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
olga_2 [115]
3 years ago
6

A balloon rises at the rate of 8 feet per second from a point on the ground 12 feet from an observer. To 2 decimal places in rad

ians per second, find the rate of change of the angle of elevation when the balloon is 9 feet above the ground. Type your answer in the space below. If your answer is a number less than 1, place a leading "0" before the decimal point

Mathematics
1 answer:
butalik [34]3 years ago
8 0

Answer:

  0.43 radians/second

Step-by-step explanation:

The relationship between the horizontal distance to the balloon (x), the vertical height of the balloon (y), and the angle (θ) is given by ...

  \dfrac{y}{x}=\tan{\theta}

Differentiating with respect to time gives ...

  \dfrac{y'}{x}=\dfrac{\theta '}{\cos^2{\theta}}\\\\\theta '=\dfrac{y'\cos^2{\theta}}{x}=\dfrac{xy'}{x^2+y^2}\\\\\theta '=\dfrac{12\cdot 8}{12^2+9^2}=\dfrac{96}{225}=0.42\overline{6}\approx 0.43 \quad\text{rad/s}

The rate of change of the angle of elevation is about 0.43 radians per second when the balloon is 9 feet high.

_____

You can obtain a good numerical approximation of this value by looking at the angles 0.05 seconds before and after the balloon is 9 ft high. Then the rate of change is approximately that difference divided by the time difference of 0.10 seconds.

  \theta '\approx\dfrac{\arctan{\dfrac{9.4}{12}}-\arctan{\dfrac{8.6}{12}}}{0.1}\approx\dfrac{0.664495-0.621824}{0.1}\\\\\theta '\approx\dfrac{0.042671}{0.1}=0.42671\approx 0.43 \quad\text{rad/s}

The key to a good approximation is to use ballon height values both below and above the nominal height. The smaller the time interval, the better the approximation. For this problem a time interval of 0.1 seconds is small enough to get sufficient accuracy for a 2 decimal place answer.

Some calculators will let you define a function that gives angle in terms of balloon height, then can calculate the numerical value of the derivative for you. (see attached)

You might be interested in
Decompose 2/6 into 4 equal lengths
anyanavicka [17]
I'm not 100% sure, but I'm pretty sure it's 1/12, 1/12, 1/12, 1/12
4 0
3 years ago
Compare these two functions, where the input variable represents days.
olga2289 [7]

Answer:

c.) 70

# carry on learning

4 0
3 years ago
jimmy is moving to a new apartment. he fills a box with 4 books, 5 DVDs, and 2 CD's. when he starts unpacking his new place he p
pav-90 [236]

Answer:

\frac{2}{11}

Step-by-step explanation:

First, find the probability of him pulling out a book.

There are 11 objects in total, and there are 4 books.

So, the probability of pulling out a book is \frac{4}{11}

Next, find the probability of him pulling out a DVD after.

Since a book was taken out, there are only 10 objects left. There are 5 DVDs.

So, the probability of pulling out a DVD is \frac{5}{10}, or simplified to \frac{1}{2}

To find the probability that this happens in order, multiply the probabilities:

\frac{4}{11} x \frac{1}{2}

= \frac{2}{11}

So, the probability is \frac{2}{11}

7 0
3 years ago
What is the slope of the line through [-1,2] and [2,-3]
Gennadij [26K]

Answer:

-3/5

Step-by-step explanation:

to figure that out you have to do y^2 - y^1/ x^2 - x^1

2-(-1)/ -3 - 2

= 3/-5

=-3/5

8 0
3 years ago
A one year membership to Metro Gym costs $460. There is a fee of $50 when you join and the rest is payed monthly. How much do th
dexar [7]
Wouldn't it be $34.17? 
5 0
3 years ago
Read 2 more answers
Other questions:
  • A right triangle has legs of length 4x and 20x. What is an expression for the hypotenuse of the right triangle
    14·1 answer
  • What is the greatest possible whole number remainder if you divide any number by 41
    11·2 answers
  • Solve 4+(-8+s)=20<br> -4<br> 16<br> 24<br> 8
    11·1 answer
  • Benito is selling T-shirts for $8 each for his school fund-raiser. So far, he has sold 16 T-shirts. How many more does he need t
    10·1 answer
  • What is the word form of the number 43.06?
    10·2 answers
  • Choose the multiplication equation that is related to the division equation seven divided by one sixth.
    9·1 answer
  • Find the value of x....
    13·1 answer
  • Clark, Kasalyn, and Carson wrote an equation of a line with a slope of -5 that passes through the point at (-2,4). Is either of
    9·1 answer
  • Help me pleeaassee :P
    15·2 answers
  • Can anyone help me solve this​
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!