A guitar string is plucked at a distance of 0.6 centimeters above its resting position and then released, causing vibration. The damping constant of the guitar string is 1.8, and the note produced has a frequency of 105 cycles per second.
a.) Write a trigonometric function that models the motion of the string.
b.) Using a graphing calculator, determine the amount of time t that it takes the string to be damped so that -0.24 y 0.24. Please be sure to show a screenshot of your graph.
Answers
Answer:
The equation that represents the motion of the string is given by:
[tex]y =Ae^{-kt}\cos(2\pi ft)[/tex] .....[1] where t represents the time in second.
Given that: A = 0.6 cm (distance above its resting position) , k = 1.8(damping constant) and frequency(f) = 105 cycles per second.
Substitute the given values in [1] we get;
[tex]y =0.6e^{-1.8t}\cos(2\pi 105t)[/tex]
or
[tex]y =0.6e^{-1.8t}\cos(210\pi t)[/tex]
(a)
The trigonometric function that models the motion of the string is given by:
[tex]y =0.6e^{-1.8t}\cos(210\pi t)[/tex]
(b)
Determine the amount of time t that it takes the string to be damped so that [tex]-0.24\leq y \leq0.24[/tex]
Using graphing calculator for the equation
[tex]y =0.6e^{-1.8x}\cos(210\pi x)[/tex]
let x = t (time in sec)
Graph as shown below in the attachment:
we get:
the amount of time t that it takes the string to be damped so that [tex]-0.24\leq y \leq0.24[/tex] is, 0.5 sec
The trigonometric function that models the motion of the guitar string is y(t) = Asin(ωt)e^(-kt). Using a graphing calculator, the time it takes for the string to be damped within the given range can be determined.
Explanation:a) The trigonometric function that models the motion of the guitar string can be represented by the equation y(t) = Asin(ωt)e^(-kt), where A is the amplitude, ω is the angular frequency, t is the time, and k is the damping constant.
b) To determine the amount of time it takes for the string to be damped so that -0.24 ≤ y ≤ 0.24, you can use a graphing calculator to plot the trigonometric function and find the values of time at which the function crosses these y-values. You should set up the equation as follows: -0.24 ≤ Asin(ωt)e^(-kt) ≤ 0.24 and solve for t.
Please see the attached screenshot for an example of the graph.
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Related Questions
Compare using the quadratic formula to find solutions to a quadratic equation having irrational roots to that of one that has rational roots.
Answers
For a general quadratic equation with rational coefficients
[tex]ax^2+bx+c=0,\,\,\,\,a,b,c \in Q[/tex]
the two solutions are:
[tex]x_{1,2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-b\pm\sqrt{D}}{2a}[/tex]
where D is the determinant.
Clearly, a solution will a rational number as long as [tex]\sqrt{D}[/tex] is rational. However, it can be shown that a square root of an integer is only rational if its value is an integer. In other words, [tex]\sqrt{D}[/tex] is rational if and only if the determinant is a perfect square, [tex]D=n^2, \,\,\,n\in N[/tex], otherwise the square root is irrational. Therefore the coefficients of quadratic equations that are to have rational solutions must satisfy the following condition:
[tex]b^2-4ac=n^2\,\,\,n\in N[/tex]
The exponential distribution is frequently applied to the waiting times between successes in a poisson process. if the number of calls received per hour by a telephone answering service is a poisson random variable with parameter λ = 6, we know that the time, in hours, between successive calls has an exponential distribution with parameter β =1/6. what is the probability of waiting more than 15 minutes between any two successive calls?
Answers
The question requires us to use the exponential distribution to calculate the probability of waiting more than 15 minutes between two successive calls given a Poisson process with β parameter 1/6. We achieve this by using the cumulative distribution function. The probability is approximately 96.4%.
Explanation:This question is essentially asking us to use the exponential distribution to calculate the probability of waiting for more than 15 minutes (which is 0.25 hour because we are dealing with hours as units of time) between two successive calls when the parameter β =1/6 is given. The exponential distribution is used to model the time between events in a Poisson process, and the probability density function is given by f(x) = βe^(-βx) for x ≥ 0.
The cumulative distribution function is F(x) = 1 - e^(-βx), so to compute the probability of waiting time that is more than 0.25 hour, we subtract the cumulative distribution function from 1.
So, P(X > 0.25) = 1 - P(X ≤ 0.25) = 1 - [1 - e^(-β * 0.25)] = e^(-β * 0.25) = e^(-(1/6) * 0.25) ≈ 0.964, or 96.4% as the answer.
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Michelle has a maximum of 4500 milliliters of water for her plants today. Each basil plant requires 350 of water, and each fennel plant requires 525 of water. Write an inequality that represents the number of basil plants (B)(B)left parenthesis, B, right parenthesis and fennel plants (F) Michelle can water today.
Answers
Answer:
Let B represents the number of basil plants and F represents the number of fennel plants,
As per the given condition: Michelle has a maximum of 4500 milliliters of water for her plants today. Each basil plant requires 350 of water, and each fennel plant requires 525 of water.
Each basil plant require 350 of water
⇒ total number of basil plant require water = 350B
Each Fennel plant require 525 of water.
⇒ total number of fennel plant require water = 525F
Since, Michelle has a maximum of 4500 milliliters of water;
then;
[tex]350B+525F \leq 4500[/tex]
Therefore, an inequality that represents the number of basil plants(B) and fennel plants can water today is; [tex]350B+525F \leq 4500[/tex]
Find the? mean, median, and mode of the set of values. Age? (years) 11 12 13 14 15 16 17 Frequency 7 9 11 10 9 4 3 Find the mean. Select the correct choice below? and, if? necessary, fill in the answer box to complete your choice. The mean is approximately nothing years. ?(Round to two decimal places as? needed.)
Answers
Answers:
Mean = 13.55
Median = 13
Mode = 13
====================================================
Explanations
Mean:
To get the mean, multiply the ages with their corresponding frequencies. Then add up the results. I show this in the attached image below. The highlighted yellow cell is the sum of the x*y column (x = age, y = frequency). This value is 718.
Once you get to 718, you divide this over the total frequency which is 7+9+11+10+9+4+3 = 53 (so there is 53 people).
The mean is therefore 718/53 = 13.5471698113208 which rounds to 13.55
----------------------
Median:
The median is the middle most value. There are 53 values here (add up the frequencies), so the midpoint is at slot 53/2 = 26.5 = 27. The first 26 values are below the median, and the last 26 values are above the median leaving 53-26-26 = 53 - 52 = 1 value in the very exact center.
Add up the frequencies starting from 7 then to 9, etc until you get to 27. So we have
7+9 = 16 which isn't 27
7+9+11 = 16+11 = 27 perfect, we landed on 27
This means that the last copy of 13 is in slot 27. This is because the last frequency added (11) corresponds to the age 13.
That is why the median is 13
----------------------
Mode:
The mode is the most frequent value. Simply record the age that has the highest frequency. In this case, the highest frequency is 11 which corresponds to age 13, pointing to the mode being 13.
If a certain cannon is fired from a height of 9.1 meters above the ground, at a certain angle, the height of the cannonball above the ground, h, in meters, at time, t, in seconds, is found by the function h(t) = -4.9t² + 27.5t + 9.1. Find the time it takes for the cannonball to strike the ground.
The cannonball will strike the ground after about ___ seconds.
(Type an integer or a decimal. Round to the nearest hundredth as needed.)
Answers
Answer:
The cannonball will hit the ground after about 5.926 seconds.
Step-by-step explanation:
h(t) = -4.9t² + 27.5t + 9.1
If you graphed the function on a graph, the cannonball would be hitting the ground when the function crossed the x-axis at 0. So, to solve this arithmetically, you just need to set h(t) equal to 0.
-4.9t² + 27.5t + 9.1 = 0 Plug this into a calculator if you have one, if not solve with the quadratic formula.
[tex]\frac{-27.5 \pm \sqrt{(27.5^2) - 4(-4.9)(9.1)} }{2(-4.9)}[/tex]
t = -.0313
t = 5.9256
Since time can't be negative, you know your answer will be 5.926 seconds.
Answer:
t =5.93 seconds
Step-by-step explanation:
h(t) represents the height of the cannon ball. Zero is when the ball will hit the ground. Substitute zero for h(t).
0 = -4.9t² + 27.5t + 9.1
This is a complicated quadratics, so we will need to use the quadratic formula to solve
-b ± sqrt(b^2 -4ac)
-----------------------------
2a
where a = -4.9 b = 27.5 and c = 9.1
-27.5 ± sqrt(27.5 ^2 -4 (-4.9) 9.1)
-----------------------------------------------
2(-4.9)
-27.5 ± sqrt(756.25 +178.36)
-----------------------------------------------
-9.8
-27.5 ± sqrt(943.61)
-----------------------------------------------
-9.8
-27.5 ± 30.57139186
-----------------------------------------------
-9.8
3.071391856/-9.8 or -58.07139186/-9.8
-.313407332 or 5.925652231
But time cannot be negative, the ball cannot land before it takes off,
so t= 5.925652231 seconds
Rounding to the nearest hundredth
t =5.93 seconds
Identify the perimeter and area of an equilateral triangle with height 12√2 cm. Give your answer in simplest radical form. HELP PLEASE!!!
Answers
Answer:
perimeter = 24√6 cmarea = 96√3 cm²Step-by-step explanation:
The edge length (s) of an equilateral triangle is (2/3)√3 times the height. This means the edge length of your triangle is ...
... s = (2/3)√3 × 12√2 = 8√6
The perimeter is 3 times the side length:
... p = 3s = 3×8√6 = 24√6 . . . cm
The area is half the product of edge length and height:
... a = (1/2)sh = (1/2)(8√6)(12√2) = 48√12 = 96√3 . . . cm²
The perimeter of the equilateral triangle is 24√6 cm, and its area is 96√3 cm².
To find the perimeter and area of an equilateral triangle with a given height, we can follow these steps:
First, let's denote the side length of the equilateral triangle as a. In an equilateral triangle, the height (h) can be found using the formula:
h = (a√3)/2
We know the height is 12√2 cm. Therefore,12√2 = (a√3)/2
To find the side length a, we can rearrange the equation: a = 2 * 12√2 / √3
Simplify the expression: a = 24√2 / √3
Multiplying numerator and denominator by √3 to rationalize the denominator: a = 24√6 / 3
a = 8√6
Now, the perimeter (P) of the equilateral triangle is given by: P = 3a
Substitute the value of a: P = 3 * 8√6
P = 24√6 cm
Next, we calculate the area (A) using the formula: A = (base * height) / 2
In an equilateral triangle, the base is equal to the side length a, and the height is given as 12√2 cm:
A = (8√6 * 12√2) / 2
Multiplying the bases and simplifying: A = (96√12) / 2
Since √12 can be simplified to 2√3: A = (96 * 2√3) / 2
A = 96√3 cm²
The perimeter of the equilateral triangle is 24√6 cm, and its area is 96√3 cm².
i need help 6 7 is due tommorw pls help
Answers
Answer:6: 8.64
7: 584
Step-by-step explanation:
48×.18=8.64
800×.73=584
Hope this helps
Answer:
Answer for question number 7 is 58.4
Step-by-step explanation:
If the speed was 65 for 1 minutes and drove at a constant speed for 5.5 minutes for a linear equation
Answers
Answer:
65,1,5.5
Step-by-step explanation:
Replace with 1 and simplify.
the equation for distance is d= st. if a car has a speed of 35 m/s for 15 seconds, how far does it go ?
A. 300m
B. 50 M
C.2.33 M
D.525 M
Answers
Answer:
D.525 M
Step-by-step explanation:
We know the equation
d = st
where d is the distance, s is the speed and t is the time
The problem gives us the speed of 35 m/s and the time of 15 seconds
s = 35 m/s and t =15 s, we can substitute them in
d =st
d = 35 * 15
d = 525 m
Answer:
525 m
Step-by-step explanation:
please help me will give braneliest
Answers
Answer:
3/40
Step-by-step explanation:
After you multiply across, you get 3/40. Since these two numbers don't have common multiples, it just stays as 3/40
Answer:
the third option
Step-by-step explanation:
1*3= 3
10*4= 40
3/40
The store is 5/6 miles .Yuri house.The bank is 2/5 of the distance from Yuri's house from the store.How many miles is the bank from the store
Answers
Answer: I think it 3/1
Step-by-step explanation: Because 2 to get to 5 is 3 and 5 to get to 6 is 3. Hope I am correct
15 boys and 16 girls took part in the basketball competition. What is the ratio of the number of girls to the number of boys who participate in the competition?
Answers
Answer:
The ratio of girls to boys is 16:15
Step-by-step explanation:
To find this, we simply start by using the words.
Girls:Boys
Now we plug in the values.
16:15
Write the equation of the given line in slope-intercept form.
Answers
Answer:
y = 2x - 1
Step-by-step explanation:
The line passes through points A(1, 1) and B(2, 3).
The slope-intercept form of the equation of a line is
y = mx + b,
where m = slope, and b = y-intercept.
We see from the graph that the y-intercept is -1, so b = -1, and we have
y = mx - 1
Going from point A to point B, we go up two units and right 1 unit. Slope = m = difference in y over difference in x, so
m = 2/1 = 2
The equation is
y = 2x - 1
Answer: 2x + -1
Step-by-step explanation: In this problem, we're asked to write the equation of the given line in slope-intercept form.
Slope-intercept form is the same thing as y = mx + b form where m represents the slope of the line and b represents the y-intercept. So our first step is to find the slope and the y-intercept of the given line.
To find the slope, we use the ratio rise over run between any two points on the line. To get from point A to point B along this line, we rise 2 units and run 1 unit. So our slope or rise over run is 2/1 or just 2.
Next, to find the y-intercept, it's important to understand that the y-intercept is the point where the line crosses the y-axis. Notice that this line crosses the y-axis at the point (0,-1) which means that the y-intercept is -1.
Therefore, m = 1/2 and b = -1 and we substitute these values into our formula for m and b to get y = 2x + -1.
So the equation of the given line in slope-intercept form is y = 2x + -1.
A total of 626 tickets were sold for the school play. They were either adult tickets or student tickets. There were 74 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answers
Answer:
351 adults and 275 students
Step-by-step explanation:
We can set-up a system of equations to find the number of adults. We know students and adults attended. We will let s be the number of students and a be the number of adults. Since 626 tickets were purchased, then s+a=626.
We also know that 74 fewer student tickets then adult tickets. So s+76=a, the number of student tickets plus 76 will be the number of adult tickets.
We will solve by substituting one equation into the other. We substitute a=76+s into s+a=626. Simplify and isolate the variable a.
s+a=626
s+76+s=626
2s+76=626
2s+76-76=626-76
2s=550
s=275
This means that 275 students attended and 351 adults attended since 275+351=626.
By setting up a system of equations and solving for the number of adult tickets, we are able to determine that 350 adult tickets were sold for the school play.
To find out how many adult tickets were sold, we can set up an equation to model the situation. Let A be the number of adult tickets and S be the number of student tickets. We are given that a total of 626 tickets were sold, which can be described by the equation A + S = 626. We are also told that there were 74 fewer student tickets sold than adult tickets, described by the equation S = A - 74.
To solve for A, we can substitute (A - 74) for S in the first equation, giving us A + (A - 74) = 626. Simplifying this, we get 2A - 74 = 626. Adding 74 to both sides, we get 2A = 700. Finally, dividing both sides by 2, we find A = 350.
Therefore, 350 adult tickets were sold.
Is the movie is 28 minutes and 58 seconds and we have watched 13 minutes and 44 second how much long do we have before we finish the movie
Answers
Final answer:
To find out how much longer we have before we finish the movie, subtract the time already watched from the total duration of the movie.
Explanation:
To find out how much longer we have before we finish the movie, we need to subtract the time we have already watched from the total duration of the movie.
Given that the movie is 28 minutes and 58 seconds long, and we have already watched 13 minutes and 44 seconds, we subtract the time already watched from the total duration:
28 minutes 58 seconds - 13 minutes 44 seconds = 15 minutes 14 seconds.
Therefore, we have 15 minutes and 14 seconds left before we finish the movie.
This question is worth Points and Brainliest!!!!! HURRY
Which of the following is the solution to the equation 4 over 5n = 20?
A) n = 4
B) n = 12
C) n = 16
D) n = 25
Answers
Answer:
D) n=25
Step-by-step explanation:
Answer:
25
Step-by-step explanation:
I didi the quiz
You will flip a coin five times. How many ways can you get three heads and two tails?
Jill is trying to solve this probability question. Which row in Pascal's Triangle would tell her the number of outcomes for this event?
A) second row
B) third row
C) fifth row
D) sixth row
Answers
Answer:
10 ways, 6th row
Step-by-step explanation:
You will flip a coin five times and get three heads and two tails. You can get it in
[tex]C_5^3=\dfrac{5!}{3!(5-3)!}=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5}{1\cdot 2\cdot 3\cdot1\cdot 2}=10[/tex]
different ways.
6th row tells her the number of favorable outcomes for this event.
Nicolas sold 20 1/2 pounds of shrimp in 4 hours. At this rate, how many pounds of shrimp did he sell i 1 hour? How many pounds of shrimp can he sell in 9 hours.
Answers
20 1/2÷4= 5 1/8 1 hour
for 9 hours = 5 1/8×9
=46 1/8
Nicolas sells 5.125 pounds of shrimp per hour.
Therefore, he can sell 46.125 pounds of shrimp in 9 hours.
Explanation:To determine how many pounds of shrimp Nicolas sold in one hour, we need to divide the total pounds of shrimp by the total number of hours.
Nicolas sold 20 1/2 pounds of shrimp in 4 hours.
Therefore, we divide 20.5 (which is the decimal equivalent of 20 1/2) by 4 to find the rate per hour:
20.5 ÷ 4 = 5.125
Nicolas sold 5.125 pounds of shrimp per hour.
To find out how many pounds he can sell in 9 hours, we multiply the rate per hour by 9:
5.125 × 9 = 46.125
Nicolas can sell 46.125 pounds of shrimp in 9 hours.
The table below shows the temperature in degrees for eight consecutive days as well as the respective number of ice cream cones an ice cream shop sold on each of these days.
Temperature: 68 77 83 85 89 94 96 99
Number of cones: 403 447 457 465 489 503 543 576
about how many ice cream cones would you expect the shop to sell if the temperature one day is 106 degrees?
Answers
Answer:
We have to find the equation of least squares regression line in order to find the number of ice cream cones that the shopkeeper to sell if the temperature is 106 degrees. We can use excel regression data analysis tool to find the equation of the regression line. The excel output is attached here.
The equation of the least squares regression line is:
[tex]\hat{y}=46.587+5.080x[/tex]
Now, if the temperature is 106 degrees, then the number of cones expected to be sold is given below:
[tex]\hat{y}=46.587+5.080 \times 106[/tex]
[tex]=585.1 \approx 585[/tex]
Therefore, the number of ice cream cones that the shopkeeper would expect to sell if the temperature is 106 degrees is 585.
Jason spends 1/3 of each day sleeping.What is the total numbers of days that jason spend awake in 1 week?
Answers
Answer:
Given the statement: Jason spends 1/3 of each day sleeping
Since, a day has 24 hours and
Jason spends each day [tex]\frac{1}{3}\times 24 = 8 hours [/tex] in sleeping .
Remaining Jason spend awake each day, 24 - 8 = 16 hours.
In 1 day = Jason spend 16 hours in awake
then;
in 1 week(i.e, 7 days) = [tex]16 \times 7 = 112 hours[/tex] spend in awake.
To find the total number of days Jason spend awake in 1 week.
Divide 112 hours by 24 we get;
[tex]\frac{112}{24} = \frac{14}{3} =4\frac{2}{3}[/tex] days
therefore, [tex]4\frac{2}{3}[/tex] is the total number of day that Jason spend awake in 1 week.
A company had a competition to see which team could load trucks at the loading dock the fastest. On day one, team one loaded 5 1?2 trucks. The leader of team three said her team could load 3 1?2 times that number when it was their turn to participate. How many trucks does that team expect to load?
Answers
Answer:25
Step-by-step explanation:cause it is
To calculate the number of trucks Team Three expects to load based on the information provided.
To calculate the number of trucks Team Three expects to load:
Calculate 3.5 times the number of trucks loaded by team one: 5.5 trucks x 3.5 = 17.5 trucks.
Therefore, team three expects to load 17.5 trucks.
PLEASE HELP ASAP!!! CORRECT ANSWERS ONLY PLEASE!!!
Use the imaginary numberi to rewrite the expression as a complex number
Answers
Answer:
i think the answer would be C
Step-by-step explanation:
since it seemed you had to divide the numbers a hundred would turn into ten and ninety would turn into nine and because you have one negative and one positive that means it would have to be a negative so it would be
10 - 9i
Answer: B
Step-by-step explanation:
[tex]\sqrt{100} + \sqrt{-81}[/tex]
[tex]\sqrt{100} = 10[/tex][tex]\sqrt{-81} = \sqrt{-1} * \sqrt{81} = i * 9 = 9i[/tex][tex]\sqrt{100} + \sqrt{-81}[/tex]
= 10 + 9i
can someone plz help with this but with work shown
Answers
Find the solution to the system of equations graphed here:
Answers
Answer: (1,3)
All you're doing is looking for where the two lines cross, or the point of intersection. What you can do is draw a vertical line through this intersection point to see that the vertical line lands on x = 1 on the x axis. At the same time, draw a horizontal line to the y axis and it will get to y = 3.
So together x = 1 and y = 3 pair up to get (x,y) = (1,3)
Miguel deposits $680 in an account that pays 3.5% simple interest. If he neither adds more money not withdraws any money, what amount will be in the account after 6 years?
Answers
Answer:835.893
Step-by-step explanation:
do the interest rate thing you learned in class
if you didn't learn it im adding a picture on the bottom for help
Liam is setting up folding chairs for a meeting. If he arranges the chairs in 4 rows of the same length, he has 3 chairs left over. If he arranges the chairs in 2 rows of that same length, he has 15 left over. How many chairs does Liam have?
Answers
47 chairs
Trig-Please help ASAP! Use either the law of cosines or the law of sines to find the measure of angle C
Answers
Answer:
C = 81.6 degrees
Step-by-step explanation:
The formula for law of sines
sin A sin B sin C
-------- = ----------- = -------------
a b c
Looking at the diagram, we know A = 54, b = 7.4 and c = 15.8
not enough to use the law of sines
We will need to use the law of cosines
a^2 = b^2 + c^2 - 2ac cos A
Using the law of cosines, we can calculate the length of a
a^2 = 7.4^2 + 15.8^ -2*7.4*15.8 cos 54
a^2 = 54.6+249.64-233.84cos54
a^2 =166.7922966
a = 12.921
Now we can use the law of sines to find C
sin 54 sin C
-------- = -------------
12.921 15.8
Using cross products
15.8 * sin 54 = 12.921 * sin C
Divide each side by 12.921
15.8 /12.921 * sin 54 = sin C
Take the arcsin of each side
arcsin (15.8 /12.921 * sin 54) = arcsin (sin C)
arcsin (15.8 /12.921 * sin 54) = C
C = 81.6 degrees
Write a function rule for the given graph. Identify the value of y when x=12.
Answers
Answer: f(x) = x + 2; f(12) = 14
Step-by-step explanation:
The parent function of f(x) = x has been shifted up 2 units
⇒ f(x) = x + 2
f(12) = 12 + 2
= 14
Christopher bought a new watch at the store when they were having a 20\%20% off sale. If the regular price of the watch was \$48$48, how much did Christopher pay with the discount? \$\
Answers
Answer:
38.4
Step-by-step explanation:
48*20%=9.6 48-9.6=38.4
Write a linear function that passes through the points (-5, -6) and (2, 8).
Answers
Answer:
y = 2x+4
Step-by-step explanation:
We need to find the slope
m = (y2-y1)/(x2-x1)
= (8--6)/(2--5)
= (8+6)/(2+5)
= 14/7
=2
Now we can use the point slope form of a line
y-y1 = m(x-x1)
y--6 = 2 (x--5)
y+6 = 2(x+5)
If we want it in slope intercept form
Distribute the 2
y+6 = 2x+10
Now subtract the 6 from both side
y+6-6=2x+10-6
y = 2x+4