The function which represents the inverse of the function f(x)=[tex]x^2[/tex]+5 is [tex]f^{(-1)(x)}[/tex] = ±[tex]\sqrt{(x - 5)}[/tex].
To find the inverse of a function, we can follow the steps:
Step 1: Replace f(x) with y. So we have:
y = [tex]x^2[/tex] + 5
Step 2: Swap x and y in the equation obtained in step 1:
x = [tex]y^2[/tex] + 5
Step 3: Solve for y in the equation obtained in step 2:
x - 5 = [tex]y^2[/tex]
y = ±[tex]\sqrt{(x - 5)}[/tex]
Note that we need to include both the positive and negative square roots to get the complete inverse function. Therefore, the inverse function of f(x) = [tex]x^2[/tex] + 5 is:
[tex]f^{(-1)(x)}[/tex] = ±[tex]\sqrt{(x - 5)}[/tex]
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In his free time, Gary spends 9 hours per week on the Internet and 9 hours per week playing video games. If Gary has five hours of free time per day, approximately what percent of his free time is spent on the Internet and playing video games?
What is the answer to 5/8 ÷ 3/4 in fraction form
Answer:
5/6
Step-by-step explanation:
Kathy purchases an eyebrow pencil and a concealer. The eyebrow pencil is priced at $6.94. If she is billed $14.13 for both the items,what is the price of the concealer?
Answer:
7.19
Step-by-step explanation:
Take the total cost and subtract the eyebrow pencil to find the cost of the concealer.
14.13
- 6.94
---------------
7.19
A/C
DVD Power Leather
Player Mirror Seats
Bucket
Seats
Power
Sunroof
Average
Model Retail
Value
4. V6 four-dr
a. $
5. V6 four-dr a.
6. V8 four-dr a.
Yes
Average
Mileage Retail
Price
35,000 b. $
58,400 b.
80,255 b.
No
Yes
Yes
No
No
Yes
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
No
7. Monica Rizzo wants to buy a
4-year-old V6 four-door sedan
that has 52,686 miles. It has
manual transmission, a DVD
player, leather seats, front
bucket seats, and power
sunroof but no air conditioning. Ee
8. Kurt Sorensen is looking at a
4-year-old V8 four-door sedan
that has 80,575 miles. It also has
a DVD player, power mirrors,
leather seats, front bucket seats,
and a power sunroof. Asunno
The cοrrect answer is Nο. The prices οf fοur-year-οld vehicles with a V6 and V8 engine vary greatly, depending οn the make and mοdel, mileage, and any additiοnal features.
What is an engine?An engine is a machine that cοnverts energy intο mechanical mοtiοn. It is a device that creates thrust οr pοwer thrοugh the cοnversiοn οf energy frοm fuel, such as gasοline, diesel, οr natural gas. Engines are used in a variety οf applicatiοns, such as in cars, bοats, planes, and electric generatοrs. They are alsο used in industrial machinery and manufacturing plants. Engines prοvide the pοwer that drives numerοus types οf transpοrtatiοn and prοductiοn.
Mοnica Rizzο's V6 fοur-dοοr sedan with a DVD player, leather seats, frοnt bucket seats, and pοwer sunrοοf but nο air cοnditiοning wοuld have an average retail value οf arοund $35,000. Kurt Sοrensen's V8 fοur-dοοr sedan with the same features, hοwever, wοuld have an average retail value οf arοund $58,400. The difference in value is largely due tο the V8 engine, which has mοre pοwer and thus requires mοre fuel. The additiοnal features alsο affect the value, but the engine is the biggest factοr.
Therefοre, Kurt Sοrensen will have tο pay significantly mοre fοr his V8 sedan than Mοnica Rizzο will have tο pay fοr her V6 sedan.
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Mrs Patel and Mrs rose both get paid 25,800 a year. Mrs Patel receives a 2. 5 % rise. Mrs rose gets a pay rise of £55 per month. Who receives the greater pay rise ?
Mrs. Rose receives the greater pay rise of £660 per year compared to Mrs. Patel's pay rise of £645.
To compare the pay rise for Mrs Patel and Mrs Rose, we need to calculate the actual amount of the pay rise for each of them.
For Mrs Patel, her pay rise is 2.5% of her current salary of £25,800:
Pay rise for Mrs Patel = 2.5% of £25,800 = £645
For Mrs. Rose, her pay rise is £55 per month, which is:
Pay rise for Mrs. Rose = £55 x 12 = £660 per year
Comparing the two pay rises, we see that Mrs. Rose receives a greater pay rise of £660 per year, compared to Mrs. Patel's pay rise of £645. Therefore, Mrs. Rose receives the greater pay rise.
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a) Write an expression for the surface area of a
cube with edge length x. Fully simplify your
answer.
b) A cube has a surface area of 1176 cm². What is
its edge length?
Give your answer in centimetres (cm) and give
any decimal answers to 1 d.p.
a) The surface area of a cube with edge length x is
6x^2b) The edge length of the cube is solved to be 14 cm.
How to find the edge length of the cubea) The surface area of a cube with edge length x is given by the formula:
SA = 6x^2
b) We are given that the surface area of the cube is 1176 cm^2. Setting this equal to the formula for surface area of a cube, we get:
6x^2 = 1176
Dividing both sides by 6, we get:
x^2 = 196
Taking the square root of both sides, we get:
x = ± 14
Since the edge length must be a positive number, we discard the negative solution and conclude that the edge length of the cube is 14 cm (to 1 decimal place).
Therefore, the edge length of the cube is 14 cm.
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For the following exercises, find the angle
in the given right triangle. Round answers to the nearest hundredth.
Provide the given right triangle
I'm sorry, but I cannot see the following exercises that you are pertaining to. Please provide the given right triangle and the specific angle you want to find so I can assist you accordingly.
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3) When 100 coins are tossed find the probability that exactly 35 will be heads (numerical answer followed by Matlab code), assuming the number of experiments is 100000.
Write a matlab code please
This program creates [tex]N=100000[/tex] trials in which [tex]n=100[/tex] coins are tossed, and it counts the trials in which precisely [tex]k=35[/tex] heads are thrown. This code's output will be the calculated probability of receiving [tex]35[/tex] heads.
What are examples and probability?It is based on the possibility that something will materialize. The fundamental underpinning of maximum possible is just the explanation of likelihood. For instance, while flipping a coin, there is a 12-percent probability that it will land on its head.
How should a novice compute probability?Determine the number of alternative ways to roll a 4 as well as multiply it by the overall number of outcomes to determine the likelihood of the event occurring.
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Find the value of x.
Answer:
[tex]\large\boxed{\textsf{x=3}}[/tex]
Step-by-step explanation:
[tex]\textsf{We are asked to find the value of x.}[/tex]
[tex]\textsf{Notice that we have Intersecting Chords inside of the circle.}[/tex]
[tex]\large\underline{\textsf{What are Chords?}}[/tex]
[tex]\textsf{Chords are line segments that have endpoints on the circumference of a circle.}[/tex]
[tex]\textsf{Note that some Chords can be Diameters, but not for this problem. There's no Center.}[/tex]
[tex]\textsf{These Chords don't create Perpendicular Lines, which means the angles aren't equal.}[/tex]
[tex]\textsf{We can identify the unknown angles with a postulate.}[/tex]
[tex]\large\underline{\textsf{What is a Postulate?}}[/tex]
[tex]\textsf{A Postulate is a statement that can never be false. No matter what, it's always}[/tex]
[tex]\textsf{true.}[/tex]
[tex]\textsf{We can use the Linear Pair Postulate to find the needed angle.}[/tex]
[tex]\large\underline{\textsf{What is the Linear Pair Postulate?}}[/tex]
[tex]\textsf{Linear Pair Postulate is a Postulate that proves 2 angles add up to 180}^{\circ}.[/tex]
[tex]\textsf{A Linear Pair is 2 Adjacent Angles that form a Straight Angle.}[/tex]
[tex]\large\underline{\textsf{For our Problem;}}[/tex]
[tex]\tt \angle HLJ \ and \ \angle ILJ \ are \ Linear \ Pairs.[/tex]
[tex]\large\underline{\textsf{Find} \ \angle \textsf{HLJ;}}[/tex]
[tex]\tt \angle HLJ + \angle ILJ = 180^{\circ}.[/tex]
[tex]\tt \angle HLJ + 85.5^{\circ} = 180^{\circ}.[/tex]
[tex]\underline{\textsf{Subtract 85.5 from both sides of the equation;}}[/tex]
[tex]\tt \angle HLJ = 94.5^{\circ}[/tex]
[tex]\textsf{Now that we have} \tt \ \angle HLJ, \textsf{we can find x.}[/tex]
[tex]\textsf{We are given 2 arc measures and we have 2 Intersecting Chords.}[/tex]
[tex]\textsf{We should use} \tt \ \angle HLJ \ \textsf{to find x.}[/tex]
[tex]\large\underline{\textsf{How?}}[/tex]
[tex]\textsf{The Angle is equal to half the sum of the 2 arcs.}[/tex]
[tex]\underline{\textsf{For our Problem;}}[/tex]
[tex]\tt \angle HJL = \frac{1}{2} (42x-6 + 15x+24)[/tex]
[tex]\tt 94.5^{\circ} = \frac{1}{2} (42x-6 + 15x+24)[/tex]
[tex]\large\underline{\textsf{Solve;}}[/tex]
[tex]\textsf{Multiply each side by 2 first.}[/tex]
[tex]\tt 189^{\circ} = 42x-6 + 15x+24[/tex]
[tex]\underline{\textsf{Combine Like Terms;}}[/tex]
[tex]\tt 189^{\circ} = 57x+18[/tex]
[tex]\underline{\textsf{Subtract 18 from both sides;}}[/tex]
[tex]\tt 171^{\circ} = 57x[/tex]
[tex]\underline{\textsf{Divide each side by 57;}}[/tex]
[tex]\large\boxed{\textsf{x=3}}[/tex]
A cell tower is 3km from the highway. It has a circular range with a radius of 7km. What length of the highway is in the range.
Answer:
Below
Step-by-step explanation:
With questions like this it is helpful to draw the scenario as in the image below. Using Pyhtagorean theorem , you can calculate 'x'..... then the highway distance in range is two times this = 12.6 km
PLEASE HELPPP MEEEEEEEE
The equation that models the relationship between the models to show Ethan's employee benefits is y = 14 + 3(x - 1).
The number of paid vacation days Ethan earned was 35 paid vacation days .
Ethan will reach the maximum number of paid vacation days in 7 years.
How to find the paid vacation days ?Let x represent the number of years he has worked for this employer and y represent the number of paid vacation days he has earned.
The equation that models the relationship between x (years worked) and y (paid vacation days earned) is:
y = 14 + 3(x - 1)
Using the equation from part a, we can find out how many paid vacation days Ethan has earned after eight years:
y = 14 + 3(8 - 1)
y = 14 + 3(7)
y = 14 + 21
y = 35
To determine when Ethan will reach the maximum number of paid vacation days, we can set y equal to 30 and solve for x:
30 = 14 + 3(x - 1)
16 = 3(x - 1)
16/3 = x - 1
x = 16/3 + 1
x = 16/3 + 3/3
x = 19/3
x = 7
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and |q1| > |q2| attract each other with a force of magnitude 90.4 mn when separated by a distance of 4.64 m . the spheres are then brought together until they are touching, enabling the spheres to attain the same final charge q.
The final charge of the spheres when they are touching after calculated using Coulomb's law , is 1.24 * 10^-6 C
According to the given information, two charged spheres with charges |q1| > |q2| attract each other with a force of magnitude 90.4 mn when separated by a distance of 4.64 m. When the spheres are brought together until they are touching, they attain the same final charge q.
We can use Coulomb's law to find the final charge q. Coulomb's law states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, Coulomb's law is expressed as:
F = k * (q1 * q2) / r^2
Where F is the force between the charged particles, k is Coulomb's constant (8.99 * 10^9 N*m^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.
We can rearrange the equation to solve for q:
q = sqrt(F * r^2 / k)
Plugging in the given values:
q = sqrt(90.4 * 10^-6 N * (4.64 m)^2 / (8.99 * 10^9 N*m^2/C^2))
q = 1.24 * 10^-6 C
Therefore, the final charge of the spheres when they are touching is 1.24 * 10^-6 C
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Sandra uses the formula, P = DB, to find her approximate six-month premium when her driver risk factor, D, is 1.0 and the basic six-month premium is $567.
What will her monthly premium be?
A.$170.00
B.$177.50
C.$94.50
D. $201.15
Therefore , the solution of the given problem of unitary comes out to be C is the correct response. $94.50.
An unitary method is what?The task can be finished by combining the data expression obtained to use this nanosection methodology with in all supplemental data from two people who employed a particular variable tactic. In plain English, this indicates that, if the desired result occurs, either the stated individual will be known or, in fact, the colour by both enormous processes would be skipped. A refundable fee of Rupees ($1.01) may be necessary for forty pens.
Here,
By dividing both sides by D, we can determine B if P = DB:
=> B = P/D
Sandra's premium P will be equal to the $567 base premium because her risk factor D is 1.0.
So,
=> B = P/D = $567/1.0 = $567
Her monthly payment will be as follows because this is her six-month premium:
=> Monthly premium = six-month premium / six,
which equals $567 / six, or $94.50.
Thus, C is the correct response. $94.50
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Solve for x.
X
16
25
20
Answer:41
Step-by-step explanation:
add 25 and 16 together
0.053053053053053 irrational or rational
Answer: It is rational
Step-by-step explanation:
ABCD is a square. A is the point (-2,0)and C is the point (6,4). AC and BD are diagonals of the square which intersect at M.
Find the coordinates of M, B, and D
The coordinates of M, B, and D of a square with vertices A(-2,0) and C(6,4) where AC and BD diagonals intersect at M are M(2,2), B(6,-8), and D(-2,8), respectively.
To locate the coordinates of M, we will first find the midpoint of AC, which is also the middle of the square. The midpoint of AC is ((-2+6)/2, (0+4)/2), which simplifies to (2,2).
Hence, M is in the middle of the square. To locate the length of a side of the square, we are able to use the distance formula.
The distance between a and C is √((6-(-2))² + (4-0)²), which simplifies to √(64+16), that is 8. Accordingly, each side of the square has a length of 8. Given that M is in the middle of the square, we can use this truth to discover the coordinates of B and D.
Because B and D are on opposite sides of M, their x-coordinates and y-coordinates should differ via 8. Seeing that a is at (-2,0), B needs to be 8 units to the proper and 8 units down, so B is at (6, -8). Further, on the grounds that c is at (6,4), D should be 8 gadgets to the left and 8 units up, so d is at (-2, 8).
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whats 18 divided by 800 solved
Answer:
Step-by-step explanation:44,4444444
Answer:
0.0225
Step-by-step explanation:
18÷800=0.0225
The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:
68% of the data will fall within one standard deviation of the mean.
95% of the data will fall within two standard deviations of the mean.
Almost all (99.7%) of the data will fall within three standard deviations of the mean.
The empirical rule is used as a rough gauge of normality. When a number of data points fall outside the three standard deviation range, it can indicate non-normal distributions.
68% of the data will fall within one standard deviation, 95% of the data will fall within two standard deviations and 99.7% of the data will fall within three standard deviations of the mean.
The empirical rule, which is also referred to as the three-sigma rule or the 68-95-99.7 rule, gives a rapid estimate of the distribution of data in a normal distribution, provided the mean and standard deviation. The empirical rule describes that in a normal distribution:68% of the data will fall within one standard deviation of the mean.95% of the data will fall within two standard deviations of the mean.99.7% of the data will fall within three standard deviations of the mean.This empirical rule is useful in determining whether the data is roughly normally distributed. When a number of data points fall outside the three standard deviation range, it indicates non-normal distributions. Consequently, the empirical rule is a valuable tool for approximating normal distributions, but it is important to remember that it is just an approximation, and not all normal distributions will follow this rule exactly.
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solve for x in the following:
2x^3+3x^2+12x+1 =0
The value of x in the equation 2x³ + 3x² + 12x + 1 = 0 when solved is x = -0.085
Given the following equation
2x³ + 3x² + 12x + 1 = 0
The equation cannot be factorized
So, we solved graphically
See attachment for the graph of the equation 2x³ + 3x² + 12x + 1 = 0
On the graph, we observe that the curve of the equation crosses the x-axis at
x = -0.085
This means that the solution for x in the equation is x = -0.085
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5/14 x4 in simplest form as a fraction
Answer:
the answer is 5 simplify
Answer:
20/14
10/7
0r 5/56
Step-by-step explanation:
i hope you find it helpful
Amanda is planning a graduation party for her son, who has just finished 8th grade. She has a party budget that is broken down into 5 categories: venue, invitations, food, entertainment, and decorations. Amanda wants to spend 5/24 of her budget on the venue, 1/16 on invitations, 1/4 on food, 3/16 on decorations, and the rest of her budget on entertainment. What fraction of amanda’s budget can she spend on entertainment?
Amanda can spend 9/48 of her budget on entertainment, which can be simplified to 3/16.
What is expression ?In mathematics, an expression is a combination of numbers, symbols, and/or operators that represents a mathematical quantity or relationship. Expressions can be composed of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, as well as functions and other mathematical constructs.
According to the given information:The fractions of Amanda's budget allocated to each category are:
Venue: 5/24
Invitations: 1/16
Food: 1/4
Decorations: 3/16
The sum of these fractions is:
5/24 + 1/16 + 1/4 + 3/16 = 15/48 + 3/48 + 12/48 + 9/48 = 39/48
This means that Amanda has allocated 39/48 of her budget to these four categories. To find out what fraction of her budget she can spend on entertainment, we need to subtract this fraction from 1 (since the remaining fraction is what she can spend on entertainment):
1 - 39/48 = 9/48
Therefore, Amanda can spend 9/48 of her budget on entertainment, which can be simplified to 3/16.
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Find a logarithmic equation that relatesyandx. (Round all numerical values to three decimal places.). ln(y)=
X: 1 2 3 4 5 6
Y: 1 0.630 0.481 0.397 0.342 0.303
ln(y) = -0.544x + 1.099
The logarithmic equation that relates x and y is given by ln(y) = -0.544x + 1.099. To get this equation, the given table of values is considered.Let's first find the value of "a" in the general logarithmic equation, which is y = a ln(x). We use the given table of values to get this value. When x = 1, y = 1, and so a ln(1) = 1, which implies that a = 1. Hence, the equation becomes y = ln(x).Now, we need to convert this to an equation in terms of y and x. That is, we need to find ln(y) in terms of x. To do this, we substitute y for x in the equation y = ln(x). That is, we get ln(y) = ln(x).Now, we substitute the given values of x and y in this equation to get the required logarithmic equation. That is, we have:ln(y) = -0.544x + 1.099 (rounded to three decimal places). Therefore, the logarithmic equation that relates x and y is ln(y) = -0.544x + 1.099.
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A quadratic function y=f(x) is plotted on a graph and the vertex of the resulting parabola
(−4,−5). What is the vertex of the function defined as g(x)= f(-x)−4?
Answer:
(-2,6)
Step-by-step explanation:
y=f(x) has vertex (-4,6)
g(x)=-f(x-2) shifts the graph 2 units to the right, not to the left, and the graph is reflected over the x-axis
therefore the vertex is (-2,6) and the reflection over the x-axis has no affect on the vertex
David bought 200 shares of Oracle stock yesterday and sold it today. His profit was $22. 0. At what price did he buy the stock yesterday?
Assuming that David sold the stock at $10 per stock today, then, we will agree that he bought the Oracle stock yesterday at a price of approximately $9.89 per share.
How do we calculate the price he bought the stock yesterday?A share, also known as a stock or equity, is a unit of ownership in a corporation. Stock investments provide investors with returns in the form of dividends and capital gains. The higher the expected returns, the more shares an investor owns.
Let the price at which David bought each share of Oracle stock yesterday be x. Then, his total cost for buying 200 shares would be: 200x
Similarly, his total revenue for selling 200 shares at $10 per share today would be:
= 200*($10)
= $2000
His profit was $22, so we can set up the equation which is $2000 - 200x = $22.
Simplifying and solving for x, we get:
$1978 = 200x
x = $1971/200
x = $9.89.
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A young boy ask his mother to get 5 game boy cartridges from his collection of 10 arcade and 5 sports games. How many ways are there that his mothercan get 3 arcades in 2 sports game
There are 1200 ways for the boy's mother to get 3 arcades and 2 sports games from his collection.
We can use the combination formula which is:
nCr = n! / (r! * (n-r)!)
where n is the total number of items, r is the number of items to choose, and ! denotes the factorial function.
In this case, we want to choose 3 arcade games and 2 sports games from a collection of 10 arcade games and 5 sports games. So, we have:
Number of ways to choose 3 arcade games from 10 arcade games = 10C3 = 120
Number of ways to choose 2 sports games from 5 sports games = 5C2 = 10
Therefore, the total number of ways to choose 3 arcade games and 2 sports games is:
120 * 10 = 1200
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3.4 You have one piece on the bar. Your opponent controls the inner 2, 4 , and 6 points and has a blot on the 5 point. It is your turn. What is the probability of being able to hit the blot on your next roll? What is the probability of your having to enter from the bar without hitting the blot?
The probability of being able to hit the blot on the next roll is 11/36, and the probability of entering from the bar without hitting the blot is 2/36 or 1/18.
The probability of being able to hit the blot on the next roll is 11/36. The probability of entering from the bar without hitting the blot is 2/36 or 1/18. The given situation can be represented by the following diagram:In the given scenario, the player has one piece on the bar, and the opponent controls the inner 2, 4, and 6 points while also having a blot on the 5 point. This means that the player needs to roll a 5 in order to hit the opponent's blot. The probability of rolling a 5 is 4/36 or 1/9.The player can also enter from the bar without hitting the blot. To do this, the player must roll a 1 or a 2. The probability of rolling a 1 or a 2 is 2/36 or 1/18.Therefore, the probability of being able to hit the blot on the next roll is 11/36, and the probability of entering from the bar without hitting the blot is 2/36 or 1/18.
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Point P is plotted on the number line. What number is represented by point P? P 5 * 10 ^ - 3; 6 * 10 ^ - 3
For a point P plotted on a given number line, the number represented by the point P is 5.80⁻³.
What is a number line?A number line is a linear visual representation of numbers and integers. This line is used to compare equally spaced numbers on an infinity line that extends horizontally or vertically on either side. A number line shows point values and equal divisions on the line. Between point 5.10⁻³ and point 6.10⁻³ there are 10 evenly spaced points.
Counting from the left, it looks like this:
5.10⁻³, 5.20⁻³, 5.30⁻³, 5.40⁻³, 5.50⁻³, 5.60⁻³, 5.70⁻³, 5.80⁻³, 5, 90⁻³, 6.00⁻³, 6.10⁻³.
Point P drops to 5.80⁻³, so the value of point P is 5.80⁻³.
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The complete question is as follows:
The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y=3X−2, where X has the density function f(x) = ¼ e^−x/4, for x > 0, f(x) = 0, elsewhere. Find the mean and variance of the random variable Y.
The mean of Y is -5/4 and the variance of Y is 144.
What is mean?In statistics, the mean is a measure οf central tendency that represents the average οf a set οf data. It is alsο knοwn as the arithmetic mean. The mean is calculated by adding up all the values in the dataset and dividing the sum by the tοtal number οf values. It is οften used tο describe the typical οr average value in a set οf data.
What is variance?Variance is a statistical measure that describes hοw much the values in a set οf data vary frοm the average value οr mean. It measures the spread οr dispersiοn οf the data pοints arοund the mean.
In the given questiοn,
Tο find the mean and variance οf the randοm variable Y, we need tο use the fοrmulas fοr the expected value and variance οf a functiοn οf a randοm variable:
The expected value (mean) of Y is given by:
E(Y) = E(3X - 2) = 3E(X) - 2
The variance of Y is given by:
Var(Y) = Var(3X - 2) = 9Var(X)
To find E(X), we need to integrate the density function f(x) over all possible values of X:
E(X) = ∫0∞ x f(x) dx
= ∫0∞ x (1/4)[tex]\rm e^{(-x/4)[/tex]dx
This integral can be solved using integration by parts, with u = x and dv/dx = (1/4)[tex]\rm e^{(-x/4)[/tex] dx.
Integrating by parts, we get:
E(X) = [-x/4 [tex]\rm e^{(-x/4)[/tex]]_0∞ + ∫0∞ (1/4)[tex]\rm e^{(-x/4)[/tex] dx
= [0 + (1/4)]/1 + [0]
= 1/4.
Therefore, the expected value of Y is:
E(Y) = 3E(X) - 2 = 3(1/4) - 2 = -5/4.
To find Var(X), we can use the formula for the variance of an exponential distribution, which is:
Var(X) = (1/λ²) = (4²) = 16.
Therefore, the variance of Y is:
Var(Y) = 9Var(X) = 9(16)
= 144.
Therefore, the mean of Y is -5/4 and the variance of Y is 144.
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The mean of Y is 10 minutes, and the variance of Y is 144 minutes squared.
What is integration?
Integration is a mathematical operation that is the reverse of differentiation. Integration involves finding an antiderivative or indefinite integral of a function.
To find the mean and variance of Y, we will first need to find the mean and variance of X.
Mean of X:
The mean of X is given by:
E(X) = integral from 0 to infinity of xf(x) dx
E(X) = integral from 0 to infinity of x * 1/4 e^(-x/4) dx
Using integration by parts with u = x and dv = e^(-x/4) dx, we can evaluate this integral to get:
E(X) = 4
Variance of X:
The variance of X is given by:
Var(X) = E(X^2) - [E(X)]^2
We will first need to find E(X^2). Using integration by parts again, we can evaluate this integral as:
E(X^2) = integral from 0 to infinity of x^2 * 1/4 e^(-x/4) dx
E(X^2) = 32
Therefore, the variance of X is:
Var(X) = E(X^2) - [E(X)]^2
Var(X) = 32 - 16
Var(X) = 16
Now, we can use the formula for the mean and variance of a linear transformation of a random variable to find the mean and variance of Y.
Mean of Y:
The mean of Y is given by:
E(Y) = E(3X - 2)
E(Y) = 3E(X) - 2
E(Y) = 10
Variance of Y:
The variance of Y is given by:
Var(Y) = Var(3X - 2)
Var(Y) = 9Var(X)
Var(Y) = 144
Therefore, the mean of Y is 10 minutes, and the variance of Y is 144 minutes squared.
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Question 4
The area of a rectangular picture frame is 54 in2. The length of the frame is 3 feet
longer than the width. What are the dimensions of the frame?
O 3 in; 18 in
O 6 in; 9 in
O4 in; 7 in
O9 in; 12 in
Using Algebraic equations, the dimensions of the frame are 6in: 9 in.
The word "area" refers to a free space. A shape's length and width are used to compute its area. Unidimensional length is expressed in terms of feet (ft), yards (yd), inches (in), etc. But a shape's area is a two-dimensional measurement. Hence, it is expressed in square units such as square inches (in2), square feet (ft 2), square yards (yd2), etc.
Given, the area of the rectangular picture frame is 54 in².
length of the frame is 3 feet longer than the width,
Let us assume width = x
length of the frame will be = x + 3,
Area of rectangle = length × breadth
⇒ 54 = (x) (x + 3)
⇒ 54 = x² + 3x
Now, factorizing the given equation,
x² + 3x - 54
x² + 9x - 6x - 54
x(x + 9) -6 (x + 9)
(x + 9) (x - 6)
The value of x will be x = -9 and x = 6
As the negative length of the rectangle is not possible, we will consider, x = 6 and width = 9,
∴ The length of the rectangle = 6 inch,
Width of the rectangle will be = 6 + 3 = 9 inch.
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One year ago, Derek joined a health club. He paid a yearly membership fee of $259. 99 that covers all the club’s services except use of the racquetball courts. Derek paid $3. 75 each time he used a racquetball court. For the entire year at a health club, Derek paid a total of $436. 24.
Part A: Write an equation to represent the problem. Use r to represent the number of times Derek used one of the racquetball courts. Explain your reasoning.
Part B: solve the equation you wrote in Part A to find how many times Derek used a racquetball court. Show your work.
Please try to explain fully of what you did and how you solved it.
Tysmm‼️
Part A: The equation to represent the problem is 259.99 + 3.75r = 436.24. The variable 'r' represents the number of times Derek used one of the racquetball courts.
Part B
Derek used a racquetball court 46.93 times
Part A: The equation to represent the problem is: 259.99 + 3.75r = 436.24. The “r” represents the number of times Derek used one of the racquetball courts. 259.99 is the yearly membership fee and the 3.75 is the cost of each time Derek used a racquetball court. 436.24 is the total amount Derek paid for the entire year at the health club.
Part B: To solve the equation, we need to isolate “r” on one side of the equation. To do this, we need to subtract 259.99 from both sides of the equation. We then get 3.75r = 176.25. We divide both sides by 3.75 to get r = 46.93. This means that Derek used a racquetball court 46.93 times.
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