate design for the investigation?
a. Each graduate was randomly assigned to one method.
b. The law schools were randomly selected.
c. Each method served as a block.
d. The sample size was less than 30.
e. Each law school served as a block, and the graduates were paired within each law school and randomly assigned to one of the two methods.
Answer:
e. Each law school served as a block, and the graduates were paired within each law school and randomly assigned to one of the two methods.
Step-by-step explanation:
Note, the matched pairs design is typically used when the experiment has two independent variables (treatment conditions), so the subjects of the experiment are grouped into pairs based on some common variable relationship they have before been randomly assigned to the treatment conditions.
In this case, the graduates were paired within each law school and randomly assigned to one of the two methods (the two treatment conditions).
Which number, when placed in the box, will make the equation true?
_^2 =square root of 64
Answers
Answer:
2.828 (the square root of 8)
Step-by-step explanation:
2.828 squared is 8, which is also the square root of 64
The value in the missing is x = 2√2 or x = -2√2 or x = ±√8 if the expression is _² = √64.
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
We have a mathematical expression:
_² = √64
Let the missing number be x:
Then, the equation becomes:
x² = √64
√64 = √(8x8) = 8
x² = 8
The above equation represents a quadratic equation, Any equation of the form [tex]\rm ax^2+bx+c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.
As we know, the formula for the roots of the quadratic equation is given by:
[tex]\rm x = \dfrac{-b \pm\sqrt{b^2-4ac}}{2a}[/tex]
x = ±√8
x = ±2√2
Thus, the value in the missing is x = 2√2 or x = -2√2 or x = ±√8 if the expression is _² = √64.
Learn more about the expression here:
brainly.com/question/14083225
#SPJ5
What is the probability that a randomly chosen four-digit integer (no leading zeros) has distinct digits and ends with the digit 7? For example, 1237 qualifies for the event space but 1227 does not.
All of the possible answers are rounded to four decimal digits.
a. 0.3484
b. 0.0448
c. 0.0498
d. 0.0720
Answers
Answer:
c. 0.0498
Step-by-step explanation:
First of all, let us find the number of four digit numbers possible.
There are 4 digits and let us have a look at the possibility of each digit.
Number of possible options for Unit's digit = 10
Number of possible options for ten's digit = 10
Number of possible options for hundred's digit = 10
Number of possible options for thousand's digit = 9 (because 0 can not be there to make it a 4 digit number)
Total number of possible outcomes = [tex]10\times 10\times 10\times 9 = \bold{9000}[/tex]
As per the given condition, unit's digit is 7.
So, number of possible options for unit's digit = 1
Number of possible options for thousand's digit = 8 (7 and 0 can not be there to make it a 4 digit number)
Number of possible options for hundred's digit = 8 (7 and one digit used in thousand's place can not be there)
Number of possible options for ten's digit = 7 (7, two digits used at thousand's and hundred's places)
Number of possible outcomes as per given conditions = [tex]1\times 8\times 8\times 7 = \bold{448}[/tex]
Formula for probability of an event E can be observed as:
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}[/tex]
[tex]P(E) = \dfrac{448}{9000} =\bold{0.0498}[/tex]
the number of ducks in a pond was 345. If 33 1/3% of the ducks flew away. how many were left
Answers
Answer: 230 ducks were left.
Step-by-step explanation:
Answer:
31 ducks
Step-by-step explanation:
345 ÷ 33 1/3% = ducks left.
345 ÷ 33 1/3% = approximately 31
So once solved, you can conclude that there are 31 ducks left in the pond.
Can u solve this one attached below
Answers
Answer:
[tex]\angle 6, \angle 4[/tex]
Step-by-step explanation:
[tex]\angle 1\ and\ \angle 2\ are\ remote\ interior\ angles\ of\ \angle 6\ as\ they\ are\ opposite\ to\\ its\ linear\ pair\ \angle 3.\\\angle 2\ and\ \angle 3\ are\ remote\ interior\ angles\ of\ \angle 4\ as\ they\ are\ opposite\ to\\ its\ linear\ pair\ \angle 1.[/tex]
A textbook costs $65.49 before tax. The tax on the textbook is 6.5%.
What is the total cost of the textbook?
Round your answer to the nearest cent.
Enter your answer in the box.
Answers
Answer:
Step-by-step explanation:
the tax will cost
65.49*6.5=4.26
the total price of the book is
65.49+4.26=69.75
Answer:
70.74
Step-by-step explanation:
took the test (im in k-12) btw
Given sinθ = 2/5 and cosθ < 0:
1. Find cscθ
2. Find cotθ
Answers
Recall that
sin²(θ) + cos²(θ) = 1
for all θ, and given that cos(θ) < 0, we find that
cos(θ) = -√(1 - sin²(θ)) = -√(1 - (2/5)²) = -√(21)/5
Now,
csc(θ) = 1/sin(θ) = 1/(2/5) = 5/2
and
cot(θ) = cos(θ)/sin(θ) = (-√(21)/5)/(2/5) = -√(21)/2
Out of 40 students in class 8,20% were absent.How many students were present?
Answers
Answer:
32 students were present.
Step-by-step explanation:
To find out how many students were present in class 8, we have to find out how many students were absent first.
As the question has already provided us, the number of students who are absent would be [tex]40 \cdot 20\%[/tex], which equals to [tex]40 \cdot \frac{20}{100}[/tex] which is 8 students in total who were absent in class 8.
Now that we have found out how many students were absent, simply subtract the number of students who were absent from the total number of students in class 8, which is (40 - 8) = 32 students, and you have the number of students that were present: 32.
Hope this helped!
1. David has $40 to spend at the state fair. The entrance fee is $15 and each
ride costs $1.25. Write an equation that best represents the situation. Then
solve to find out how many rides he can go on.
Answers
Answer:
let he takes x ride in total
1
$15 + 1.25x = $40
on solving
1.25x = 40 -15
1.25x=25
x=25/1.25 = 20
He can go on 20 rides.
Step-by-step explanation:
I did not do as good as I hoped on my math model :(
Answers
Answer:
you're going to get an A
Step-by-step explanation:
.............
The TV repairman charges $40 to come out to your house and check the TV
plus $25 per hour of labor. Write an equation to find the cost of repairing a TV.
Answers
Answer:
The equation to find the cost of repairing a TV is:
y = 25x + 40
Step-by-step explanation:
We know that the slope-intercept form of the line equation is
[tex]y=mx+b[/tex]
where m is the slope or rate of change and b is the y-intercept
Given that the TV chairman charges $40 to come out to your house and check the TV plus $25 per hour of labor.
It means $40 is the y-intercept because it represents the starting point/condition of the repairman.
Let 'x' be the number of hours and y be the cost.
The rate of change = 25$ per hour of labor
The rate of change can also be termed as the slope.Thus, m = 25
Hence, the slope-intercept form of the line equation becomes
y = mx+b
substituting m = 25 and b=40
y = 25x + 40
Therefore, the equation to find the cost of repairing a TV is:
y = 25x + 40
Validation:
y = 25x + 40
putting x = 1 to determine the cost if labor works for 1 hour.
y = 25(1) + 40 = $65
putting x = 2 to determine the cost if labor works for 2 hours.
y = 25(2) + 40 = $95
CITIES: The distance between New York City, New York, and Boston, Massachusetts, is 187 miles, and the distance between New York City and
Hartford, Connecticut, is 97 miles. Hartford, Boston, and New York City form a triangle on a map. What must the distance between Boston and
Hartford be greater than?
Answers
Answer:
Greater than 90 miles
Step-by-step explanation:
Use the triangle inequality theorem, since the cities form a triangle.
This theorem states that the sum of any 2 sides of a triangle has to be greater than the length of the third side.
We can find what the length of the third side has to be greater than by subtracting 97 from 187:
187 - 97
= 90
So, the distance has to be greater than 90 miles.
Literal equations
Show work
Solve for x
-3(x + n) = x
Answers
Answer:
x=-3/4n
Step-by-step explanation:
first, remove the parenthesis.
-3(x + n) = x
-3x -3n = x
collect like terms.
-3x - x = 3n
divide both sides.
-4 = 3n
and therefore,
x= -3/4n
<8 and <6 are ___________.
0443e860-aba2-49e1-952c-862061d9e91a?w=400&h=400
a
Same-side Interior Angles
b
Alternate Exterior Angles
c
Alternate Interior Angles
d
Corresponding Angles
Answers
which fraction has an equivalent decimal value that repeats
Answers
Answer:
The answer is 2/3
Step-by-step explanation:
2/3 = 0.6666666666... with it repeating forever
Answer:
2/3
Step-by-step explanation:
The answer you selected is correct.
The decimal form of 2/3 is 0.66666666 (it repeats forever), unlike the other three:
3/4 = 0.75
4/5= 0.80
7/8= 0.875
What is the range of the function y = |x| - 9 ?
Answers
Answer:
Step-by-step explanation:
(a)
Percentage of girls in the school.
(c)
There are 1400 students in a school out of which 780 are girls, find the,
Number of boys in the school.
(b) Percentage of boys in the school.
Answers
Answer:
(a) Percentage of girls = 55[tex]\frac{5}{7}[/tex]
(b) Percentage of boys = 44[tex]\frac{2}{7}[/tex]
Step-by-step explanation:
To find percentage, we multiply the fraction out of the total by 100%.
Percentage of girls = [tex]\frac{780}{1400}[/tex] × 100%
= 55[tex]\frac{5}{7}[/tex]
Percentage of boys = [tex]\frac{1400-780}{1400}[/tex] × 100%
= 44[tex]\frac{2}{7}[/tex]
HELP ASAP! 100 POINTS + MARKED BRAINLIEST‼️‼️ WHATS THE ANSWER
Answers
Answer:
-4.834 / 73
Step-by-step explanation:
Two points: (2007, 84.134) and (2080, 79.300)
Use the slope formula.
m = y2-y1/x2-x1
m = 79.300 - 84.134 / 2080 - 2007
m = -4.834 / 73
Find the quotient and the remainder when you divide f ( x ) = x 3 + 2 x 2 + 4 x + 6 by x − 2
Answers
Answer:
x^2 + 4x + 12 + [tex]\frac{30}{x-2}[/tex]
Step-by-step explanation:
idrk how to explain this one and im on a computer so i cant take a picture of my work but just divide lol
divide like you would with normal division kinda lol
hope this helps <3
what is the absolute value of the product of all the integers from -6 to 3inclusive
Answers
This product involves zero. So, you don't have to bother and compute all the factor, because if you multiply anything by zero you get zero.
What is the area of this. Pls help.
Answers
Answer: 49cm squared *tiney 2 above cm*
Step-by-step explanation:
100 POINTS AND BRAINLIST
Does anyone know how to do Polygons on the Coordinate Plane
I need your email
Answers
Answer:
You can use the coordinates of a figure to find its dimensions by finding the distance between two points. To find the distance between two points with the same x-coordinates, subtract their y-coordinates. To find the distance between two points with the same y-coordinates, subtract their x-coordinates.
Step-by-step explanation:
__% of 80 is 40 what is the =
Answers
PLEASE HELP!! PLEASE SHOW WORK OR EXPLAIN!! I WILL GIVE BRAINLIEST!!
Answers
Answer:
-4
Step-by-step explanation:
The equation for slope is as follows:
[tex]\frac{y2 - y1}{x2 - x1}[/tex]We can use point P and point V's coordinates to find the slope.
Point P is at (2,4)
Point V is at (3,0)
Let's define all the variables:
y2 = 0 (from point V)
y1 = 4 (from point P)
x2 = 3 (from point V)
x1 = 2 (from point P)
Now that we have all the variables we can plug it into the equation:
[tex]\frac{0-4}{3-2} = \frac{-4}{1} = -4[/tex]
Therefore, the slope of line PV is -4.
I hope this helps!!
Feel free to ask any questions!
- Kay :)
need help please, i don’t get it
Answers
This is the same as saying x = 3 and x = 7.
========================================
Work Shown:
Plug x = 3 into f(x)
f(x) = (1/2)*|x-3| + 3
f(3) = (1/2)*|3-3| + 3
f(3) = 0.5*|0| + 3
f(3) = 0.5*0 + 3
f(3) = 0 + 3
f(3) = 3
-----------------------
Plug x = 5 into f(x)
f(x) = (1/2)*|x-3| + 3
f(5) = (1/2)*|5-3| + 3
f(5) = 0.5*|2| + 3
f(5) = 0.5*2 + 3
f(5) = 1 + 3
f(5) = 4
-----------------------
Plug x = 7 into f(x)
f(x) = (1/2)*|x-3| + 3
f(7) = (1/2)*|7-3| + 3
f(7) = 0.5*|4| + 3
f(7) = 0.5*4 + 3
f(7) = 2 + 3
f(7) = 5
-----------------------
Plug x = 3 into g(x)
g(x) = sqrt(x-3) + 3
g(3) = sqrt(3-3) + 3
g(3) = sqrt(0) + 3
g(3) = 0 + 3
g(3) = 3
-----------------------
Plug x = 5 into g(x)
g(x) = sqrt(x-3) + 3
g(5) = sqrt(5-3) + 3
g(5) = sqrt(2) + 3
g(5) = 1.414214 + 3
g(5) = 4.414214 ..... which is approximate
-----------------------
Plug x = 7 into g(x)
g(x) = sqrt(x-3) + 3
g(7) = sqrt(7-3) + 3
g(7) = sqrt(4) + 3
g(7) = 2 + 3
g(7) = 5
-----------------------
To summarize everything, refer to the table below.
The rows highlighted in yellow refer to when f(x) = g(x)
This happens twice when x = 3 and x = 7
If x = 3, then f(x) = g(x) = 3
If x = 7, then f(x) = g(x) = 5
Therefore, x = 3 and x = 7 are two solutions to f(x) = g(x)
100 Points and Brainliest to whoever helps me!
Answers
step-by-step explanation:
this would make it the y intercept.
Answer:
thirsty
Step-by-step explanation:love pionts
2x-y=5 in slope intercept form?
Answers
Answer:
y = 2x - 5
I hope this helps!
need help I don’t understand this.
Answers
what is the angle of elevation to the nearest tenth of a degree to the top of a 35ft building from 75ft away?
Answers
Answer:25°
Step-by-step explanation:
tanθ= 35/72