The answer on Probability to draw probability tree and calculate the final solution the answers are (a) the tree is given below , (b) i) P(R1 ∩ W2) means the probability of drawing a red marble on the first draw and a white marble on the second draw , ii) The solution is P(R1) * P(W2|R1) = (4/9) * (5/8) = 5/18. (C) 17/36. (d) the tree is given below.
What is Experiment?In probability theory, experiment is process or situation that produces set of possible outcomes or events, each with associated probability of occurrence.
In experiment, the set of all possible outcomes is called sample space, denoted by symbol Ω. The outcomes in sample space can be either discrete, meaning they are countable and separate, or continuous, meaning they are uncountable and form range or interval.
a) Probability tree:
P(W1) = 5/9
/ \
P(W2|W1) P(R2|W1) = 4/9
/ \ / \
P(W3|W1W2) P(R3|W1W2) P(W3|R1W2) P(R3|R1W2)
/ \ / \ / \ / \
P(W4|W1W2W3) P(R4|W1W2R3) P(W4|R1W2W3) P(R4|R1W2R3)
b)
i) P(R1 ∩ W2) means the probability of drawing a red marble on the first draw and a white marble on the second draw.
ii) The solution is P(R1) * P(W2|R1) = (4/9) * (5/8) = 5/18.
c)
i) There are two ways to get at least one red ball: drawing a red ball on the first draw or drawing a white ball on the first draw and a red ball on the second draw.
ii) Angela is not correct. To calculate the probability of drawing at least one red ball, we need to add the probabilities of the two ways to get at least one red ball:
P(at least one red ball) = P(R1) + P(W1) * P(R2|W1) = (4/9) + (5/9) * (4/8) = 17/36.
d) If the first ball is replaced before the second draw, then the probability tree would have the same probabilities for each draw. The new probability tree would be:
P(W1) = 5/9
/ \
P(W2) P(R2) = 4/9
/ \ / \
P(W3) P(R3) P(W3) P(R3)
/ \ / \ / \ / \
P(W4) P(R4) P(W4) P(R4) P(W4) P(R4)
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A carpenter has a box of nails of various
different lengths. You decide to practice your
weighted averaging skills to figure out the
average length of a nail in the box. You grab
two handfuls of nails and count out the
number of each type of nail. You record your
data in the table below.
Sample
Type
Short nail
Medium nail
Long nail
Number Abundance
of Nails
(%)
67
18
10
[?]
Nail Length
(cm)
2.5
5.0
7.5
What is the percent abundance of the
medium nails in your sample?
The percent abundance of medium nails in the sample is approximately 18.95%.
What is weighted averaging?An average value for a collection of data is calculated statistically via weighted averaging, where each data point is assigned a weight based on its relative relevance or frequency. This method is frequently applied when certain data points are more important than others or when the sample sizes for several sets of data vary. Each data point is multiplied by its weight in weighted averaging, and the resulting products are added up and divided by the overall weight.
The number of each type of nail is:
Abundance = 67 + 18 + 10 = 95
The weighted average is:
Weighted Average = (Abundance1 x Value1 + Abundance2 x Value2 + ... + AbundanceN x ValueN) / Total Abundance
For different types of nails we have:
Short nail: Abundance1 = 67, Value1 = 2.5
Medium nail: Abundance2 = 18, Value2 = 5.0
Long nail: Abundance3 = 10, Value3 = 7.5
Substituting the values:
Weighted Average = (67 x 2.5 + 18 x 5.0 + 10 x 7.5) / 95
Weighted Average ≈ 3.53
The percent abundance is:
Percent Abundance = (Abundance2 / Total Abundance) x 100%
Substituting the values:
Percent Abundance = (18 / 95) x 100%
Percent Abundance ≈ 18.95%
Hence, the percent abundance of medium nails in the sample is approximately 18.95%.
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A home has gone up in value over several
decades and is now worth 1354% of its
original sale price of $23,000. What is the
value now?
Answer:
$31,142
Step-by-step explanation:
To convert a percentage into a decimal, you move the decimal two places to the left. 1354% converted into a decimal is 13.54.
$23,000 * 13.54 = $31,142
Bill buys candy that costs $5 per pound. He will spend at least $40 on candy. What are the possible numbers of pounds he will buy? Use p for the number of pounds Bill will buy. Write your answer as an inequality solved for p.
Answer: 5p = 40
Step-by-step explanation:
5 X p = 40
5/5 40/5
p = 8
a coin is toosed and a number cube is rolled. sketch a list or tree digrams to represnts the sample space
The first of these two events is the number displayed on the cube. The result of the coin toss is the second. There should be two columns in this tree, one for each occurrence.
a list or tree digrams to represnts the sample space in the given below figure . Observe the first column first. Let the number cube's result be. The following six values are possible:. Six branches, one for each outcome, result from that proceed to the following column. Let the result of the coin toss be. There would always be two possible results for, regardless of the number obtained for.
The first of these two events is the number displayed on the cube. The result of the coin toss is the second. There should be two columns in this tree, one for each occurrence.
Observe the first column first. Let N represent the number cube's result. Six values are possible: 1, 2, 3, 4, 5, and 6. Six branches, one for each outcome, result from that. proceed to the following column. Let the result of the coin toss be. There would always be two possible results for, regardless of the number obtained for.
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the actual question is :
The number cube is rolled and a coin is tossed . How can you sketch a tree diagram to represent the sample space ?
4^(-x)=1/256
I believe it is x=4, but I need how to work it out pls thxxx
Answer:
x = 4
Step-by-step explanation:
using the rule of exponents
• [tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex] , then
[tex]4^{-x}[/tex] = [tex]\frac{1}{4^{x} }[/tex]
and 256 = [tex]4^{4}[/tex]
then
[tex]\frac{1}{4^{x} }[/tex] = [tex]\frac{1}{4^{4} }[/tex]
so
[tex]4^{x}[/tex] = [tex]4^{4}[/tex]
since bases on both sides are equal, bot 4 then equate exponents
x = 4
find a parameterization of each of the following surfaces, in terms of sines, cosines, and hyperbolic sines and cosines
Parameterizing a surface over a rectangle Parameterizing the surface z = x²+2y² over the rectangular region R defined by -3 ≤ x ≤ 3, −1 ≤ y ≤ 1 are falls under the range of R.
Let's start by expressing x and y as functions of u and v. Since x varies between -3 and 3 over R, we can use the following parameterization for x:
x = u
where u varies between -3 and 3. Similarly, since y varies between -1 and 1 over R, we can use the following parameterization for y:
y = v
where v varies between -1 and 1.
Next, we can use these parameterizations for x and y to express z as a function of u and v. Substituting x = u and y = v into the equation z = x² + 2y², we get:
z = u² + 2v²
So, the parameterization of the surface z = x² + 2y² over the rectangular region R is given by:
x = u, y = v, z = u² + 2v²
where -3 ≤ u ≤ 3 and -1 ≤ v ≤ 1.
The parameterization allows us to study various properties of the surface z = x² + 2y² over the rectangular region R.
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Complete Question:
Parameterizing a surface over a rectangle Parameterizing the surface z = x²+2y² over the rectangular region R defined by -3 ≤ x ≤ 3, −1 ≤ y ≤ 1.
Trigonometric Functions
pls help and be sure to show all your work !! the second pic is the graph that needs to be plotted for the first question.
A) Values of y will be 1, 0, -1, 0, 1
B) and C) plot the values as shown in below figure.
Define the term Trigonometric Function?Trigonometric functions are a set of mathematical functions that relate the angles and sides of right triangles.
A) for f(x) = cos x; the values of x-y chart are:
x y = f(x)
0 1
[tex]\frac{\pi }{2}[/tex] 0
[tex]\pi[/tex] -1
[tex]\frac{3\pi }{2}[/tex] 0
[tex]2\pi[/tex] 1
B) the points are plotted as given figure below.
C) Connects the dots to show the graph of y = cos x; sown in figure below please check it.
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Desmond makes online purchase of R395 from Lineup store . he gets 46 and half % discount due to loyality points. calculate the amount of discount Desmond receive
Desmοnd receives a discοunt οf R183.675. We can rοund this tο R183.18 οr R183.18, depending οn the desired level οf precisiοn.
What is discοunt?We all are eager abοut availing discοunts, but we barely knοw hοw we get these discοunts. There is a prοper way tο calculate discοunt and that we dο by using a discοunt fοrmula. The discοunt rate fοrmula can either be extracted by subtracting the selling price οf the prοduct frοm its marked price οr by multiplying the discοunt rate οffered and the marked price οf the prοduct. In terms οf Mathematics, the fοrmula fοr discοunt is represented as belοw,
Accοrding tο given infοrmatiοn,
Tο calculate the discοunt Desmοnd receives, we first need tο find 53.5% οf the οriginal price. We can dο this by multiplying the οriginal price by 0.535:
R395 x 0.535 = R211.325
Sο, Desmοnd pays R211.325 fοr his purchase after the discοunt.
Tο find the amοunt οf discοunt, we need tο subtract the discοunted price frοm the οriginal price:
R395 - R211.325 = R183.675
Therefοre, Desmοnd receives a discοunt οf R183.675. We can rοund this tο R183.18 οr R183.18, depending οn the desired level οf precisiοn.
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the simplest form of the expression sqr3-sqr6/sqr3+sqr6?
Answer:
1 - [tex]\frac{2\sqrt{2} }{3}[/tex]
Step-by-step explanation:
[tex]\frac{\sqrt{3}-\sqrt{6} }{\sqrt{3}+\sqrt{6} }[/tex]
rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
the conjugate of [tex]\sqrt{3}[/tex] + [tex]\sqrt{6}[/tex] is [tex]\sqrt{3}[/tex] - [tex]\sqrt{6}[/tex]
= [tex]\frac{(\sqrt{3}-\sqrt{6})(\sqrt{3}-\sqrt{6}) }{(\sqrt{3}+\sqrt{6})(\sqrt{3}-\sqrt{6}) }[/tex] ← expand numerator/ denominator using FOIL
= [tex]\frac{3-\sqrt{18}-\sqrt{18}+6 }{3-\sqrt{18}+\sqrt{18}+6 }[/tex]
= [tex]\frac{9-2\sqrt{18} }{3+6}[/tex]
= [tex]\frac{9-2(3\sqrt{2}) }{9}[/tex]
= [tex]\frac{9-6\sqrt{2} }{9}[/tex]
= [tex]\frac{9}{9}[/tex] - [tex]\frac{6\sqrt{2} }{9}[/tex]
= 1 - [tex]\frac{2\sqrt{2} }{3}[/tex]
Find the associated z-score or scores that represent the following standard normal areas(hint use the excel function =NORM.S.INV()
A. Middle 50 percent
B. Lowest 5 percent
C. Middle 90%
Answer all questions please(URGENT
The z-scores that represent the middle 50% of the standard normal distribution are between -0.6745 and 0.6745, the lowest 5% of the standard normal distribution is -1.645, and the 90% of the standard normal distribution is between -1.645 and 1.645.
What is the definition of standard normal variation?The mean and variance of a standard normal distribution are both 0. A z distribution is another name for this.
Yes, here are the z-scores for the given standard normal areas:
A. Middle 50%: The area between the 25th and 75th percentiles corresponds to the middle 50% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:
-0.6745 is the z-score corresponding to the 25th percentile.
The 75th percentile z-score is 0.6745.
As a result, the z-scores representing the middle 50% of the standard normal distribution range between -0.6745 and 0.6745.
B. Lowest 5%: The area to the left of the 5th percentile corresponds to the lowest 5% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-score that corresponds to that percentile as follows:
The z-score associated with the fifth percentile is -1.645.
As a result, the z-score representing the bottom 5% of the standard normal distribution is -1.645.
C. Middle 90%: The area between the 5th and 95th percentiles corresponds to the middle 90% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:
The z-score associated with the fifth percentile is -1.645.
The z-score associated with the 95th percentile is 1.645.
As a result, the z-scores representing the middle 90% of the standard normal distribution range between -1.645 and 1.645.
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10 POINTS NEED HELP ASAP!!!! PLEASE HELP ME FIND THE AREA AND THE PERIMETER!!
The area of the composite shape using the area formula for the different shapes is 460.48ft².
What are composite shapes?The area of composite shapes refers to the space occupied by any composite shape. A composite shape is a shape that is made by connecting a few polygons to form the required shape.
These figures or shapes can be built from a wide range of shapes, such as triangles, squares, quadrilaterals, etc. Divide a composite item into basic forms such a square, triangle, rectangle, or hexagon to get its area.
Now in the question,
First let us find the area of the semi-circle.
Area of semi-circle = πr²/2
= [3.14 × (16/2) ²]/2
= (3.14 × 8²)/2
= 200.96/2
= 100.48ft²
Now coming to the rectangle,
area of the rectangle = l × b
= 20 × 15
= 300ft²
Now for calculating the area of the triangle,
area = 1/2 × b × h
= 1/2 × 12 × 10
= 60ft²
Therefore, the area of the total figure = 100.48 + 300 + 60 = 460.48ft².
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The retail price of a television set is $4,500. If the buyer pays by cash, the price is 10% below the retail price. If the set is bought on higher purchase, the buyer pays a down payment of $675 and 24 monthly installments of $212.50. Calculate for the television the:
cash price to the buyer, if he pays by cash.
amount payable.
outstanding balance.
hire purchase price.
interest.
difference between the hire purchase and the cash price.
percentage interest charged.
Cash price to the buyer, if he pays by cash:
The cash price would be 10% less than the retail price of $4,500:
Cash price = $4,500 - (10% * $4,500)
Cash price = $4,500 - $450
Cash price = $4,050
Amount payable:
If the buyer chooses to purchase the TV on higher purchase, the amount payable would be the sum of the down payment and the total amount of monthly installments:
Amount payable = Down payment + (24 * Monthly installment)
Amount payable = $675 + (24 * $212.50)
Amount payable = $5,175
Outstanding balance:
The outstanding balance would be the difference between the amount payable and the down payment:
Outstanding balance = Amount payable - Down payment
Outstanding balance = $5,175 - $675
Outstanding balance = $4,500
Hire purchase price:
The hire purchase price would be the sum of the down payment and the outstanding balance:
Hire purchase price = Down payment + Outstanding balance
Hire purchase price = $675 + $4,500
Hire purchase price = $5,175
Interest:
The interest charged on the hire purchase would be the difference between the hire purchase price and the cash price:
Interest = Hire purchase price - Cash price
Interest = $5,175 - $4,050
Interest = $1,125
Difference between the hire purchase and the cash price:
The difference between the hire purchase price and the cash price would be the interest charged:
Difference = Interest
Difference = $1,125
Percentage interest charged:
To calculate the percentage interest charged, we can divide the interest by the hire purchase price and multiply by 100:
Percentage interest charged = (Interest / Hire purchase price) * 100
Percentage interest charged = ($1,125 / $5,175) * 100
Percentage interest charged = 21.74%
Therefore, the cash price to the buyer is $4,050, the amount payable is $5,175, the outstanding balance is $4,500, the hire purchase price is $5,175, the interest charged is $1,125, the difference between the hire purchase and the cash price is $1,125, and the percentage interest charged is 21.74%.
Given x-intercepts of (-1,0), (6, 0) and point (-2, 2) that lies on the
quadratic graph, which equation below represents the correct quadratic
equation in standard form?
Answer:
y = (-1/3)x^2 - 2x + 5/3
Step-by-step explanation:
To find the quadratic equation that passes through these points, we can start by using the standard form of a quadratic equation:
y = ax^2 + bx + c
We know that the graph passes through the point (-2,2), so we can substitute these values into the equation:
2 = a(-2)^2 + b(-2) + c
Simplifying this equation, we get:
4a - 2b + c = 2
We also know that the graph has x-intercepts at (-1,0) and (6,0). This means that when x = -1 and x = 6, the value of y (i.e., the height of the graph) is 0. We can use these two points to write two more equations:
0 = a(-1)^2 + b(-1) + c
0 = a(6)^2 + b(6) + c
Simplifying these equations, we get:
a - b + c = 0
36a + 6b + c = 0
Now we have a system of three equations:
4a - 2b + c = 2
a - b + c = 0
36a + 6b + c = 0
We can solve for a, b, and c using any method of solving systems of equations. One way is to use substitution:
From the second equation, we get:
c = b - a
Substituting this into the other two equations, we get:
4a - 2b + (b - a) = 2
36a + 6b + (b - a) = 0
Simplifying these equations, we get:
3a - b = 1
35a + 7b = -b
Multiplying the first equation by 7 and adding it to the second equation, we eliminate b:
21a = -7
a = -1/3
Substituting this value of a into the first equation, we can solve for b:
4(-1/3) - 2b + (b - (-1/3)) = 2
-4/3 - b + b + 1/3 = 2
b = -2
Finally, we can use either of the first two equations to solve for c:
c = a - b = (-1/3) - (-2) = 5/3
So the quadratic equation in standard form is:
y = (-1/3)x^2 - 2x + 5/3
Pleasee help!
Highlight the vertex, and name the angle on the image below
Answer:
Highlight "x" then it's an acute angle
Step-by-step explanation:
You have three dice in your pocket, one is 4 -sided and two are 6-sided. You randomly select two of the dice, roll them, and record the sum. a. What's the probability the sum is 7 ? b. Given that the sum of the dice is 7 , what's the probability both dice are 6 -sided?
The probability that sum of two randomly selected dice is 7 is 1/2. The given that the sum of the dice is 7, the probability that both dice are 6-sided is 1/3.
a. Each pair has a probability of being selected of 1/3 * 1/2 = 1/6. Now sum of the probabilities of the three pairs that have a sum of 7: P = P(4-6) + P(6-4) + P(6-6) = 1/6 + 1/6 + 1/6 = 1/2
b. Given that the sum of the dice is 7, there are three possible pairs that could have been rolled: (1-6), (2-5), or (3-4). Of these, only one pair has both dice as 6-sided.
P = (1/3 * 1/2) / (1/2) = 1/3
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1. Solve. 4x+4=4
O 1
0 0
O 3
O
-3
Answer:
x = 0Step-by-step explanation:
4x+4 = 4
Subtract 4 from both sides:-
4x+4-4 = 4-4
4x = 0
Divide both sides by 4:-
4x/4 = 0/4
x = 0
Suppose that your mass is 86 kg How fast (in m/s) would vou have t0 run to have the same linear momentum as 1490 kg car moving at 2.7 km/h? Number Azo Units mfs The number of _ significant digits is set to 2; the tolerance is +/-3%/
The velocity with which we have to move to have the momentum equal to the car is 46.78 km/h.
Linear momentum is the vector quantity and described as the product of the mass of an object, m, and its velocity, v. It is represented by the symbol "p," which is also used to denote momentum. Please be aware that the body's momentum always points in the same direction as its vector of motion.
We are given:-
The mass of the car (M1) = 1490kg.
The velocity of the car (V1V1)= 2.7 km/h.
The formula to calculate linear momentum:- linear momentum= mass of the body* Velocity of the body (P= mvmv)
therefore, the momentum (P1) of the car = M1V1
P1= 1490*2.7= 4023
let momentum of yourself is P2 which is equal to P1 according to the question, Hence P2= 4023.
M2= 86 kg.
V2= ?
P2= M2V2
V2= P2/M2
V2 = 4023/86
V2= 46.78 km/h.
hhence toto the velocity with which we have to run to get momentum equal to the car is 46.78 km/h.
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Converting
What is 3.4 hours in hours and minutes?
By answering the presented questiοn, we may cοnclude that 0.4 οf an expressiοns hοur is 0.4 x 60 = 24 minutes in this situatiοn. As a result, 3.4 hοurs equals 3 hοurs and 24 minutes.
What is expressiοn ?An expressive in mathematics is a collection of numbers, variables, and complicated mathematical operations (such as addition, subtraction, multiplication, division, multiplicands, and others) that convey a quantity's value. Expressions can be as straightforward as "3 + 4" or as complex as "(3x2 - 2) / (x + 1)". They could also include functions like "sin(x)" or "lg(y)".
The real way to evaluate an expression is to insert the variables with their values and carry out the arithmetic operations in the order specified. For instance, the phrase "3x + 5" means 3(2) + 5 = 11 if x = 2. In mathematics, expressions are typically used to describe actual situations, build equations, and decompose challenging mathematical diagrams.
3 hοurs and 24 minutes is cοmparable tο 3.4 hοurs.
Yοu may separate the full number (which represents the hοurs) frοm the decimal tο cοnvert decimal hοurs tο hοurs and minutes (which represents the fractiοn οf an hοur in minutes).
0.4 οf an hοur is 0.4 x 60 = 24 minutes in th
is situation. As a result, 3.4 hours equals 3 hours and 24 minutes.
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You run 3 laps around a rectangular field. The field is 100 meters long and 97 meters wide. How many meters do you run?
what is the range of function m?
The range of the function m is [1, ∞). Thus, B is correct option.
What are a function's range and domain?The dοmain οf a functiοn is the cοllectiοn οf values that can be plugged intο it. This set represents the x values in a functiοn like f. (x). A functiοn's range is the set οf values that the functiοn can take οn. This is the set οf values returned by the functiοn when we enter an x value.
Tο determine the range οf functiοn m, we must first determine the set οf all pοssible functiοn οutput values.
Looking at the function graph, we can see that the lowest point is (3, 1) and the highest point is (∞, ∞).
As a result, the function m has a range of [1, ∞).
We can write: in interval notation:
m ∈ [1, ∞) range.
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let z be a standard normal random variable. find the values of a such that p({z > 1.35}) or {z > 2.34})
The values of a such that p({z > 1.35}) or {z > 2.34}) is 0.0981.
Let z be a standard normal random variable.
The standard normal distribution is a probability distribution that is symmetric around a mean of zero and has a standard deviation of one. It is also referred to as the Gaussian distribution or the normal distribution. This distribution is commonly used in the field of statistics to measure the likelihood of certain outcomes.
We have to determine the values of a such that p({z > 1.35}) or {z > 2.34}).
p(z > 1.35) + p(z > 2.34) = p(z > a)
After using the standard normal distribution table.
0.0885 + 0.0096 = p(z > a)
p(z > a) = 0.0981
p(z > 1.29) = 0.0981
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Let f(x)=-3x-1 and g(x)=x - 4 Find (fxg)(-1).
The value of the equation (fxg)(-1) is 10.
What is Equation?An equation is a mathematical statement that shows that two expressions are equal. It usually includes variables, which are represented by letters or symbols, and constants, which are fixed values.
What are the different types of Equations?There are several types of equations in mathematics. Here are some of the most common types:
Linear equation: An equation of the form "ax + b = c", where "a", "b", and "c" are constants and "x" is the variable. The graph of a linear equation is a straight line.Quadratic equation: An equation of the form "ax² + bx + c = 0", where "a", "b", and "c" are constants and "x" is the variable. The graph of a quadratic equation is a parabola.
Cubic equation: An equation of the form "ax³ + bx² + cx + d = 0", where "a", "b", "c", and "d" are constants and "x" is the variable. The graph of a cubic equation is a curve that can have one or two humps.
In the given question,
To find (f x g)(-1), we need to evaluate the product of f(-1) and g(-1).
First, we find f(-1):
f(-1) = -3(-1) - 1 = 2
Next, we find g(-1):
g(-1) = -1 - 4 = -5
Now, we can find the product (f x g)(-1):
(f x g)(-1) = f(-1) x g(-1) = 2 x (-5) = -10
Therefore, (f x g)(-1) = -10.
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under the normal distribution, if the mean of the distribution of raw scores is equal to 100, then its equivalent z-score is equal to
The equivalent z-score for a raw score of 115 under a normal distribution with a mean of 100 and a standard deviation of 15 is 1.
Under the normal distribution, if the mean of the distribution of raw scores is equal to 100 and the standard deviation is known, we can use the z-score formula to calculate the equivalent z-score for any given raw score.
The z-score formula is given by:
z = (x - μ) / σ
where x is the raw score, μ is the mean of the distribution, σ is the standard deviation of the distribution, and z is the corresponding z-score.
Since the mean of the distribution is 100, we have μ = 100. To calculate the z-score, we need to know the standard deviation of the distribution or have information about the distance of the raw score from the mean in terms of standard deviations.
If we assume that the standard deviation is 15, which is a common value used in educational testing, and the raw score is 115, then the corresponding z-score would be:
z = (115 - 100) / 15 = 1
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Locate the absolute extrema of the function on the closed interval
Answer:
The absolute extrema is minimum at (-1, 2/9)
Step-by-step explanation:
Absolute extrema is a logical point that shows whether a the curve function is maximum or minimum.
Forexample a curve in the image attached. A, B and C are points of absolute maxima or absolute maximum. and P and Q are points of absolute minima or minimum.
Remember A, B, C, P, Q are critical points or stationary points.
How do we find absolute extrema?
The find the sign of the second derivative of the function.
From the question;
[tex]{ \sf{g(x) = \sqrt[3]{x} }} \\ \\ { \sf{g(x) = {x}^{ \frac{1}{3} } }} \\ [/tex]
Find the first derivative of g(x)
[tex]{ \sf{g {}^{l}(x) = \frac{1}{3} {x}^{ - \frac{2}{3} } }} \\ [/tex]
Find the second derivative;
[tex]{ \sf{g {}^{ll} (x) = ( \frac{1}{3} \times - \frac{2}{3}) {x}^{( - \frac{2}{3} - 1) } }} \\ \\ { \sf{g {ll}^{(x)} = - \frac{2}{9} {x}^{ - \frac{5}{3} } }}[/tex]
Then substitute for x as -1 from [-1, 1]
[tex]{ \sf{g {}^{ll}(x) = - \frac{2}{9} ( - 1) {}^{ - \frac{5}{3} } }} \\ \\ = \frac{ - 2}{9} \times - 1 \\ \\ = \frac{2}{9} [/tex]
Since the sign of the result is positive, the absolute extrema is minimum
Rewrite without absolute value for the given condition: |(square root 2) +3 -5|
Answer: We can simplify the expression inside the absolute value first:
|(sqrt(2) + 3 - 5)|
= |(sqrt(2) - 2)|
Since sqrt(2) > 2, we know that sqrt(2) - 2 is negative. Therefore, we can rewrite the absolute value as a negative:
|sqrt(2) - 2| = -(sqrt(2) - 2)
So the expression without absolute value is:
-(sqrt(2) - 2)
Step-by-step explanation:
how do you simplify 7/8 + 3/4?
Answer:
13/8
Step-by-step explanation:
You first find the lcm of 8 and 4 which is 8
Then make a dividing line (/) with the lcm as the numerator.
After which you start diviing the lcm by each denominator of the two terms, and then multiplying with their numerators:
8/8 x 7 + 8/4 x 3 all over the lcm which is 8
(1 x 7 + 2 x 3)/8
(7 + 6)/8
13/8
2) The table shows the number of minority officers in the U.S. military in 2000.
Army
Marines
Air Force Total
9162
1341
4282
18309
2105
914
1518
7269
4075
599
3823
11150
15342
2854
9623
36728
Assume that one person will be randomly selected from the group described in the table. Find the
probability of selecting an officer who is in the Navy, given that the officer is African American. (Do
not reduce.)
3524
A)
8909
African Americans
Hispanic Americans
Other Minorities
Total
B)
Navy
3524
2732
2653
8909
8909
18,309
C).
3524
18,309
3542
14785
D).
2)
The probability of selecting an officer who is in the Navy given that the officer is African American is 0.192.
What is the probability?Probability calculates the likelihood that an event would happen. The likelihood that an event would occur has a value that lies between 0 and 1 or 0 and 100. The more likely an event would happen, the closer the probability value would be to 1 or 100.
The probability of selecting an officer who is in the Navy given that the officer is African American = number of African Americans that are in the Navy / total number of African Americans
Total number of African Americans = 9162 + 3524 + 1341 + 4282 = 18,309
The probability = 3524 / 18,309 = 0.192
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TRUE/FALSE. When calculating probabilities, combinations and permutations are used to find the number of outcomes in a sample space.
TRUE. Combinations and permutations are used in probability calculations to determine the number of possible outcomes in a sample space.
When calculating probabilities, combinations and permutations are used to find the number of outcomes in a sample space. Why?When order is important, permutations are utilized, and when order is not important, combinations are used. The formula P(n,r) = n!/(n-r)!, where n! signifies the factorial of n, and the expression C(n,r) = n!/r!(n-r)!, give the number of permutations of n items taken r at a time and the number of combinations of n objects taken r at a time, respectively.
It is possible to count the number of ways that a subset of items can be chosen from a larger set using both permutations and combinations. These ideas are crucial for estimating probabilities because they let us know how many different outcomes could satisfy a certain condition.
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What does the point (5, 10) represent on the
graph?
Answer:
It means the point x = 5 and y = 10
Answer: The place 5 units right and 10 units up from the center of the graph (the origin).
Step-by-step explanation:
Starting at the origin, the 5 represents moving right 5, and the 10 represents going up 5.
Find, correct to four decimal places, the length of the curve of intersection of the cylinder4x^2 + y^2 = 4and the planex + y + z = 6
The length of the curve of intersection, correct to four decimal places, is 27.7128 units.
To find the length of the curve of intersection, we need to first parameterize the curve.
From the equation of the cylinder, we have:
4x² + y² = 4
Dividing both sides by 4, we get:
x²/1 + y²/4 = 1
This is the equation of an ellipse centered at the origin with semi-major axis a=1 and semi-minor axis b=2.
From the equation of the plane, we have:
x + y + z = 6
Solving for z, we get:
z = 6 - x - y
So the equation of the curve of intersection is:
r(t) = <t, 2sin(t), 6-t-2sin(t)>
where 0 <= t <= 2π.
To find the length of the curve, we need to integrate the magnitude of its derivative with respect to t:
L = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
Taking the derivative of r(t), we get:
r'(t) = <1, 2cos(t), -1-2cos(t)>
So the magnitude of r'(t) is:
|r'(t)| = √(1² + (2cos(t))² + (-1-2cos(t))²) = √(9+9cos(t)²)
Therefore, the length of the curve is:
L = ∫_0⁽²π⁾ √(9+9cos(t)²) dt
This integral can be solved using the substitution u = tan(t/2), which gives:
L = 18∫_0∞ 1/√(1+u²) du
This integral can be evaluated using trig substitution, letting u = tanθ, which gives:
L = 18∫_0⁽π/²⁾ secθ dθ = 18 ln|sec(π/4)+tan(π/4)| = 18 ln(1+√2)
Using a calculator, we get:
L ≈ 27.7128
Therefore, the length of the curve of intersection, correct to four decimal places, is 27.7128 units.
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