3. Trigonometric Function a. Describe two real-world situations that could be modelled by a trigonometric function. Cannot be Ferris Wheel ride, tides, hours of daylight. Cite any Internet source you may have used for reference. b. Clearly define all variables in the relationship. c. Clearly justify why this model fits the real applications with specific reference to key features of the function. d. Your justification should also include reference to the graphical and algebraic models. e. Accurately describe what changes to the base function y = sin x would be necessary to fit both real applications.
Answers
For this problem, we need to describe a real-life situation where trigonometric functions can be used to model the problem.
Let's assume that a certain vehicle's position is controlled by the speeds of the wheels on each side of the car. Whenever the speeds on the left wheels and right wheels are equal, then the car moves forward, if the speed on the left side is greater than the one on the right side the car goes right, and if the speed on the right side is greater, then the vehicle goes to the left side. This type of car is called a differential drive car, and it's very common on remote-controlled (RC) vehicles.
If we want to model the speed of the car in a two dimensional grid, such as below:
We need to assume that the vehicle will have two components of velocity, one that is parallel to the x-axis and one that is parallel to the y-axis. These will form the linear velocity for the vehicle. We also need an angular velocity, which is the rate at which the angle of the vehicle changes.
If we assume that the wheels of the vehicles are at a distance of "L" apart from each other, then we can model the angular velocity of the vehicle as:
[tex]\omega=\frac{v_r-v_l}{L}[/tex]Where "vr" is the speed on the right wheel, and "vl" is the speed on the left wheel. The movement will happen with the center of the car as the center of the movement, with this we can assume that the velocity of the vehicle on the two axes should be:
[tex]\begin{gathered} v_x=\frac{1}{2}(v_r+v_l)\cdot cos(\theta)\\ \\ v_y=\frac{1}{2}(v_r+v_l)\cdot sin(\theta) \end{gathered}[/tex]Therefore we can describe the vehicle speed with the following equations:
[tex]\begin{gathered} \omega=\frac{v_{r}-v_{l}}{L}\\ \\ v_x=\frac{1}{2}(v_r+v_l)cos(\theta)\\ \\ v_y=\frac{1}{2}(v_r+v_l)s\imaginaryI n(\theta) \end{gathered}[/tex]The input variables are "vr" and "vl" which are the speeds of each wheel and the angle of the vehicle "theta", the output is the speed at the x coordinate and the speed at the y coordinate, and the angular speed.
This works very well because if the vehicle is moving parallel to the x-axis, the angle will be 0, the cosine of 0 is 1, therefore the speed on the y axis will be 0 and the speed on the x-axis will be given by 0.5(vr+vl). The opposite happens when the vehicle is moving parallel to the y-axis.
Related Questions
write a part to part and a part to whole ratio for each problem situationof the 250 students surveyed 142, prefer carrots and 97 prefer peas
Answers
Here, we want to write ratios in part to part and in part to whole ratio
For the part to part
Writing carrot to peas, we have;
[tex]142\colon97[/tex]For the parts to whole ratio, we have;
[tex]\begin{gathered} \text{Carrot} \\ \frac{142}{250}\text{ = 142:250} \\ \text{Peas} \\ \frac{97}{250}\text{ = 97:250} \end{gathered}[/tex]18)Betsy is collecting data on the amount of time shoppers spend inside of a particular large department store. She stands outside the department store and surveys every 10th shopper who exits. What type of sampling is used? Explain your answer.
Answers
Consider the 5 main types of sampling: Random, Systematic, Convenience, Cluster, and Stratified.
In the case of systematic sampling, every kth element of the data set is taken.
In our case, consider all the shoppers and imagine that they can be ordered in a line; then, Betsy selected the 10th shopper in the line, the 20th one, and so on.
This is analogous to systematic sampling; the answer is systematic sampling.In a recent survey of dog owners, it was found that 901, or 34%, of the owners take their dogs on vacation with them. Find the number of dog owners in the survey that do NOT take their dog on vacation with them rounded to the nearest whole number
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we have that
34% represents 901 owners that take their dogs on vacation with them
so
the percentage of dog owners in the survey that do NOT take their dog on vacation is equal to
100%-34%=66%
Applying proportion
901/34=x/66
solve for x
x=(901/34)*66
x=1,749 ownersIf there are 40 seats per row how many seats are in 90 rows?
Answers
Answer:
3,600 seats
Step-by-step explanation:
If you have 40 seats in a row, and there are 90 rows, you simply take the amount of seats, and multiply that by the amount of rows.
-Hope this helps
Answer:
Step-by-step explanation:
3600
If you were to multiply 40 seats by 90 rows, you would result with 3600 seats!
Find the length of the given side for the congruent triangles ABC and A'B'C'.
The length of side AC
The length of side AC is ____.
Answers
ABC and A'B'C' are congruent triangle. The length of the side AC is 5.31cm.
congruent triangle - Congruent triangles are those whose two sides and included angle match those of another triangle's matching sides and angle.
rules -
Each of the three sides is equal.A comparable side and two angles share the same properties.The angle between the two sides is also equal, and there are two equal sides.(given)
AB = 17 cm
BC = 34 cm
Using the trigonometric functions
cos x = B/H
cos(81°) = B/34
B = Cos(81°) x 34
B = 5.31.
ABC and A'B'C' are congruent triangle. The length of the side AC is 5.31cm.
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can you help me with key attributes of quadratic function
Answers
The shape of a quadratic function is a parabola.
The domain of a quadratic function is the set of all real numbers.
The range of the quadratic function is the set of all y values at or above the vertex for a parabola open upwards.
In the given parabola, y=0 is the y coordinate of the vertex of the parabola.
Therefore, the range is R=[0, ∞).
The domain is (-∞, ∞).
Ronald purchased a brand new phone for $450.00. Since phones are taxablehe had to pay a sales tax of 45%. much sales tax did Ronald pay for the phone ?
Answers
How many different arrangements of 5 be formed if the first must Work (of allowed?
Answers
ANSWER
There are 913,952 different 5-letter combinations that can be formed.
EXPLANATION
Recall that there are 26 letters in the English Alphabet.
From the question, we are to find the arrangement of 5 letters with the first letter being either W or K, and repetition of letters is allowed.
The possibilities for the 1st letter is 2 since the 1st letter can be either W or K;
More so, the possibilities for the 2nd letter is 26;
The possibilities for the 3rd letter is 26;
The possibilities for the 4th letter is 26, and
The possibilities for the 5th letter is 26;
The possibilities of arranging 5 letters = 2 x 26 x 26 x 26 x 26 = 913,952.
Hence, a total of 913,952 different 5-letter combinations can be formed.
Home Liquidators marks up its merchandise 35% on cost. What is the company’s equivalent markup on selling price?
Answers
Home Liquidators marks up its merchandise 35% on cost, the company's equivalent markup on the selling price is 25.9%.
What is an equivalent value?An equivalent value represents equality.
Equivalent values show that two paired mathematical expressions are equal.
How is the markup on the selling price determined?The markup on the selling price can be derived by equating the markup on cost with the markup on the selling price as follows:
Markup on cost = 35%
Cost = 100%
Selling price = 135%
Markup on selling price = 25.9% (0.35/1.35 x 100)
Thus, we can describe Home Liquidators' markup on the selling price as equivalent to 25.9% or simply 26%.
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A hot air balloon was descending at a rate of 25 feet per minute and was known to be at an altitude of 425 feet above the ground 21 minutes after it began its descenta) determine the slope-intercept form of the equationb) How high was the balloon when it began its descent (0 minutes)c) How many minutes did it take to land?
Answers
We can model the problem as a linear equation of the form:
[tex]y=mx+b[/tex]Where:
m = Slope (Rate of change)
b = y-intercept (Initial value)
a)
Since it is descending at a rate of 25ft per minute, the slope is:
[tex]m=-25[/tex]So, the equation is:
[tex]y=-25x+b[/tex]b) We know that the ballon was 425ft above the ground 21 minutes after it began its descent, so:
[tex]\begin{gathered} y=425,x=21 \\ so\colon \\ 425=-25(21)+b \\ 425=-525+b \\ b=950 \end{gathered}[/tex]Therefore, the balloon was 950ft when it began its descent, so, we can conclude that the y-intercept is 950, now the equation is complete
[tex]y=-25x+950[/tex]c) We need to know for which value of x, y is equal to 0, so:
[tex]\begin{gathered} y=0 \\ 0=-25x+950 \end{gathered}[/tex]Solve for x:
[tex]\begin{gathered} 25x=950 \\ x=\frac{950}{25} \\ x=38 \end{gathered}[/tex]The balloon will land after 38 minutes
Jackson bought a Ford Mustang for $40,000 and it depreciates in value 9% per year. Write an equation that
models the value of Jackson's car.
Answers
Answer:
[tex]v = 40000( {.91}^{x} )[/tex]
1. There are three car manufacturing factories A, B and C, and they are producing the same type
of cars. They are employing 1000, 2000 and 3000 men and producing 10, 15 and 25 cars per
month respectively. Find the labor productivity of each firm and the production of each firm
per year.
Answers
Is my answer correct help me please
Answers
3(-2) + 4
-6 + 4
whichever number has the greater value, that’s the sign you use. In this case it’s 6
6-4=2
Keep the negative
-2 is the answer
You have a huge review project for English where you have to answer 300 questions. You start on it after school at 5:00. At 6:30, you have answered 55 questions.What time will you finish your assignment if you don't take any breaks?
Answers
Given :
The total questions = 300
Time of starting = 5:00
At 6:30 the number of answered questions = 55
The time spent to answer 55 questions = 6:30 - 5:00 = 1:30 hours
Which is equal to : 1 hour and 30 minutes = 90 minutes
So, the rate of answering the questions =90/55 = 18/11 minute per question
So, for solving the 300 questions, the time will be :
[tex]300\cdot\frac{18}{11}=490\frac{10}{11}\min [/tex]So, 490 10/11 minutes = 8 hours and
Which function is the result of vertically stretching ƒ(x) = x2 by a factor of 2 and translating it 4 units upward?Question 8 options:A) y = –4x2 + 2B) y = 2x2 + 4C) y = 2x2 – 4D) y = 4x2 + 2
Answers
To find the result function,
First stretch it, multiplying it by the factor we want to stretch.
So f(x) = x²
Stretching the function by a factor of 2:
g(x) = 2* f(x)
g(x) = 2* x²
On the other hand, to translate a function 4 units upward, we need to sum it to the function, so the result function is:
g(x) = 2x² + 4
The answer is option B.
Suppose that $250 is deposited into an account that pays 4.5% interest compoundedquarterly. Using A = P(1 +r/n)nt where t is the number of years, r the interestrate as a decimal, and n the number of times interest is compounded per year, find outhow many years it takes (to the nearest whole year) to reach $1000, and type youranswer into the box.
Answers
31 years
Explanation:We would apply the compound interest forula:
[tex]A=P\mleft(1+\frac{r}{n}\mright)^{nt}[/tex]A = future amount = $1000
P = principal = $250
r = rate = 4.5% = 0.045
n = compounded quarterly = 4 times
n = 4
t = time = ?
Inserting the values into the formula:
[tex]\begin{gathered} 1000\text{ = 250(1 + }\frac{0.045}{4})^{4\times t} \\ 1000=250(1+0.01125)^{4t} \\ \text{divide through by 250} \\ \frac{1000}{250}=\text{ }(1+0.01125)^{4t} \\ 4\text{ = (1}.01125)^{4t} \end{gathered}[/tex][tex]\begin{gathered} \text{Taking log of both sides:} \\ \log 4=log(1.01125)^{4t} \\ \log 4=4t\lbrack log(1.01125)\rbrack \\ 0.6021\text{ = 4t(}0.0049) \end{gathered}[/tex][tex]\begin{gathered} 0.6021\text{ = }0.0196t \\ \text{divide both sides by 0.0196} \\ \frac{0.6021}{0.0196}=\frac{0.0196t}{0.0196} \\ 30.72\text{ = t} \\ To\text{ the nearest whole number, t = 31 years} \end{gathered}[/tex]It takes 31 years to reach $1000.
Translate to an algebraic expression.10 more than dThe translation is
Answers
10 more than d is the same as d plus 10, so the algebraic expression is:
d + 10
Answer: d + 10
Esmeralda rents a car from a company that rents by the hour. she has to pay an initial fee of $50, and they charge her $8 per hour. she has $150 available to spend on car rental. what is the greatest number of hours for which she can rent the car?A. 18 hoursB. 12.5 hoursC. 12 hoursD. 13 hours
Answers
Let the number of hours be x,
Initial fee is $50,
Amount charged per hour is $8
The charge for a number of x hours is'
[tex]x\times8=8x[/tex]The total amount Esmeralda has is $150,
Therefore,
[tex]8x+5\leq150[/tex]Solving for x to find the greates number of hours,
[tex]\begin{gathered} 8x+50\leq150 \\ \text{Collect like terms,} \\ 8x\leq150-50 \\ 8x\leq100 \\ \text{Divide both sides by 8} \\ \frac{8x}{8}\leq\frac{100}{8} \\ x=12.5\text{ hours} \end{gathered}[/tex]Hence, the greates number of hours is 12.5 hours.
Option B is the right answe
help I'm practicing
Answers
We have the next formula to find the volume of a triangular prism-
[tex]V=B\times h[/tex]where
B= area of the base
h= height
in our case
B=18 square inches
H= 5 inches
[tex]V=18\times5=90in^3[/tex]the volume of the triangular prism is 90 cubic inches
Finding the intercepts, asymptotes, domain, and range from the graph of a rational function
Answers
From the given graph
The asymptotes are the dotted lines in the graph, then
The vertical asymptote is x = 3
The horizontal asymptote is y = 1
The domain is all values of x that make the function defined
Since x can not equal 3, then
The domain is
[tex]D=(-\infty,3)\cup(3,\infty)[/tex]The range is all values of y corresponding to the values of the domain (x)
Since y can not equal 1, then
The range is
[tex]R=(-\infty,1)\cup(1,\infty)[/tex]The x-intercept is the value of x at the graph intersecting the x-axis
Since the graph intersects the x-axis at the point (6, 0), then
The x-intercept is 6
The answer is the first choice 6
The y-intercept is the value of y at the graph intersection the y-axis
Since the graph intersects the y-axis at point (0, 2_, then
The y-intercept is 2
The answer is the second answer 2
1. Sally uses 312 cups of flour for each batch of cookies.
How many cups does she need to make 4 batches of cookies?
Answers
In the picture below, angle 2 = 130 degrees, what is the measurement of angle 1?
Answers
Answer:
50°
Step-by-step explanation:
[tex]\angle 1[/tex] and [tex]\angle 2[/tex] form a linear pair, and are thus supplementary (meaning they add to 180°).
The point P is on the unit circle. If the y-coordinate of P is −3/5, and P is in quadrant IV, then
x = _________
Answers
Answer:
[tex]\frac{4}{5}[/tex]
Step-by-step explanation:
Knowing that [tex]{-\frac{3}{5}}^{2} + {\frac{4}{5}}^{2} = 1, {-\frac{3}{5}}^{2} + {-\frac{4}{5}}^{2} = 1[/tex],
so x can be positive or negative 4/5,
and we know that x coordinate of any point in quadrant IV is positive,
so x = 4/5.
Writing an Equation Assume that the ball rebounds the same percentage on each bounce. Using the initial drop height and the height after the first bounce, find the common ratio,r.Note: Round r to three decimal places. Use this formula:common ratio = height on first bounce/initial heightheight on first bounce = 54 in Dropped from 72in (6 feet)
Answers
The common ratio = 0.750 (3 decimal places)
Explanation:
Initial drop height = 72 inches
Height after the first bounce = 54 inches
common ratio = r = height on first bounce/initial height
r = 54/72
r = 0.75
The common ratio = 0.750 (3 decimal places)
By hu Background: Given I go to the mobile application as Logged user on the US market with active Brainly Tutor subscription And I type the question And I submit valid subject (e.g: "[mathematics]") And tutor for chosen subject is available in tutor application (e.g: "[mathematics]") Scenario: When I click "[Ask question]" button Then following helping options are presented: | "Ask Tutor" | | "Ask Community" | When I click "[Ask Tutor]" button And I submit "[Ask your question]" button Then there is a redirection to the waiting screen When I am connected with the tutor Then "[We have found one]" screen is presented And the tutor's avatar is presented And privacy policy regulations are presented
Answers
43524
[tex]\frac{d32423657565\mod{43242}}{dx}_{564564}[/tex]4tet
54
Eddie has already written 23 pages, and he expects to write 1 page for every additional hour spent writing. After spending 21 hours writing this week, how many pages will Eddit have written in Total?
Answers
EXPLANATION
Eddie has written ----->23 pages
He expects to write---> 1 page/additional hour
After spending 21 hours writing this week:
Eddy will have written:
Total pages are written =
Pages already written + Total additional hours*Pages/hours
Total pages written = 23 + 21 hours * 1 page/hour
Total pages written = 23 + 21
Total pages written = 44 pages
Answer: Eddie will have written 44 pages this week.
Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)
P(x) = x3 − 2x2 − 15x
x =
Answers
Answer:
-3, 0, 5
Step-by-step explanation:
You want the zeros of P(x) = x³ − 2x² − 15x using the factored form.
Factored formWe notice right away that x is a factor of every term. Factoring that out gives us a quadratic to factor:
P(x) = x(x² -2x -15)
To factor this, we need two factors of -15 that have a sum of -2. The factors -5 and +3 have those properties. That means our factored form is ...
P(x) = x(x +3)(x -5) . . . . factored form
ZerosThis product will be zero when any of its factors is zero. Considering them one at a time, we find the zeros of P(x) to be ...
x = 0
x +3 = 0 ⇒ x = -3
x -5 = 0 ⇒ x = 5
The zeros of P(x) are -3, 0, 5.
25, -34, -2, 56, 8,-7 greatest to least
Answers
To arrange this from the greatest to the least
we will first look out for the positive numbers
Among the positive numbers, 56 comes first
then 25 and finally 8
Then we move to the negative numbers
-2 comes first
then -7 and then -34
Hence
56, 25, 8, -2, -7, -34
In the diagram below, FG is parallel to CD. If the length of CD is the same as the length of FE, CE = 26, and FG = 11, find the length of FE. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.
Answers
Answer:
The length of FE is √286 units.
Explanation:
Let the length of FE = x
Since FG is parallel to CD, then triangles EFG and ECD are similar triangles.
The ratio of the corresponding sides are:
[tex]\frac{FE}{CE}=\frac{FG}{CD}[/tex]Substitute the given values from the diagram above:
[tex]\frac{x}{26}=\frac{11}{x}[/tex]We then solve the equation for x.
[tex]\begin{gathered} \text{ Cross multiply} \\ x^2=26\times11 \\ \text{ Take the square root of both sides} \\ x=\sqrt{26\times11} \\ x=\sqrt{286} \\ \implies FE=\sqrt{286}\text{ units} \end{gathered}[/tex]The length of FE is √286 units (in simplest radical form).
A spherical iron ball is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 8 mL/min, how fast is the outer surface area of ice changing when the outer diameter of the ball with ice on it is 24 cm?
Answers
When the outer diameter of the ball with ice on it is 24 cm then the outer surface area of ice changes at the rate of 1/90cm²/sec.
As given in the question,
Spherical ball is coated with uniform thickness.
Ice melts at the rate of 8ml/min
= (8/60)cm³/sec
Consider r as the radius of given spherical ball
Diameter = 24cm
⇒Radius 'r' =12cm
Volume 'V' = (4/3)πr³
⇒ dV /dt = (4/3)π(3r²r')
⇒-(8/60) = (4/3)π(3 ×12²×r')
⇒r' =-(8/60)×(1/576π)
⇒r' = - 1/ 4320π
Rate at which change in surface area
A =4πr²
⇒A' = 4πrr'
⇒A' = 4π (12) ( - 1/ 4320π)
= -1/90 cm²/sec.
Decrease in surface area = 1/90 cm²/sec.
Therefore, when the outer diameter of the ball with ice on it is 24 cm then the outer surface area of ice changes at the rate of 1/90cm²/sec.
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