A cabinet has three drawers. In the first drawer there are 3 gold balls. In the second drawer there are 3 silver balls. In the third drawer there are 3 silver balls and 2 gold balls An experiment consists of choosing a drawer, and then choosing a ball inside it. Each drawer is equally likely to be chosen, and within each drawer, each ball is equally likely to be chosen. For example, P(A gold ball is chosen | Drawer 3 is chosen) 2/5 Given that a gold ball is chosen, what is the probability that the drawer with 3 gold balls was selected? Now what if all the balls are numbered from 1-3 based on which drawer they were in, and then taken from the drawers and placed into a large pile of 11 balls? Each ball in this pile is now equally likely to be chosen, unlike in the previous part. Given that a gold ball is chosen, what is the probability that the gold ball came from the drawer with 3 gold balls?
Answers
Solution :
Let \($D_1,D_2,D_3$\) be the events that the first, second and the third drawers are selected respectively.
Therefore, \($P(D_1)=P(D_2)=P(D_3)=\frac{1}{3}$\)
Now the G shows that the events that the gold ball is selected, so
\($P(G|D_1)=1,P(G|D_2)=0,P(G|D_3)=\frac{2}{5},$\)
By probability, he gold ball is selected is :
\($P(G)=P(G|D_1)P(D_1)+P(G|D_2)P(D_2)+P(G|D_3)P(D_3)$\)
\($=1.\frac{1}{3}+0.\frac{1}{3}+\frac{2}{5}.\frac{1}{3}$\)
\($=\frac{7}{15}$\)
Now the required probability is :
\($P(D_1|G)=\frac{P(G|D_1)P(D_1)}{P(G)}$\)
\($=\frac{1/3}{7/15}$\)
\($=\frac{5}{7}$\)
= 0.71
Now out of 11 balls, 3+2 = 5 balls are gold balls.
Therefore, \($P(G)=\frac{5}{11}$\)
The probability that a gold ball and the first drawer is selected is :
\($P(G \text{ and first drawer})= \frac{3}{11}$\)
Now the required probability is :
\($P(\text{first drawer}|G)=\frac{P(\text{ G and first drawer })}{P(G)}$\)
\($=\frac{3/11}{5/11}$\)
\($=\frac{3}{5}$\)
Related Questions
Mary borrowed $12,000.00 for starting a bakery shop at $11.5% per year at simple interest. What is the amount of money that she yas to pay at the end of 3 yesrs to clear her debt ?
Answers
The amount of money that she yas to pay at the end of 3 yesrs to clear her debt is: $16,140.
How to find the interest rate?First step is to find the interest
Interest = ( Principal × Interest rate × time)
Interest = $12,000 × 11.5% × 3
Interest = $4,140
Now let find the amount she use to clear the debt
Amount = Principal + Interest
Amount = $12,000 + $4,140
Amount = $16,140
Therefore she will use $16,140 to clear the debt.
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Stacy is thinking of a number which she calls N. She triples it and adds 10
Answers
Answer:
3N + 10
Step-by-step explanation:
Answer:
3N+10
Step-by-step explanation:
In the real world, functions are mathematical representations of input-output situations. A vending machine is one such example. The input is the money combined with the selected button. The output is the product.
Here is another example: The formula for converting a temperature from Fahrenheit to Celsius is a function expressed as:
C = (5/9)*(F - 32), where F is the Fahrenheit temperature and C is the Celsius temperature.
If it is 77 degrees Fahrenheit in Phoenix Arizona, then what is the equivalent temperature on the Celsius thermometer?
Our input is 77.
C = (5/9)*(77 - 32)
C = (5/9)*(45)
C = 25
The equivalent temperature is 25 degrees Celsius.
To complete the Discussion activity, please do the following:
Choose your own function or choose from the list below and then provide a unique example of a function and evaluate the function for a specific input (like the example above).
Arm length is a function of height.
The circumference of a circle is a function of diameter.
The height of a tree is a function of its age.
The length of person's shadow on the ground is a function of his or her height.
Weekly salary is a function of the hourly pay rate and the number of hours worked.
Compound interest is a function of initial investment, interest rate, and time.
Supply and demand: As price goes up, demand goes down.
Answers
The correct answer is John's weekly salary is $240 based on his hourly pay rate and the number of hours worked.
Let's choose the function "Weekly salary is a function of the hourly pay rate and the number of hours worked."Example: John works as a part-time employee at a grocery store. His hourly pay rate is $12, and he worked for 20 hours in a week. We can evaluate the function to find his weekly salary.
Weekly salary = Hourly pay rate * Number of hours worked
Weekly salary = $12/hour * 20 hours
Weekly salary = $240
So, John's weekly salary is $240 based on his hourly pay rate and the number of hours worked.
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If 7 people share 17 muffins, how many does each person get?
O 1
02
3/7
O 7/17
O 1/2
Answers
Explanation:
The largest multiple of 7 that is smaller than 17 is 14
14/7 = 2
Each person gets 2 whole muffins
In addition, they get 3/7 of a muffin since the remainder is 17-14 = 3.
Put together, each person gets 2 & 3/7 muffins.
100 Points! Geometry question. Find x and y. Photo attached. Please show as much work as possible. Thank you!
Answers
The calculated values of x and y in the figure are x = 2 and y = 4
How to calculate x and y in the figurefrom the question, we have the following parameters that can be used in our computation:
The figure
Where, we have
equal side lengths
This means that
2y - 1 = 3y - 5
Evaluate
y = 4
Next, we have
x + 3 = 3/2x + 2
So, we have
1/2x = 1
This gives
x = 2
Hence, the values of x and y in the figure are x = 2 and y = 4
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Determine the turning points and distinguish between them when necessary y=x³ - 3x - 9x + 4
Answers
The turning points of the function y = x³ - 3x² - 9x + 4 are (3, -23) and (-1, 9).
To determine the turning points of the given function y = x³ - 3x² - 9x + 4, we need to find the critical points where the derivative of the function is equal to zero.
1. Find the derivative of the function:
y' = 3x² - 6x - 9
2. Set the derivative equal to zero and solve for x:
3x² - 6x - 9 = 0
3. Factorize the quadratic equation:
3(x² - 2x - 3) = 0
4. Solve the quadratic equation by factoring or using the quadratic formula:
(x - 3)(x + 1) = 0
This gives us two possible values for x: x = 3 and x = -1.
5. Substitute these critical points back into the original function to find the corresponding y-values:
For x = 3:
y = (3)³ - 3(3)² - 9(3) + 4
= 27 - 27 - 27 + 4
= -23
For x = -1:
y = (-1)³ - 3(-1)² - 9(-1) + 4
= -1 - 3 + 9 + 4
= 9
6. Therefore, the turning points are (3, -23) and (-1, 9).
Note: It appears that there was a typo in the original equation, where the term "-9x" should have been "-3x²". The above solution assumes the corrected equation.
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marking as brainlist pls help
Answers
Explanation:
For this problem, we can use the law of cosines to find the length C. We rearrange the formula for C and solve:
c^2 = a^2 + b^2 - 2ab cos(C)
c = sqrt(a^2 + b^2 - 2ab cos(C))
So for the values, use the following from our image:
a = 61
b = 63
C = 142 degrees
c = sqrt( 61^2 + 63^2 - 2(61)(63)cos(142))
c = sqrt( 3721 + 3969 - (-6056.65))
c = sqrt( 13746.65 )
c = 117.246
So, we have found the value of c to be 117.246.
To round to the nearest tenth, we will have 117.2
Cheers.
Which is an exponential decay function
F(x)= 0.2
Answers
The function F(x)=0.2F(x)=0.2 is not an exponential decay function. It is a constant function where the output value is always equal to 0.2, regardless of the input value xx.
An exponential decay function is typically in the form f(x)=a⋅bxf(x)=a⋅bx, where aa and bb are constants and bb is a value between 0 and 1. As xx increases, the exponential decay function decreases exponentially.
For example, an exponential decay function could be f(x)=0.2⋅(0.5)xf(x)=0.2⋅(0.5)x, where the base 0.50.5 represents the decay factor. As xx increases, the value of the function decreases rapidly.
In summary, an exponential decay function follows the pattern of decreasing values as xx increases, while the function F(x)=0.2F(x)=0.2 represents a constant value regardless of the input xx.
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Write the equation of the line in a slope-intercept form
Answers
Answer:
It is roughly a 13/32 slope.
Step-by-step explanation:
Rise 40, Run 2
Answer: 40/2
Step-by-step explanation:
9. Environmental Glass Products, Inc. wants to build a new centralized facility to receive household, commercial, and industrial glass for recycling. This center will be supplied by trucks coming from four "collection points," where recyclable glass is dropped off by individuals and businesses. The volume and the map coordinates for the four collection centers are shown below. Where should the collection center be located?
Collection point Load (X,Y) Coordinates
A 9,000 (4,8)
B 4,000 (7,2)
C 2,000 (4,1)
D 5,000 (7,3)
Answers
Answer:
the collection center should be located at Collection Option (D).
Step-by-step explanation:
To determine the best location for the collection center, you can calculate the average coordinates of the four collection points and choose the location that is closest to this average.
To calculate the average coordinates, you can add the coordinates of each collection point and divide the result by the number of collection points. For example, to find the average X coordinate, you can add the X coordinates of all four collection points and divide by 4. Similarly, to find the average Y coordinate, you can add the Y coordinates of all four collection points and divide by 4.
Using this method, the average coordinates for the four collection points are (5.75,3.25). The collection point that is closest to these average coordinates is Collection Point D, which is located at (7,3). Therefore, the collection center should be located at Collection Point D.
I hope this helps to clarify the process for determining the best location for the collection center. Please let me know if you have any other questions.
Alguien q me ayude es para hoy antes de las 9 quien me ayude le doy corona
Answers
The answers of the divisions are 5.71, 0.75,2.72,0.07,24,5.33,1.17,0.44.
According to the statement
we have to give the perfect answers of the given statements with the help of the division.
So, For this purpose, we know that the
Division is the mathematical process of finding out how many times one number is contained in another.
From the given information:
The statements are:
1) 5/3 ÷ 2/7
1.6 ÷ 0.28
5.71
Then
2) 1/3 ÷ 4/9
0.33 ÷ 0.44
0.75
Then
3) 3/5 ÷ 2/9
0.6 ÷ 0.22
2.72
Then
4) 2/15 ÷ 5/3
0.13 ÷ 1.66
0.07
Then
5) 3 ÷ 1/8
3÷ 0.125
24
Then
6) 4 ÷ 3/4
4÷ 0.75
5.33
Then
7) 8/9 ÷ 3/4
0.88÷ 0.75
1.17
Then
8) 16/9 ÷ 4
1.77 ÷ 4
0.44
So, The answers of the divisions are 5.71, 0.75,2.72,0.07,24,5.33,1.17,0.44.
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Jermaine kicked a soccer ball at a speed of 24 feet per second. If the ball never leaves the ground, then it can be represented by the function H(t) = −16t2 + 24t. Determine the time the ball traveled. (1 point) t = 24 seconds t = 8 seconds t = 1.5 seconds t = 0.67 seconds
Answers
The time that the ball traveled is given as follows:
1.5 seconds.
How to obtain the time traveled by the ball?The quadratic function determining the ball's height after t seconds is given as follows:
H(t) = -16t² + 24t.
The roots of the quadratic function in this problem are given as follows:
-16t² + 24t = 0.
16t² - 24t = 0
8t(2t - 3) = 0.
Hence we apply the factor theorem as follows:
8t = 0 -> t = 0.2t - 3 = 0 -> 2t = 3 -> t = 1.5.Hence the time is given as follows:
1.5 - 0 = 1.5 seconds.
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Solve the inequation x - 12 ≤ 3 - 2x and graph its solution.
Let's do these one-by-one.
The solution to x - 12 ≤ 3 - 2x is:
Answers
Answer:
Step-by-step explanation:
x - 12 ≤ 3 - 2x
3x ≤ 15
x ≤ 5
To graph this, just draw a VERTICAL line passing through x = 5, then shade the area to the LEFT
2. A(n) is NOT an example of an agreement. (1 point)
O lease
O month-to-month
O annual
O fine print
Answers
An Annual is not an example of an agreement.
Choose any number, say 20, and divide 729 by 20 ignoring any remainders. " 729/20 = 36 " Find the average of 20 and 36. 20 + 36 / 2 = 28. " Repeat the process until the estimate is the same as the quotient (until the remainder is 0). " 729 / 27 = 27.
Answers
Answer:
Answer is solved in the explanation below.
Step-by-step explanation:
Solution:
We have to repeat the process until the estimate is the same as the quotient and remainder = 0. Which can be done only in the case of value = 27 (remainder = 0)
729/27 = 27 (remainder 0)
Let's I am starting this process from 19
So,
729 divided by 19 = 38.3684
Ignoring the remainders
729 divided by 19 = 38
Now, average of 19 and 38 = (19+38)/2 = 28
729 divided by 28 = 26.03
ignoring the remainders = 26
Now, average of 28 and 26 = (28+26)/2= 54/2 = 27
So,
729 divided by 27 = 27
Hence,
estimate is the same as quotient ( and remainder = 0)
825 use each digit once. make the smallest 3digit number
Answers
Step-by-step explanation:
Given: To make smallest 3-digit number of 825.
To find: The smallest 3-digit number of 825.
Solution: We can make the smallest 3-digit number of 825 by separating the numbers and arranging it to ascending order. The given number is 825. ...
Final answer: The smallest 3-digit number of 825 is 258.
hope it helps
Answer:
258
Step-by-step explanation:
We are given 3 numbers:
8 2 5
And we are asked to find the smallest 3 digit number using those 3 digits above.
To make the smallest number, place the numbers in value from least to greatest:
2 5 8
This is your 3 digit number: 258.
Hope this helps! :)
What is the sin B?
/21
B
5
2
sin (B) =
[?]
![What is the sin B?/21B52sin (B) =[?]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/bJqEBQdNtkeNzRPqszdQkxPqXUYlMPak.png)
Answers
Answer:
Step-by-step explanation:
sin (B) = \(\frac{2}{5}\)
Melissa earns a 2% commission on the first $750 of sales she makes each week. She earns a 3% commission on the amount of her sales that are greater than $750 each week. Melissa earned $37.50 in commissions last week.
How much were her sales last week?
Enter your answer as a number, like this: 42
(not multiple choice)
Answers
0.02 * min(x, 750) + 0.03 * max(x - 750, 0) = 37.5
Here, the first term represents the commission on the first $750 of sales, and the second term represents the commission on any sales above $750. The max function ensures that the second term is only calculated for sales above $750, while the min function ensures that the first term is only calculated for the first $750 of sales.
We can solve this equation for x:
0.02 * min(x, 750) + 0.03 * max(x - 750, 0) = 37.5
0.02 * 750 + 0.03 * max(x - 750, 0) = 37.5
15 + 0.03 * max(x - 750, 0) = 37.5
0.03 * max(x - 750, 0) = 22.5
max(x - 750, 0) = 750
Since the maximum function is equal to 750, we know that x - 750 = 750, which implies that x = 1500.
Therefore, Melissa's total sales last week were $1500.
Eliana loves music, especially 80's punk rock. Eliana looked at the number of songs in her itunes account and she has 6 less than 4 times her best friend. Eliana's classmate David has 3 more than double her best friend. When talking to her Social Studies teacher, Eliana found out social study teacher has 7 more than two times the number of songs in his library than Eliana. Define a variable, then write and solve an equation. If the total number of songs is 382, how many songs does Eliana, her best friend, David and her social study teacher have?
Answers
Answer
Step by step explantion:
bsf= 26
Eliana:98
SS teacher=203
David=55
Step-by-step explanation:
m= music
bsf: m
eliana:4m-6
David:2m+3
SS teacher: 2(4m-6)+7
m+4m-6+2m+3+8m-12+7
382=15m-7
7+ +7
389=15m
/15m /15m
m=26
multiply 26 to each one of them
Last year, Jeffrey grew 5/8 of an inch and his brother grew 1/2 of an inch. How much more
did Jeffrey grow than his brother?
Simplify your answer and write it as a fraction or as a whole or mixed number.
Answers
Answer:
Jeffrey grew 1/8 inch more than his brother
Step-by-step explanation:
5/8 - 1/2 = 5/8 - 4/8 = 1/8
Answer:
\(\frac{5}{4}\) \(or\) \(1.25\)
Step-by-step explanation:
\(\frac{5}{8}\) ÷ \(\frac{1}{2}\) = \(?\)
\(\frac{5}{8}\) = \(0.625\)
\(\frac{1}{2}\) = \(0.5\)
\(\frac{0.625}{0.5}\) = \(\frac{5}{4}\)
\(Therefore,\) \(\frac{5}{4}\) \(is\) \(the\) \(answer.\)
Hope this helps! <3
A bag of marbles has 12 green marbles, 5 red marbles, 8 blue marbles and 7 yellow marbles. What is the probability of randomly selecting a blue marble?
Answers
Probability = # of desired options / # of total options
Our desired option is blue and there are 9 blue marbles in the bag.
There are 32 total marbles (options) in the bag.
P(blue) = 9 / 32 = 0.28125 = 28.125%
Hope this helps!
Pretty please help this work is hard
Answers
Answer:
Parentheses, exponents, multiplication and division, and addition and subtraction
Step-by-step explanation:
You can remember this order by using PEMDAS. PEMDAS stands for parentheses, exponents, multiplication, division, addition, and subtraction, in that order. First up should be parentheses, then exponents, then multiplication and division, and finally addition and subtraction.
A scientist has two solutions what she has labeled solution a and solution b each contains salt she knows that solution a is 60% salt and solution b is 85% salt she wants to obtain 70 ounces of a mixture that is 80% salt how many ounces of each solution should she use
Answers
Solving a system of equations we can see that:
She needs to use 56 oz of the 85% solution.She needs to use 14 oz of the 60% solution.How many ounces of each solution should she use?We can use the variables:
x = mass of the 60% solution.
y = mass of the 85% solution.
We can write a system of equations.
x + y = 70
x*0.6 + y*0.85 = 70*0.8
We can isolate x on the first equation to get:
x = 70 - y
Replace that in the other one:
(70 - y)*0.6 + y*0.85 = 70*0.8
Now we can solve this for y.
y*0.25 = 70*0.8 - 70*0.6
y = 14/0.25 = 56
Then he needs to use 56 ounces of the 85% solution, and 14 ounces of the 60% solution.
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A committee must be formed with 2 teachers and 3 students. If there are 10 teachers to choose from, and 13 students, how many different ways could the committee be made?
Answers
Answer:
12870 ways
Step-by-step explanation:
10c2*13c3
10!/(10-2)!2! *13!/(13-3)!3!
[(10*9*8!)/8!*2*1] *[(13*12*11*10!)/10!*3*2*1]
45*286
12870 ways
Paul opens a savings account with $350. After 4 months of making regular monthly deposits, his balance is $550. Write the linear model to represent the total amount of money paul deposits into his account after x months After how many months will paul have saved $2000.?
Answers
Answer: 8.25 months
Step-by-step explanation:
Let's start by finding the slope of the line, which represents the rate at which Paul is depositing money into his savings account each month. We can use the given information to calculate the slope:Slope = (Ending balance - Starting balance) / (Number of months)
Slope = ($550 - $350) / (4 months)
Slope = $200 / monthSo, the linear model to represent the total amount of money Paul deposits into his account after x months is:y = 200x + 350where y is the total amount of money Paul has saved after x months.To find out after how many months Paul will have saved $2000, we can substitute y = $2000 into the equation and solve for x:2000 = 200x + 3501650 = 200xx = 8.25
Therefore, after 8.25 months, or about 8 months and 1 week, Paul will have saved $2000.
David found a vintage watch his father lost 38 years ago. The watch’s date showed that it had worked for over two years after his father lost it. When David found the watch, it was 3 times as old as when his father lost it. How old was the watch when David’s father lost it?
Answers
Answer:
the watch was 1 year old when David's father lost it.
Step-by-step explanation:
Let's break down the information given:David found the vintage watch 38 years after his father lost it.The watch's date showed that it had worked for over two years after his father lost it.When David found the watch, it was 3 times as old as when his father lost it.
Let's assume the age of the watch when David's father lost it as X years.According to the information, the watch worked for over two years after his father lost it. Therefore, we can represent the age of the watch when David found it as (X + 2) years.We are also given that when David found the watch, it was 3 times as old as when his father lost it. So we can write the equation:(X + 2) = 3XSimplifying the equation:
2 = 3X - X
2 = 2X
X = 1
At December 31, 2019, the records of Hoffman Company reflected the following balances in the shareholders’ equity accounts:
Common shares: par $14 per share; 51,000 shares outstanding
Preferred shares: 9 percent; par $10 per share; 7,650 shares outstanding
Retained earnings: $225,500
On January 1, 2020, the board of directors was considering the distribution of a $68,500 cash dividend. No dividends were paid during 2018 and 2019.
Required:
Determine the total and per-share amounts that would be paid to the common shareholders and to the preferred shareholders under two independent assumptions:
Answers
Answer:
a) Total preferred dividend = $37,020; total common stock dividend = $26,980.
b) Total preferred dividend = $12,340; total common stock dividend = $51,660.
C) see explanation section.
Explanation:
Requirement - A
If the preferred dividend is cumulative.
Common stock = $12 x 50,000 = $600,000
Preferred stock = $19.5 x 7,910 = $154,245
Given,
Retained Earnings = $240,000, and Dividend = $64,000.
Generally, cash dividend is distributed to the preferred and common stockholders. However, after paying the preferred stock, dividend is paid to the common stockholders.
Therefore, preferred dividend = $154,245 x 8% = $12,340 for 2019.
As the preferred dividend is cumulative, the preferred stockholders will get the dividend for 2017 and 2018 if there remains any extra dividend after giving for 2019.
For 2017 = $12,340
For 2018 = $12,340
For 2019 = $12,340
Total preferred dividend = $37,020.
Therefore, the total common stock dividend = $(64,000 - 37,020) = $26,980.
Requirement - B
If the preferred dividend is non-cumulative.
Therefore, preferred dividend = $154,245 x 8% = $12,340 for 2019.
As the preferred dividend is non-cumulative, the preferred stockholders will not get the dividend for 2017 and 2018 despite remaining extra dividend after providing for 2019.
Therefore, preferred dividend = $12,340
Common stock dividend = $(64,000 - 12,340) = $51,660.
Requirement - C
The common stock dividend is more in non-cumulative as the company does not pay the previous years' dividends to the preferred dividend. On the other hand, if the company pays the dividend for the last years', which is known as cumulative, the dividend will become less. Therefore, the common stockholders will get fewer dividends because of paying the previous years' dividend to the preferred stockholders.
The factors would cause a more favorable dividend for the common stockholders if preferred dividends were non-cumulative, not in arrears, and there is an increase of distribution of dividends.
Hey can anyone help me understand how to graph this?
Answers
Answer:
Step-by-step explanation:
?
Identify a, h, and k for each function.y = 27 - 4) - 5 a-hek-y = (+4)anhekey = -(2 - 6)² + 3 amheke
Answers
We can identify a, h and k by looking at the general formula of a parabola in vertex form and comparing with the equations that you got:
\(y=a(x-h)^2+k\)Where a is the stretches of the parabola, h is the x coordinate of the vertex and k is the y-coordinate of the vertex.
In te equation:
\(y=2(x-4)^2-5\)h=4, since is the value that we are subtracting from x inside the parentheses, a=2, since it's the number that is multiplying by (x-h)^2 and k= -5, since this is the value that we add to the first term.
then a=2, h=4 and k= -5
In the equation:
\(y=(x+4)^2\)h= -4, so when you subtract -4 from x you get x-(-4)=x+4 inside the parentheses, a=1, since the term (x-h)^2 hasn't a coefficient multiplying by it, and k= 0, since we are adding nothing to the first term.
then a=1, h= -4 and k= 0
In the equation:
\(y=-(x-6)^2+3\)h=6, since is the value that we are subtracting from x inside the parentheses, a= -1, since it's the number that is multiplying by (x-h)^2 and k= 3, since this is the value that we add to the first term.
then a= -1, h=6 and k= 3
A marching band performs on the football field at half-time. As they perform, the members of the band stand in
the shape of a sinusoidal function. While playing, they move, but still maintain the sinusoidal function,
transforming it in different ways.
Darla is a member of the marching band. As the band begins to play she is positioned in the exact center of the
field. The person closest to her on the same horizontal line, stands 10 yards away. The sinusoidal function
extends to the ends of the playing field.
The playing area of football field measure 300 feet by 160 feet. Place the playing area of a football field on the
coordinate plane such that the origin is the lower left comer of the football field.
(Score for Question 1: of 2 points)
1. What is the period and the amplitude of the sine function representing the position of the band members as
they begin to play?
Answer.
Answers
The period of the sine function representing the position of the band members is 60 feet, and the amplitude is approximately \(170.3 feet.\)
What is the coordinate plane?Since the band members are standing in the shape of a sinusoidal function, we can assume that their positions can be represented by the equation:
\(y = A sin(Bx)\)
where y is the vertical position of a band member, x is the horizontal position on the field, A is the amplitude, and B is the period.
Since Darla is positioned in the exact center of the field, and the person closest to her on the same horizontal line stands 10 yards away, we can assume that the sinusoidal function has a phase shift of 0. This means that the midline of the function passes through the point (0, 0).
To find the amplitude, we need to determine the maximum and minimum heights of the function.
Since the playing area of the football field measures 300 feet by 160 feet, the distance between the two farthest points on the field is the diagonal distance, which can be calculated using the Pythagorean theorem:
\(sqrt(300^2 + 160^2) \approx 340.6 feet\)
Since the distance between the farthest points on the field is equal to the distance between two peaks or two valleys of the sinusoidal function, the amplitude is half of this distance:
\(A = 340.6/2 \approx170.3 feet\)
To find the period, we can use the fact that the distance between two consecutive peaks or valleys is equal to the period of the function.
Since the person closest to Darla stands 10 yards away, or 30 feet away, we can assume that this person is standing at a peak or a valley of the function. This means that the period is twice the distance between Darla and the person closest to her:
\(P = 2(30) = 60 feet\)
Therefore, the period of the sine function representing the position of the band members is 60 feet, and the amplitude is approximately \(170.3 feet.\)
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