Identify the simplest polynomial function having integer coefficients with the given zeros.
Answers
The polynomial for the given zeros is p(x) = x⁴ + 4x³ - 3x² - 12x. The correct answer is option (B).
What is factor theorem?The factor theorem for a polynomial p(x) states that for some real number a if p(a) = 0, then (x - a) is one of the factors of p(x).
Given that,
The zeros for a polynomial are 0, -4 and√3.
As per the factor theorem the polynomial for zeros a, b, c and d can be written as,
p(x) = (x - a)(x - b)(x - c)(x - d)
Thus the polynomial for the given zeros is,
(x - 0)(x - (-4))(x - √3)(x + √3)
= x(x + 4)( x² - 3)
= x(x³ + 4x² - 3x - 12)
= x⁴ + 4x³ - 3x² - 12x
Hence, the required polynomial is p(x) = x⁴ + 4x³ - 3x² - 12x.
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Related Questions
A standard normal distribution has a mean of _____ and a standard deviation of _____. a. 1 and 1b. 0 and 1c. 1 and 0d. 0 and 0
Answers
The correct statement is: 'A standard normal distribution has a mean of 0 and a standard deviation of 1 '
The correct answer is an option (b)
In this question, we need to complete given statement.
We know that the normal distribution is a symmetrical and bell-shaped distribution.
In standard normal distribution the mean, median and mode are all equal. Also, it always has a mean of zero and a standard deviation of one.
So, A standard normal distribution has a mean of 0 and a standard deviation of 1 .
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What are the 4 ways to solve a system?.
Answers
Answer:
I know 3
substitution and elimation method and graphing (sorry i was never taught a fourth method ):
hopes this helps please mark brainliest
a pollster is going to sample a number of voters in a large city and construct a 90% confidence interval for the proportion who support the incumbent candidate for mayor. find a sample size so that the margin of error will be no larger than 0.05. be sure to round up to the next whole number.
Answers
Value of n for confidence interval 90% is 66.
What is confidence interval?
A confidence interval, in statistics, refers to the chance that a population parameter can fall between a collection of values for a particular proportion of times.
Main body:
Finding Sample Size for Estimating a Population Proportion:
n = [tex]\frac{z}{M}^{2} }[/tex]p(1-p)
M = is the margin of error
p = is an estimated value of the proportion
We want to construct a 90% confidence interval for with a margin of error equal to 5%.
Because there is no estimate of the proportion given, we use p = 0.50 for a conservative estimate.
For a 90% confidence interval, z = 0.816
n = (0.816/0.05)^2 *0.5(1-0.5)
n = 266.3424*0.5*0.5
n = 66 approximately
Hence value of n = 66 .
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Factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
Answers
Using the definition of divisibility, 2p is the greatest common factor.
In the given question we have to factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
The given polynomial is 2p^3+6p.
Now taking 2p common from both terms
=2(p^2+3)
By the definition of divisibility, we get
2p | 2p^3 and 2p|6p, 2| 2p^3 and 2|6p also p|2p^3 and p|6p.
So, 2,p and 2p are common factors of 2p^3 and 6p.
Hence, 2p is the greatest common factor.
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The right answer is:
Factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
2p^3+6p
TRUE/FALSE. we use anova to test for differences between population means by examining the amount of variability between the samples relative to the amount of variability within the samples.
Answers
We use ANOVA to test for differences between population means by examining the amount of variability between the samples relative to the amount of variability within the samples. Hence, it is TRUE
What is ANOVA ?The statistical analysis tool known as analysis of variance (ANOVA) divides the observed aggregate variability present in a data set into two categories: systematic variables and random components. The random factors have no statistical impact on the presented data set, whereas the systematic factors do. The ANOVA test is used by analysts to ascertain how independent factors in a regression analysis affect the dependent variable.
ANOVA basically stands for analysis of the variance.
Analysis of variance, or ANOVA, is a statistical method that separates observed variance data into different components to use for additional tests.A one-way ANOVA is used for three or more groups of data, to gain information about the relationship between the dependent and independent variables.If no true variance exists between the groups, the ANOVA's F-ratio should equal close to 1Hence, according to above statements, given fact is true.
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use a quadratic equation to find two real numbers with a sum of 31 and a product of 210. two real numbers with a sum of 31 and a product of 210 are and
Answers
Answer:
The numbers are 10 and 21.
Step-by-step explanation:
x + y = 31
xy = 210
x = 31 - y
(31 - y)y = 210
31y - y² = 210
y² - 31y + 210 = 0
(y - 21)(y - 10) = 0
y - 21 = 0 or y - 10 = 0
y = 21 or y = 10
The two real numbers are 10 and 21.
What is a quadratic equation?Quadratic Equations are algebraic expressions of degree 2 in one variable and are of the form ax² + bx + c = 0.
In simpler terms, a quadratic equation is be defined as a polynomial equation of degree 2.
Given that, there are two real numbers with a sum of 31 and a product of 210.
We need to find their values using the concept of quadratic equation,
So, according to the question,
x + y = 31 (eq 1)
xy = 210 (eq 2)
x = 31 - y
(31 - y) y = 210
31y - y² = 210
y² - 31y + 210 = 0
(y - 21) (y - 10) = 0
y - 21 = 0 or y - 10 = 0
y = 21 or y = 10
Hence, the two real numbers are 10 and 21.
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Nerida Kyle is thinking of buying a car to avoid taking Lyft to work. She is using the cost-benefit principle to evaluate this decision and is calculating the costs and benefits to owning the car over the next year. She's gathered the following information to help her make her decision: The car costs $15,000 to purchase, but she can resell it after a year of use for $13,500. She thinks gas will cost her about $1,200 for the year. The annual insurance premium for her car is $800. Maintenance and repairs will cost about $300 for the year. Using Lyft to get to work would cost her $4,000 for the year. The cost of the car for the year is:
Answers
The cost of the car for the year is $3800 for Nerida Kyle and it will cost less than lyft.
The cost of the car for the year will be calculated based on the amount she will spend in total on the car. The formula to be used is -
Total cost = Cost price of the car - Selling price
Cost price of car will be sum of all the expenses, calculated as follows -
Cost price of the car = 15000 + 1200 + 800 + 300
Performing addition on Right and Side of the equation
Cost price of the car = $17,300
Total cost for the year = 17,300 - 13,500
Performing subtraction on Right and Side of the equation
Total cost for the year = $3,800
Hence, Nerida Kyle will spend $3,800 on car which is better than lift.
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There are 8 cats at the pet store. 2/3 of the cats are brown. How many cats are not brown
Answers
Answer:
3 cats are not brown.
Step-by-step explanation:
First, take the 2/3 and divide 2 by 3 on a calculator to get 0.66. Then, set up this equation on the calculator:
0.66 x 8 = 5.28. We will round this to 5 cats.
Now, we take the 5 cats and subtract it from the total of 8 cats:
8 - 5 = 3 cats.
3 cats are not brown.
May I have Brainliest please? My next rank will be the highest one: A GENIUS! Please help me on this journey to become top of the ranks! I only need 12 more brainliest to become a genius! I would really appreciate it, and it would make my day! Thank you so much, and have a wonderful rest of your day!
Given that
x
= 7.9 m and
θ
= 54°, work out AB rounded to 3 SF.
Answers
Answer:
6.39 m
Step-by-step explanation:
( for a right triangle such as this)
Sin (theta) = opp leg / hypotenuse = AB / x
sin (54) = AB / 7.9
7.9 * sin 54 = AB = 6.39 m
which of the following sets of correlations correctly shows the highest to lowest degree of relation?
a. -0.91, +0.83, -0.03, -0.10
b. -0.91, +0.83, +0.10, -0.03
c. +0.83, +0.10, -0.91, -0.03
d. +0.83, +0.10, -0.03, -0.93
Answers
Option B is the correct answer, The set of correlations which correctly shows the highest to lowest degree of relation is -0.91, +0.83, +0.10, -0.03.
What is correlation?
In statistics, the correlation is a technique used to determine the relationship between the two variables. The correlation's range is from -1 to +1. There are three primary categories of correlation: no correlation, positive correlation, and negative correlation. When the correlation value is zero, there is no association; the correlation value that is closest to zero indicates the weakest degree of relationship.
We must select the set of correlations in the question that accurately displays the highest to the lowest degree of relation.
Option B is the best choice since we must take the absolute value into account in order to determine the relationship's degrees. Here, the absolute values drop from 0.91 to 0.03.
Option A is incorrect since the absolute values decline from 0.91 to 0.03 and then climb to 0.10 in this situation.
Because the absolute values are not listed in decreasing order, Option C is incorrect.
Because the absolute values in this situation are not ordered in decreasing order, Option D is also a incorrect choice.
Therefore, option B, is the correct option.
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constant of proportionality of y=5x-1/3
Answers
The equation given is linear, not proportional, and thus, there is no constant of proportionality.
What is the constant of proportionality?
Here we have the following linear equation:
y = 5*x - 1/3
The general linear equation is:
y = m*X + b
Where m is the slope and b is the y-intercept, so in this case we have:
slope = 5
y-intercept = -1/3
Partocularly, when the y-intercept is 0 we have a proportional relation:
y = m*x
And then m is called the "constant of proportionality"
In this case the y-intercept is not zero, thus, there is no constant of proportionality, just a slope.
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Step 1: Subtract 3 from both sides of the inequality.
Step 2: __________
Step 3: Divide both sides of the inequality by the coefficient of x.
What is the missing step in solving the inequality 5 – 8x < 2x + 3?
Add 2x to both sides of the inequality.
Subtract 8x from both sides of the inequality.
Subtract 2x from both sides of the inequality.
Add 8x to both sides of the inequality.
Answers
Answer:
D. Add 8x to both sides of the inequality.
Step-by-step explanation:
1. A polynomial equation of the 5th degree has ___________ 5 roots.
2. A depressed equation is a/an ____________, when equated to zero.
3. A root of multiplicity 5 means that _______.associate the mechanisms of plate boundaries?
Answers
The Fundamental Theorem of Algebra states that a polynomial of fifth degree will have five roots, which is the case with this particular polynomial.
What are the principles governing plate boundaries?The theory that underlies plate boundaries. The primary characteristics of plate tectonics are: The ocean floors are constantly shifting, extending outward from the center, settling outward at the edges, and regenerating. The crustal plates are moved in various directions by convection currents beneath the plates.The Fundamental Theorem of Algebra states that a polynomial of fifth degree will have five roots, which is the case with this particular polynomial.5 roots are required for a polynomial of fifth degree. Since all Complex roots are paired, if 5i is a root, then -5i must also be a root. 1-3i must be a root in order for 1+3i to also be a root.A depressed equation is quadratic when equated to zero.To learn more about Fundamental Theorem of Algebra refer to:
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a single player is dealt 8 cards from a standard 52 deck what is the probability that the 5th card was the htird and last jack dealt
Answers
The probability that the 5th card was the third and last jack is 0.016%
How to calculate the probability of cards?Standard card deck only have 4 jack cards and 48 other cards is not a jack. So, to calculate the probability is simply by dividing the number of favorable outcomes by the possible outcomes.
It is very important to note, wherever the first and second jacks are located it will not affect the calculation for the third jack.
So, you can calculate with this formula,
P(5th card is third and last jack) = (48/52)×(47/51)×(4/50)×(3/49)×(2/48)×(46/47)×(45/46)×(44/45)
or this formula,
P(5th card is third and last jack) = (4/52)×(3/51)×(48/50)×(47/49)×(2/48)×(46/47)×(45/46)×(44/45)
After we simplified calculation, we get
= (4 × 3 × 2 × 44) ÷ (52 × 51 × 50 × 49)
= 0.00016 or 0.016%
Thus, the probability is 0.016% for the third and last jack on 5th card.
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The masts for the Millennium Dome were held up during construction by wire ropes as shown in the diagram. A is 106 m above the ground, C is vertically below A, the gradient of AB is 1 and CD is 53 m. a Find the gradient of AD. b Find the length of BD.
Answers
a)The gradient of AD will be 2.
b)The length of BD. will be 159 m.
What is trigonometry?Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a right-angle triangle.
It is given that, A is 106 m above the ground, C is vertically below A, the gradient of AB is 1 and CD is 53 m.
Suppose the length BC is x
Gradient of AB = 1
AB/BC=1
106/x=1
x=106
BC=106 m
CD = 53 m
The gradient of AD is,
m = BC/CD
m=106/53
m=2
b)
The length of BD is found as,
BD=BC+CD
BD=106+53
BD=159 m
Thus, the gradient of AD will be 2 and the length of BD. will be 159 m.
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A political researcher takes a survey of 300 randomly selected registered voters in atlanta, and each person was asked who they plan on voting for in the 2020 mayoral election. 101 said they plan on voting for candidate a, 184 said they plan on voting for candidate b, and 15 were unsure or plan to vote for another candidate. In question 8, the political researcher estimated p, the population proportion of all registered voters in atlanta who plan to vote for candidate a, with a 95% confidence interval. The researcher now wants to estimate p with a margin of error of 3%. What is the minimum sample size needed?.
Answers
The minimum sample size needed is 93.
What is minimum sample size?In order to obtain survey findings that accurately represent a population you are researching while maintaining your chosen sampling error and confidence, you must have a required sample size.
A political scientist polls 300 random selection Atlanta votes cast, asking each one who those who currently support in the city's mayoral race in 2020. While 184 people indicated they would vote for candidate b, 184 people said they would pick for option a, and 15 people stated they felt undecided or would vote for a different candidate.
The community proportion of all Atlanta electors who intend to vote for candidate a was calculated with a 95% confidence level in query 8 by the political researcher. Furthermore, the researcher wants to determine p with a 3% error margin.
n = 101
p = 0.5
e = 5%=0.05
z score at 95% = 1.96
[tex]z score = (observed value - mean of the sample) / standard deviation of the sample[/tex]
On auditing the value to the given equation:
n = 93
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3. Which statement illustrates the inverse property
of addition?
A. a + 4 = a + (-4)
B. a + (-7 + 7) = a +0
C. a + (a + 2) = a + (-a-2)
D. -a(5+3) = -5a + (-3a)
Answers
Answer:
B
Step-by-step explanation:
B. a + (-7 + 7) = a +0
When you remove a from both sides, you're left with -7+7=0 which shows that the opposites of the numbers add up to 0.
please answer asap (25 points)
only write down what I gotta put in the box and explanation
Answers
Answer:
12
Step-by-step explanation:
Use 5 for m
Use -3 for x
Use 7 for y
y =mx + b
7 = (5)(-3) + b
7 = -5 +b
12 = b
a recent article posted to the journal of nutritional science surveyed dog owners to ask about their dog walking habits in order to find out whether the dogs were getting enough exercise. of course, the necessary amount of exercise varies by breed; results were categorized according to breed. read (or glance through) the article below; the study design and results sections are the most relevant.
Answers
The correct answer will be, An observational study of dog owners logged into the Channel 4 website
What is Sampling?
If the data are distributed normally, a z test can be used to determine whether the means of two populations are different from one another. This calls for the determination of the value of the z test statistic, along with the setups of the null hypothesis and alternative hypothesis. The decision-making standard is based on the z critical value for Sampling . The z critical value used during hypothesis testing divides the distribution graph into acceptance and rejection zones. The test statistic must fall inside the rejection region in order for the null hypothesis to be rejected; otherwise, it cannot.
a recent article posted to the journal of nutritional science surveyed dog owners to ask about their dog walking habits in order to find out whether the dogs were getting enough exercise.
Using a sample and normally distributed data in this case
Hence, we can say An observational study of dog owners logged into the Channel 4 website
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What is the length of RS in the diagram TQ is 18 units in length?.
Answers
The length of RS in the diagram is 25 units.
Given:
TQ = 18 units
From the figure:
TQ = RT
2x + 10 = 18
2x = 18 - 10
2x = 8
divide by 2 on both sides
2x/2 = 8/2
x = 4 units
Both the triangles are congruent by right angled hypotenuse side criteria.
so RS = QS
= 9x - 11
= 9*4 - 11
= 25 units
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Full question:
What is the length of RS in the diagram TQ is 18 units in length?
Diagram is in the image uploaded.
Write in the space provided the property or properties that each relation possesses (reflexive, symmetric, transitive). For example, write r,s if the relation is reflexive and symmetric.1.) R is a relation on the set of all people. (a,b) is in R if a is taller than b.2.) R is a relation on the set of integers > 0. (a,b) is in R if a divides b. (eg. 3 divides 3 and 3 divides 15 etc.)3.) R is a relation on the set of all people. (a,b) is in R if a is married to b.4.) R is a relation on the set of all nonnegative integers. (a,b) is in R if a and b have the same remainder when divided by 5. (eg. (7,12) is in R)
Answers
The relation in option (a) is Transitive , in option (b) is Reflexive and Transitive and option (c) is Symmetric .
In the question ,
it is given that ,
Option(a) ,
R is a relation on set of all people where (a,b) is in R if a is taller than b .
Reflexive : NO , because "a" cannot be taller than "a" itself .
Symmetric : NO , because if "a" is taller than "b" , then "b" will be shorter than "a" .
Transitive : YES , because if "a" is taller than "b" and "b" is taller than "c" , then "a" is taller than "c" .
Option(b)
R is a relation on set of integers > 0 where (a,b) is in R if a divides b .
Reflexive : YES , because every number divides itself .
Symmetric : NO , because if "a" divided "b" , then "b" may not divide "a" .
Transitive : YES , because if "a" divided "b" and "b" divides "c" , then "a" must divide "c" .
Option(c)
R is a relation on set of all people where (a,b) is in R if a is married to b .
Reflexive : NO , because marriage is between two people , and "a" cannot marry "a" .
Symmetric : YES , because if "a" is married to "b" , then "b" is married to "a" .
Transitive : NO , because if "a" is married to "b" and "b" is married to "c" , then "a" is not married to "c" .
Therefore , The relation in option (a) is Transitive , in option (b) is Reflexive and Transitive and option (c) is Symmetric .
The given question is incomplete , the complete question is
Write in the space provided the property or properties that each relation possesses (reflexive, symmetric, transitive).
(a) R is a relation on the set of all people, (a,b) is in R if a is taller than b .
(b) R is a relation on the set of integers > 0 , (a,b) is in R if a divides b .
(c) R is a relation on the set of all people , (a,b) is in R if a is married to b .
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. what is the probability that a number selected at random from the set {2, 3, 7, 12, 15, 22, 72, 108} will be divisible by both 2 and 3 ? individual question
Answers
The probability that a number selected at random from the set {2, 3, 7, 12, 15, 22, 72, 108} will be divisible by both 2 and 3 is 3/8.
Given set:
random from the set {2, 3, 7, 12, 15, 22, 72, 108}
From the set:
The number that are divisible by both 2 and 3 is:
12,72,108.
So number of favorable outcomes is = 3
Number of possible outcomes = 8
we know that
probability = favorable outcomes / possible outcomes
= 3/8
= 37.5%
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Which expressions are equivalent to 78 • 7
Answers
Question:
Which expressions are equivalent to 78 • 7? Select all that apply. A. 73 • 73 B. 71879 C. (73)3 D. 74 + 75 E. 74 • 75
None of the expressions are equivalent to 78 • 7
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a complete sentence. Any one of the following mathematical operations can be used. A sentence has the following structure: Number/variable, Math Operator, Number/Variable is an expression.
None because 78 * 7 = 546
None of the options equal to 546.
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can someone explain how to do this with the answer pls
Answers
Answer:
x = 23
Step-by-step explanation:
The sum of two interior angles is equal to an exterior angle that is supplementary to the third interior angle.
We can write the following equation based on this information to find the value of x:
2x + 3 + 51 = 100 add like terms
2x + 54 = 100 subtract 54 from both sides
2x = 46 divide both sides by 2
x = 23
Change the Cartesian integral
Integral from nothing to nothing Subscript 0 Superscript 3 Baseline Integral from nothing to nothing Subscript 0 Superscript y Baseline x dx dy∫30∫y0x dx dy
into an equivalent polar integral. Then evaluate the polar integral.Change the Cartesian integral SSX x dx dy into an equivalent polar integral. Then evaluate the polar integral. Choose the correct equivalent polar integral below. O A. sin e 2 cose dr de OB. sin So r2 cose dr de 1/4 Sza So ? So O c. 3/ sin e OD. c/2 3/ sine so So ? ? cos e dr de ? cose dr de The value of the double integral is
Answers
The value of the double integral is [tex]I = \int\limits^{\pi/2}_{\pi/4} \int\limits^{3/sin0}_{r=0} r^{2}cos0drd0[/tex].
Let z* be the common (finite) optimal value of P and D.
Given integral is
[tex]I = \int\limits^3_{y = 0} \int\limits^y_{x=0} {x} \, dxdy[/tex]
The region of integration lies between
y = 0, y = 3, x = 0 & x = y
Let x = rcosθ, y = rsinθ,
[tex]x^{2} +y^{2}= r^{2}[/tex]
At y = 3, rsinθ = 3
Thus r varies from 0 to 3/sinθ
θ varies from π/4 to π/2
Now, dxdy ----> rdrdθ
Therefore,
[tex]I = \int\limits^{\pi/2}_{\pi/4} \int\limits^{3/sin0}_{r=0} r^{2}cos0drd0[/tex]
Hence the answer is The value of the double integral is [tex]I = \int\limits^{\pi/2}_{\pi/4} \int\limits^{3/sin0}_{r=0} r^{2}cos0drd0[/tex].
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a reflection in the x-axis followed by a reflection in the y-axis leaves a point in its original location. Is it always true, sometimes true or never true? Explain clearly.
Answers
It's always true that a reflection on the x-axis followed by a reflection on the y-axis leaves a point in its original location.
What is a coordinate axis?
A coordinate system in geometry is a system that uses one or more numbers, or coordinates, to determine the position of points or other geometric elements on a manifold such as Euclidean space.
When a reflection in the x-axis followed by a reflection in the y-axis leaves a point in its original location. Is it always true?
The answer is true.
Reason: For example, piece of graph paper into 4 pieces, draw this same thing on all four:
In this condition reflection in the x-axis followed by a reflection in the y-axis leaves a point in its original location of different color is yes.
Hence, it's always true that a reflection on the x-axis followed by a reflection on the y-axis leaves a point in its original location.
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What does 5 less than N mean?.
Answers
5 less than N means we subtract 5 to the given variable N. So the expression will be N - 5.
What is Algebra ?Algebra shows the numbers by alphabets in a equation including mathematical operations. In algebra we used alphabet which shows a variable unknown quantity, unknown variables which helps in changing a mathematical statement into an equation.
In the given question a mathematical statement is given and we have to change the mathematical equation into an algebraic expression by using the information.
What does 5 less than N means ?The statement 5 less than N means 5 is being subtracted from N. The mathematical expression for 5 less than N is equal to N - 5.
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Tracy buys 150 shares of upton group and 100 shares of whi health. what is tracy’s total investment? a. $4,940.00 b. $5,040.00 c. $5,914.50 d. $5,988.00
Answers
The correct option representing Tracy's total investment is a. $4,940.00.
She bought 150 shared of Upton group that offers each share at the price of $18.96. So, the cost of 150 shares = 150 × 18.96
Performing multiplication on Right and Side of the equation
The cost of 150 shares from Upton group = $2844
The cost of 100 shares of WHI health = 100 × 20.96
Performing multiplication on Right and Side of the equation
The cost of 100 shares of WHI Health = $2096
The total investment will be sum of the cost of these shares.
Total investment = 2844 + 2096
Performing addition on Right Hand Side
Total investment = $4940
Hence, the total investment of Tracy is $4940.
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The complete question is -
Use the information in the table below to answer the following question.
Name of Fund
NAV
Offer Price
Upton Group
$18.47
$18.96
Green Energy
$17.29
$18.01
TJH Small-Cap
$18.43
$19.05
WHI Health
$20.96
NL
Tracy buys 150 shares of Upton Group and 100 shares of WHI Health. What is Tracy's total investment?
a.
$4,940.00
b.
$5,040.00
c.
$5,914.50
d.
$5,988.00
You have a bag of 17 marbles. Four are blue, 6 are green, 2 are red, and the others are yellow. What is the probability of drawing a green marble, putting it aside, and then drawing another green marble?.
Answers
the probability of drawing a green marble, putting it aside, and then drawing another green marble is 3/68
Total 17 marbles
4 - blue marbles
6 - green marbles
2 - red marbles
The rest - yellow marbles
Probability of drawing red first:
P(red) = number of red / total number = 2/17
Probability of drawing green second, without replacement:
P(green) = number of green / number left in the bag = 6/16 = 3/8
Probability of red and green in order:
P = P(red)*P(green) = 2/17*3/8 = 3/68
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he length of the hypotenuse of a right triangle is hh , and the radius of the inscribed circle is rr. the ratio of the area of the circle to the area of the triangle is
Solution by e_power_pi_times_i
Since rs = A, where r is the inradius, s is the semiperimeter, and A is the area, we have that the ratio of the area of the circle to the area of the triangle is $\frac{\pi r^2}{rs} = \frac{\pi r}{s}$. Now we try to express s as h and r. Denote the points where the incircle meets the triangle as X,Y,Z, where O is the incenter, and denote AX = AY = z, BX = BZ = y, CY = CZ = x. Since XOZB is a square (tangents are perpendicular to radius), r = BX = BZ = y. The perimeter can be expressed as 2(x+y+z), so the semiperimeter is x+y+z. The hypotenuse is AY+CY = z+x. Thus we have s = x+y+z = (z+x)+y = h+r. The answer is (B) 3,14r/h+r
Answers
The ratio of the area of the circle to the area of triangle is πr : h+r.
We know that rs = A
where r is the radius , s is the semi perimeter and A is the area.
We have that the ratio of the area of the circle to the area of the triangle is
πr²/rs = πr/s.
Now we will express s as h and r. Denote the points where the in-circle meets the triangle as X,Y,Z where O is the in-center and denote
AX = AY = z, BX = BZ = y, CY = CZ = x.
Since XOZB is a square
Therefore r = BX = BZ = y.
The perimeter can be expressed as 2(x + y + x), so the semi perimeter is
x + y + z.
The hypotenuse is AY + CY = z + x.
Thus we have s = x + y + z
= (z + x) + y
s = h + r
Therefore the ratio of the area of circle to the area of the triangle will become πr : h+r.
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