The infinite product measure [tex]$\mu$\\[/tex] is uniquely defined on the product measure space[tex]$(\Omega, \mathcal{A}, \mu)$[/tex] by Dynkin's theorem, since [tex]$\mu$\\[/tex] is finitely additive, countably subadditive, and [tex]$\sigma$[/tex]-finite.
Let [tex]$\mathcal{A}_1, \mathcal{A}_2, \ldots$[/tex] be a sequence of sigma algebras defined on [tex]$\Omega_1, \Omega_2, \ldots$[/tex] respectively. Let [tex]$(\Omega, \mathcal{A}, \mu)$[/tex] be the product measure space, where [tex]\Omega = \prod_{i=1}^{\infty} \Omega_i$ and $\mathcal{A} = \bigotimes_{i=1}^{\infty} \mathcal{A}_i$[/tex].
We want to prove that the infinite product measure [tex]$\mu$[/tex] is uniquely defined. To do this, we will use Dynkin's theorem.
According to Dynkin's theorem, a measure [tex]$\mu$[/tex] is uniquely determined on a product measure space [tex]$(\Omega, \mathcal{A}, \mu)$[/tex] if the following conditions are satisfied:
1. [tex]$\mu$[/tex] is finitely additive on [tex]$\mathcal{A}$[/tex]
2. [tex]$\mu$[/tex] is countably subadditive on [tex]$\mathcal{A}$[/tex]
3. [tex]$\mu$[/tex] is [tex]$\sigma$[/tex]-finite on [tex]$\mathcal{A}$[/tex]
It is easy to see that the infinite product measure [tex]$\mu$[/tex]satisfies these conditions, since it is defined as a product of finitely additive, countably subadditive, and [tex]$\sigma$[/tex]-finite measures. Thus, by Dynkin's theorem,[tex]$\mu$[/tex] is uniquely determined on the product measure space [tex](\Omega, \mathcal{A}, \mu)$.[/tex]
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An experiment subject only to random fluctuation within a normal distribution collects the following set of unitless numerical values: 15.8, 17.8, 19.5, 20.0, 14.7, 16.6, 16.8, 19.4, 18.3
A number of hypotheses are put forward for the theoretical value of what is measured here. Select all of the theoretical values given below that can be ruled out by our usual 1-sigma uncertainty criterion. A. 15.6 B. 15.9 C. 19.5 D. 19.7 E. None of these theoretical values is confirmed by this data.
A one-sigma uncertainty criterion means that if a theoretical value falls within one standard deviation of the measured data, it is considered consistent with the data.
Using this criterion, we can rule out theoretical value D. 19.7, since it is outside the range of measured values.
The other theoretical values A. 15.6, B. 15.9, and C. 19.5 fall within one standard deviation of the measured data, so they cannot be ruled out by this criterion.
Therefore, the only theoretical value that can be ruled out by our usual 1-sigma uncertainty criterion is D. 19.7
It's worth noting that this method just gives a rough idea of the consistency of a theoretical value with the data.
Other methods, such as the chi-squared test, or Bayesian analysis could be used to find the best-fit theoretical value and its uncertainty, and confirm or rule out other theoretical values.
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Write the data in order from least to greatest and circle the median.
Of 3.50 7.99 3.25 4.00 3.50 7.50 3.65 3.50 4.00
Answer:
The answer to your questionis 3.65
Answer:
3.25, 3.50, 3.50, 3.50, 3.65, 4.00, 4.00, 7.50, 7.99
3.65 is the median.
Step-by-step explanation:
You're welcome.
A friend has a 81% average before the final exam for a course. That score includes everything but the final, which counts for 25%
The best grade that can be earned is given as 0.8575
How to solve for the gradeIf he is able to get the sum of 100 percent in the final exam, trhis would count as 25 percent of the total grade that he has
100 - 25 = 75%
Then we would have
0.75 * 81 + 25 / 100
= 85.75 / 100
= 0.8575
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Question
A friend has a 81% average before the final exam for a course. That score includes everything but the final, which counts for 15% of the course grade.
What is the best course grade your friend can earn?
are births equally likely across the days of the week? a random sample of 150 births give the following sample distribution:
Expected values are = 150/7 .Ci-square distribution is Day of the week observed count Expected Chi-square = O E (O -E )2/E
testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis. Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Statistical hypothesis: A statement about the nature of a population. It is often stated in terms of a population parameter.
Null hypothesis: A statistical hypothesis that is to be tested. : The alternative to the null hypothesis.
Test statistic: A function of the sample data.
The critical value for the chi-square statistic is determined by the level of significance (typically . 05) and the degrees of freedom. The degrees of freedom for the chi-square are calculated using the following formula: df = (r-1)(c-1) where r is the number of rows and c is the number of columns.
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Expected values are = 150/7 .Ci-square distribution is Day of the week observed count Expected Chi-square = O E (O -E )2/E
testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis. Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Statistical hypothesis: A statement about the nature of a population. It is often stated in terms of a population parameter.
Null hypothesis: A statistical hypothesis that is to be tested. : The alternative to the null hypothesis.
Test statistic: A function of the sample data.
The critical value for the chi-square statistic is determined by the level of significance (typically . 05) and the degrees of freedom. The degrees of freedom for the chi-square are calculated using the following formula: df = (r-1)(c-1) where r is the number of rows and c is the number of columns.
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Let ABCD be a parallelogram. Let M be the midpoint of line AB and N be the midpoint of line AD. Diagonal BD intersects line CM and line CN at P and Q, respectively. Find PQ/BD.
The value of PQ / BD of given parallelogram where Diagonal BD intersects line CM and line CN at P and Q, respectively = 1/3
Let's consider the two triangles BMP and DCP.
Here,
∠DPC = ∠BPM (because they are vertically opposite angles)
∠DCP = ∠BMP (because they are alternate interior angles of transversal CM with DC // BM)
∴△BMP∼△DCP (By AA theorem)
Thus,
ABCD is a parallelogram. Therefore AB = DC and AD = BC.
It is given that M is the midpoint of AB
BD = DQ + BQ BD = BP + PD
BD = 2BQ + BQ BD = 2PD + PD
BD = 3BQ BD = 3PD
therefore
BQ = PD And
Since DQ = 2BQ then DQ = 2PD and
PD + PQ = 2PD subtract PD from each side
PQ = PD
so BQ = PD = PQ
BD = PD + PQ + BQ
BD = PQ + PQ + PQ
BD = 3PQ
Therefore the value of PQ / BD of given parallelogram is PQ / BD = 1/3.
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Homework 3x-5; x-5; an al 2x-14;.:. arithmetic sequence. Calculate the Value of x Find the Which term the First three are in Determine the general term of the Sequence Soth term the sequence is equal to -62 terms of
The value of x is found to be 3 and the arithmetic sequence is found to be 4, -2, -8, ... .
What is arithmetic sequence?
The nth term of an arithmetic progression is determined using the arithmetic sequence formula. The series where the common difference between any two next words stays constant is known as an arithmetic sequence.
The arithmetic sequence is 3x - 5, x - 5, 2x - 14, ....
If the numbers are in an arithmetic sequence, then the difference between the first and second term must be the same as the difference between the second and third term.
(x - 5) - (3x - 5) = (2x - 14) - (x - 5)
x - 5 - 3x + 5 = 2x - 14 - x + 5
-2x = x - 9
-2x - x = -9
-3x = -9
x = 3
The first three terms can be obtained by substituting the value of x in the terms -
The first term - 3(3) - 5 = 4
The second term - 3 - 5 = -2
The third term = 2(3) - 14 = -8
Therefore, the arithmetic sequence is 4, -2, -8, .....
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root24 solve the initial value problem with . to solve this, we should use the substitution help (formulas) help (formulas) enter derivatives using prime notation (e.g., you would enter for ). after the substitution from the previous part, we obtain the following linear differential equation in . help (equations) the solution to the original initial value problem is described by the following equation in . help (equations)
The solution equation gives us y = 5x/2 - 2, so when x = 0, we get y = -2, which matches the initial condition.
The solution to the initial value problem can be obtained by using the substitution method. We first substitute the given function into the differential equation, which gives us the linear differential equation in y:
y' + 2y = 5x
We can then solve this linear differential equation by integrating both sides and applying the initial condition. This gives us the solution in y:
y = 5x/2 + C
where C is an arbitrary constant. To calculate the value of C, we plug in the initial condition y(0) = 2 into the equation. This gives us C = -2, so the solution to the initial value problem is given by:
y = 5x/2 - 2.
To illustrate this, let's consider the initial condition y(0) = 2. Plugging this into the solution equation gives us y = 5x/2 - 2, so when x = 0, we get y = -2, which matches the initial condition.
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A construction company charges $15 per for hour for debris removal, plus a one time fee for the use of a trash dumpster. The total fee for 9 hours of service is $195. Write an equation of a line, in slope intercept form, to find the total fee y for any number of hours x
An equation of a line, in slope-intercept form, for the total fee y for any number of hours x is y = 15x + 60.
What is the slope-intercept form?In Mathematics, the slope-intercept form of a line can be calculated by using this linear equation:
y = mx + c
Where:
x represents the number of hours.y represents the total fee.m represents the slope of a line.c represents the y-intercept of a line.Since the total fee that is being charged by this construction company is denoted by y and it charges $15 per hour for debris removal, plus a one time fee, an equation that models this situation is given by;
y = 15x + c
Next, we would determine the y-intercept as follows;
195 = 15(9) + c
y-intercept, c = 195 − 135
y-intercept, c = 60.
Therefore, the required equation is y = 15x + 60.
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Alyson deposits $500 in the bank for 12 years. The bank offers her a 4%
simple interest rate. How much money will be in her account at the end of
the 12 years? (Remember to round your answer to the nearest cent if
necessary.)
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$500\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &12 \end{cases} \\\\\\ A = 500[1+(0.04)(12)] \implies A = 740[/tex]
Write an algebraic equation for the following.
Three times the sum of a number and seven is two less than five times the number.
Answer:
We dissect each statement from the problem above to fully understand it.
the sum of a number and 7 is (x + 7)
3 times this is 3(x + 7)
is means the equal sign here 3(x + 7) =
Two less then 5 times the number is 5x - 2.
Therefore, the answer is B, 3(x + 7) = 5x - 2
(50 Points and BRAINLYEST)
The prism has a square base. That base is inscribed in the circular base of the cylinder. Draw an image of the base of the composite shape.
Answer:
Step-by-step explanation:
At a given point 67.7ft at ground level of a covered court, the angle of elevation of a roof worker from the bottom is 38.6⁰ and the angle of elevation of the roof from the top is 42.4 degrees find the height of the roof worker and the distance of the ø if someone is observing below.
The height is 114.83 ft and the distance of the ø if someone is observing below is 84.8.
What is trigonometry?The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.
The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.
Given that at a given point of 67.7ft at the ground level of a covered court, the angle of elevation of a roof worker from the bottom is 38.6⁰ and the angle of elevation of the roof from the top is 42.4 degrees.
The base will be calculated as:-
tan(38.6) = P / B
tan(38.6) = 67.7 / B
B = 67.7 / tan(38.6)
B = 54.8 ft
The height will be calculated as:-
Cos(42.4) = B / H
H = 84.8 / Cos(42.4)
H = 114.83 ft
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8. Find a when 120 ÷ a = 12.
a =
Answer:
10
Step-by-step explanation:
120/a = 12
120=12a
120/12 = a
a = 10
Answer:
1,440
Step-by-step explanation:
120 x 12 = 1,440
120 / 1,440 = 12
if we take a sample of reee's pieces has 1000 pieces. the manufacturer says that exactyl 40% of the candies are organe. if we select a sample of 50 pieces, how many will be oragne. let x
If we take a sample of 50 Reece's Pieces candies, we would expect about 20 of them to be orange.
The manufacturer says that exactly 40% of the candies are orange. We can use this information to find the expected value of the number of orange candies in a sample of 50 candies.
Expected value is found by multiplying the number of trials (in this case, 50) by the probability of success (in this case, 40% or 0.40)
x = (50)(0.40) = 20 candies.
So, if we take a sample of 50 Reece's Pieces candies, we would expect about 20 of them to be orange.
It's worth noting that this is an expectation, the actual number of orange candies in the sample of 50 candies may be different from the expected value.
The correct question of the following problem is:
If a sample of 1000 Reese's Pieces contains 40% orange candies, and a random sample of 50 Reese's Pieces is selected, how many orange candies can be expected to be in the sample of 50 Reese's Pieces? Let x be the number of orange candies in the sample of 50 Reese's Pieces.
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a manufacturer buys a new machine costing $180000 dollars. it is estimated that the machine has a useful lifetime of ten years and a salvage value of 6000 dollars at that time. assume the value of the machine varies linearly during this time. do not include dollar signs in your answers.
A manufacturer buys a new machine costing $180000 dollars. The value of the machine at the end of 10 years is $6000.
Calculate the annual depreciation of the machine.
Depreciation = (Purchase Price - Salvage Value) / Useful Life
Depreciation = (180000 - 6000) / 10
Depreciation = 174000 / 10
Depreciation = $17400 per year
Calculate the value of the machine at the end of each year.
Year 1: 180000 - 17400 = $162600
Year 2: 162600 - 17400 = $1452000
Year 3: 1452000 - 17400 = $1277600
Year 4: 1277600 - 17400 = $1103200
Year 5: 1103200 - 17400 = $928800
Year 6: 928800 - 17400 = $754400
Year 7: 754400 - 17400 = $580000
Year 8: 580000 - 17400 = $405600
Year 9: 405600 - 17400 = $231200
Year 10: 231200 - 17400 = $6000
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If a laborer weighing 895N and carrying bricks weighing 575N to scaffold up a ladder 3.1 m above the ground how much work has he done?
The amount of work that the laborer has done, given the weight of the bricks and the laborer is 4, 577 Joules .
How to find the work done ?To calculate the amount of work done by the laborer, we can use the formula:
work = force x distance x
where:
force is the force applied,
distance is the distance moved,
The force applied by the laborer is the combined weight of the laborer and the bricks:
= 895N + 575N
= 1, 470N
Therefore, the work done by the laborer is:
Work = 1, 470N x 3.1m
= 4577 Nm or Joules (J)
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Use the formula F =9 over 5 C + 32 to answer the following
Maha checked the outside temperature at noon one day and found it was 26° C. At 10 PM when she checked, the outside temperature was 18° C. How much, in degrees Fahrenheit, did the temperature drop between noon and 10 PM? Round your answer to the nearest tenth.
The temperature drop between noon and 10 PM is 14.4° F.
What is the conversion of Celsius to Fahrenheit?Celsius to Fahrenheit is the conversion of temperature from Celsius unit to Fahrenheit unit. Temperature scales provide a way of measuring how hot or cold a body is. Fahrenheit and Celsius are temperature scales.
The formula to convert Celsius to Fahrenheit is given by: °F = °C × (9/5) + 32.
Given that, Maha checked the outside temperature at noon one day and found it was 26° C.
So, the temperature in Fahrenheit is °F =26° ×(9/5)+32
= 46.8+32
= 78.8° F
At 10 PM when she checked, the outside temperature was 18° C.
So, the temperature in Fahrenheit is °F =18° ×(9/5)+32
= 32.4+32
= 64.4° F
Difference in temperature =78.8-64.4
= 14.4° F
Therefore, the temperature drop between noon and 10 PM is 14.4° F.
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I need to calculate the percentages of how much in increased/decreased
The percent change for each year is given as follows:
1845: 0%.1846: 12.30%.1847: -4.38%.1848: 10.69%.How to obtain the percent change?The percent change is obtained applying the proportions in the context of this problem.
The equation that defines the percent change is the change in the year divided by the initial population during the year.
Hence the percent change for each year during the period is given as follows:
1846: 15/122 = 12.30%.1847: -6/137 = -4.38%.1848: 14/131 = 10.69%.More can be learned about proportions at https://brainly.com/question/24372153
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Angle
(Choice B)
B
Line
(Choice C)
C
Line segment
(Choice D)
D
Point
(Choice E)
E
Ray
(Choice F)
F
None of theseAngle
(Choice B)
B
Line
(Choice C)
C
Line segment
(Choice D)
D
Point
(Choice E)
E
Ray
(Choice F)
F
None of these
The correct answer is, the figure is a line segment.
What is a straight line?A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.
here, we have,
from the given figure,
we get,
AB is a line segment.
Hence, The correct answer is, the figure is a line segment.
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In a company, 85% of the workers are women. If 315 people work for the company who aren't women, how many workers are there in all? Use pencil and paper. Show two different ways that you can solve this problem.
There are 2100 workers in the company
How many workers are there in all?
Word problems are sentences describing a 'real-life' situation where a problem needs to be solved by way of a mathematical calculation.
Since 85% of the workers are women, we can say:
The percentage of worker that aren't women = 100% - 85% = 15%
Let N represent the number of all workers. If 315 people work for the company who aren't women. We can say:
15% of N = 315
15/100 × N = 315
0.15N = 315
N = 315/0.15
N = 2100 workers
Method 2:
We can write:
N - 0.85N = 315
0.15N = 315
N = 315/0.15
N = 2100 workers
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Find the average rate of change of the function on the interval specified for real number h.
p(x) = 2x + 5 on [2, 2 + h]
The average rate of change on the interval [2, 2 + h] is r = 2.
How to find the average rate of change?Wewant to find the average rate of change of p(x) = 2x + 5 on the interval [2, 2 + h]
Remember that the average rate of change on an interval [a, b] is:
r = ( p(b) - p(a))/(b - a)
So on this case, we have:
r = ( p(h + 2) - p(2))/(2 + h - 2)
r = ( 2*(x + h) + 5 - 2*x - 5)/h
r = (2h)/h
r = 2
The average rate of change is 2.
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Factor out the greatest common factor from the given polynomial 28X -2
The factorised for. of the given Polynomial expression using the greatest common factor is; 2 (14x - 1).
What is the factorised form of the given Polynomial expression?It follows from the task content that the factorised form of the given Polynomial expression is to be determined using the greatest common factor.
Since the given expression is; 28x - 2.
By careful observation of the two terms (28x and -2) in the polynomial expression; the greatest common factor is; 2.
Hence, the factorised form of the polynomial expression as required is; 2 (14x - 1).
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Figure 1 and figure 2 are right triangles.
two right triangles labeled figure 1 and figure 2, where the legs of figure 1 are 3 units long and the legs of figure 2 are 4 units long
What scale factor was used to produce Figure 2 from Figure 1?
A. 3
B. 4
C. 3 over 4
D. 4 over 3
Answer:
4
Step-by-step explanation:
because its the factor
The scale factor of the dilation of triangles is k = 4/3
What is Dilation?Resizing an item uses a transition called Dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. Dilation transformations ensure that the shape will stay the same and that corresponding angles will be congruent
The result of dilation is that the shapes and boundaries of objects in the input image are expanded or thickened. It is often used in conjunction with other morphological operations such as erosion, opening, and closing to manipulate and enhance images.
Given data ,
Let the scale factor of the dilation be represented as k
Now , the measure of leg of the first triangle is a = 3 units
The measure of leg of the second triangle is b = 4 units
So , the dilated triangle is given by
3 k = 4
Divide by 3 on both sides , we get
k = 4/3
Hence , the dilation factor is k = 4/3
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Barbara ran
after lunch.
3
of a mile before lunch and of a mile
8
12
How far did she run in all?
Answer:
Step-by-step explanation:
1/2 is your answer
Question 2
An ecosystem has algae as the main producer.
What most likely is the location of this ecosystem?
airair
desertdesert
mountainsmountains
seasea
Question 2
Facts:
Meerkats are small carnivores. They live in Africa. Meerkats live in extended families called gangs.
Situation:
A gang of eight meerkats lives in a desert.
Based on the given facts, what ecological level describes this gang?
Responses
biomebiome
communitycommunity
habitathabitat
population
Question 3
Kaylee is studying abiotic factors in a sea ecosystem.
What factors could she include in her study?
Select all that apply.
Responses
pressurepressure
temperaturetemperature
number of fishnumber of fish
growth of algae
Question 4,
What is a common limiting factor for populations living in desert biomes?
Responses
lack of top predatorslack of top predators
lack of waterlack of water
presence of sunlightpresence of sunlight
lack of soil nutrients
question 5.
What is carrying capacity?
Responses
the highest population of a species an environment can support over a long periodthe highest population of a species an environment can support over a long period
the highest biodiversity an environment can support for a short whilethe highest biodiversity an environment can support for a short while
the highest population of a species an environment can support for a short whilethe highest population of a species an environment can support for a short while
the highest biodiversity an environment can support over a long periodthe highest biodiversity an environment can support over a long period
a) The ecosystem is in the sea. Option D
b) The ecological level is population. option D
c) The factors in the study are pressure, and temperature
d) The common limiting factor in biomes is lack of water
e) The carrying capacity is the highest population an environment can support over a long period. Option A
What is the ecosystem?We have to note that the ecosystem has to do with the part of the habitat that is composed of the living and the non living parts. The non living parts of the ecosystem are called the abiotic factors while the living parts of the ecosystem are called the biotic factors.
In the ecosystem, we have the producers, the consumers and the decomposers and these are called the trophic or the feeding levels in the system.
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select the two values of x that are roots of this equation x^2 + 3x - 6=0
x = -2 and x = -3 are the two values of x that are roots of the equation x^2 + 3x - 6 = 0
What is the root of the equation?
A root of an equation is a value of the variable that makes the equation true. An equation is a statement that asserts the equality of two expressions. A root of an equation is also known as a solution or a zero of the equation.
The two values of x that are roots of the equation x^2 + 3x - 6 = 0 are x = -2 and x = -3.
To find the roots of the equation, we can use the factoring method, which consists of factoring the polynomial on the left side of the equation.
x^2 + 3x - 6 = 0
(x-2)(x+3) = 0
So, x - 2 = 0, x = 2 and x + 3 = 0, x = -3
Alternatively, we can use the Quadratic formula, which states that for any quadratic equation in the form ax^2 + bx + c = 0, the roots are given by:
x = (-b ± √(b^2 - 4ac))/ 2a
So, by plugging in the values of a, b and c we have:
x = (-3 ± √(3^2 - 41-6))/ 2*1 = (-3 ± √(9+24))/2 = (-3 ± √33)/2 = (-3 ± √33)/2 = -2, -3
Hence, x = -2 and x = -3 are the two values of x that are roots of the equation x^2 + 3x - 6 = 0
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in 2015, the national highway traffic safety administration reported the number of pedestrian fatalities in san francisco county was 24 and that the number in los angeles county was 209. can we conclude that pedestrians are safer in san francisco than in los angeles? why or why not? if you answered no, what additional data would allow us to make a conclusion about which county is safer for pedestrians?
No, we cannot conclude that pedestrians are safer in San Francisco than in Los Angeles based on the data provided. The number of pedestrian fatalities in a given county is only one factor in determining the safety of pedestrians in that county.
Other factors, such as the number of pedestrians in each county, the number of vehicles in each county, the number of traffic laws in each county, and the number of traffic enforcement officers in each county, all play a role in determining the safety of pedestrians in a given county.
We would also need to consider other factors such as the number of pedestrian-friendly infrastructure in each county, the number of pedestrian-friendly policies in each county, and the number of pedestrian-friendly initiatives in each county. With this additional data, we would be able to make a more informed conclusion about which county is safer for pedestrians.
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You are standing 143 feet away from the base of a building and your clinometer measures 17° when it’s looking at the top of the building. (This angle is the one between the ground and the top of the building). Please calculate the height of the building.
Answer:
The height of the building is 43.7 feet.
Step-by-step explanation:
Attached is a sketch of the problem.
We can use SOH CAH TOA to find our answer.
In this acronym, O is the opposite side, A is the adjacent side, and H is the hypotenuse. S is for the SIN function. C is for the COS function. T is for the TAN function.
We can calculate the height of the building by using the TAN function.
So we can say the tangent of angle x is the opposite side divided by the adjacent side.
[tex]tan(x)=\frac{O}{A}[/tex]
Lets solve for [tex]O[/tex].
Multiply both sides by [tex]A[/tex].
[tex]tan(x)*A=\frac{O}{A}*A[/tex]
Simplify the right side by cancelling the common factor of [tex]A[/tex].
[tex]tan(x)*A=O\\O=A*tan(x)[/tex]
Now lets evaluate the height of the building.
In this example we are given
[tex]x=17\\A=143[/tex]
[tex]O=143*tan(17)[/tex]
[tex]O=43.7195[/tex]
a population numbers 11,000 organisms initially and grows by 16.3% each year. suppose represents population, and the number of years of growth. an exponential model for the population can be written in the form where
The exponential model for the population can be written as P(t) = 11,000(1.163)t, where P(t) represents the population size in t years and 1.163 is the growth rate.
We can calculate the population size at any time t by multiplying the initial population size (11,000) by the growth rate (1.163) raised to the power of t. For example, after 10 years the population size would be P(10) = 11,000(1.163)10 = 22,906.7. We can also use the model to predict the population size in future years. For example, the population size in 20 years will be P(20) = 11,000(1.163)20 = 40,803.7. For example, if the population is measured after 10 years, the equation would be P(10) = 11,000(1.163)^10. Solving for P(10) yields a population of 21,945 organisms. Thus, the population has grown by 16.3% per year over the 10-year period and has increased by 10,945 organisms.
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Your grandparents started putting money away for you when you were born. If they put $100
each month into an account earning 3% compounded monthly, how much will be in the
account when you turn 18?
now, the assumption being that your granpa's are depositing the money at the beginning of the month.
[tex]~~~~~~~~~~~~\stackrel{\textit{deposits at the beginning of the period}}{\textit{Future Value of an annuity due}} \\\\ A=pmt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right]\left(1+\frac{r}{n}\right)[/tex]
[tex]\begin{cases} A=\textit{accumulated amount} \\ pmt=\textit{periodic deposits}\dotfill & 100\\ r=rate\to 3\%\to \frac{3}{100}\dotfill &0.03\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &18 \end{cases}[/tex]
[tex]A=100\left[ \cfrac{\left( 1+\frac{0.03}{12} \right)^{12 \cdot 18}-1}{\frac{0.03}{12}} \right]\left(1+\frac{0.03}{12}\right) \\\\\\ A=100\left[ \cfrac{(1.0025)^{216}-1}{0.0025} \right](1.0025) \implies \boxed{A \approx 28665.52}[/tex]