c. The mean of the random variable X is calculated as:
mean(X) = (-5)(1/4) + (0)(1/2) + (5)(1/4) = 0
The variance of X is calculated as:
var(X) = (-5 - 0)^2(1/4) + (0 - 0)^2(1/2) + (5 - 0)^2(1/4) = 25/2
d. The mean of the random variable X is calculated as:
mean(X) = (-5)(.01) + (0)(.98) + (5)(.01) = 0
The variance of X is calculated as:
var(X) = (-5 - 0)^2(.01) + (0 - 0)^2(.98) + (5 - 0)^2(.01) = 50.25
e. The mean of the random variable X is calculated as:
mean(X) = (-50)(.0001) + (0)(.9998) + (50)(.0001) = 0
The variance of X is calculated as:
var(X) = (-50 - 0)^2(.0001) + (0 - 0)^2(.9998) + (50 - 0)^2(.0001) = 500
g. The mean of the random variable X is calculated as:
mean(X) = (0)(1/2) + (2)(1/2) = 1
The variance of X is calculated as:
var(X) = (0 - 1)^2(1/2) + (2 - 1)^2(1/2) = 1
h. The mean of the random variable X is calculated as:
mean(X) = (.01)(.01) + (1.01)(.99) = 1
The variance of X is calculated as:
var(X) = (.01 - 1)^2(.01) + (1.01 - 1)^2(.99) = .098
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X is a Poisson RV with parameter 4. Y is a Poisson RV with parameter 5. X and Y are independent. What is the distribution of X+Y? A. X+Y is an exponential RV with parameter 9 B. X+Y is a Poisson RV with parameter 4.5 C. X+Y is a Poisson RV with parameter 9
The distribution of C) X+Y is a Poisson RV with parameter 9.
This is because the sum of two independent Poisson distributions with parameters λ1 and λ2 is also a Poisson distribution with parameter λ1 + λ2. Therefore, X+Y follows a Poisson distribution with parameter 4+5 = 9.
Option A is incorrect because an exponential distribution cannot arise from the sum of two Poisson distributions. Option B is also incorrect because the parameter of X+Y is not the average of the parameters of X and Y. Option C is the correct answer as explained above.
In summary, the distribution of X+Y is Poisson with parameter 9.
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A man saves Rs.7,500 in first year. In each year after the first, he saves Rs. 2,000 more than he does in the preceding year. When will he be saved the total amount Rs.1,65,000? Find it.
Answer:
aeqtq34t
Step-by-step explanation:
q34t35t134
How to find x? I am not sure what equation to use to get the correct answer?
Two percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 95% detection rate for carriers and a 3% detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual. A. What is the probability that both tests yield the same result?
The probability that both tests yield the same result is 7.7%.
Simply put, probability is the likelihood that something will occur. When we don't know how an occurrence will turn out, we can discuss the likelihood or likelihood of various outcomes. Statistics is the study of occurrences that follow a probability distribution.
It is predicated on the likelihood that something will occur. The justification for probability serves as the primary foundation for theoretical probability. For instance, the theoretical chance of receiving a head when tossing a coin is 12.
Let's break it down:-
90% don't have of those 99%
5% will be positive
1% positive of those 1%
90% positive
10% negative.
Well we need it to be the same, so 99*(.05*.05+.95*.95)+.01*(.9*.9+.1*.1)= 90.4%.
If both tests are positive, we have:-
0.99*0.05*0.05 and 0.01*0.9*0.9 for being positive, so :-
[tex]\frac{carrier}{positive} = \frac{0.01*0.9*0.9}{(0.99*0.05*0.05+0.01*0.9*0.9)} = 7.7[/tex]
hence, the probability of the two tests yield the same result is 7.7%.
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The length of a rectangle is five times its width. If the permiteter of the rectangle is 72 m, find it’s area
Let's the width of the given rectangle be x. Then the length will be 5x.
We know that,
[tex] \bf \implies Perimeter_{( Rectangle)} = 2 ( Length + Width) [/tex]
[tex] \sf \implies 2( x+5x) = 72 [/tex]
[tex] \sf \implies 2\times 6x = 72 [/tex]
[tex] \sf \implies 12x =72 [/tex]
[tex] \bf \implies x = 6 [/tex]
Hence, the width of the rectangle is 6 m and the length is 5*6 =30 m
[tex]\bf\implies Area_{( Rectangle) }= Length \times Width [/tex]
[tex] \bf \implies Area _{( Rectangle)} = 30 \times 6 [/tex]
[tex] \bf \implies Area _{( Rectangle) }= 180 m^2 [/tex]
Therefore, the area of the given rectangle is 180 metre square.
Exponential for (0,35), (1,50), (2,100), (3,200), (4,400)
The exponential equation that fits the data points (0,35), (1,50), (2,100), (3,200), and (4,400) is y = 35 * (10/7)^x.
To find an exponential equation that fits the given data points, we can use the general form of an exponential equation:
y = a * b^x
where y is the dependent variable (in this case, the second coordinate of each data point), x is the independent variable (the first coordinate of each data point), a is the initial value of y when x is 0, and b is the growth factor.
Using the given data points, we can create a system of equations:
35 = a * b^0
50 = a * b^1
100 = a * b^2
200 = a * b^3
400 = a * b^4
The first equation tells us that a = 35, since any number raised to the power of 0 is 1. We can then divide the second equation by the first equation to get:
50/35 = b^1
Simplifying, we get:
10/7 = b
We can now substitute a = 35 and b = 10/7 into the remaining equations and solve for y:
y = 35 * (10/7)^x
This is the exponential equation that fits the given data points. We can use it to find the value of y for any value of x. This equation gives us a way to predict the value of y for any value of x.
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what is the probability of reaching into the box and randomly drawing a chip number that is smaller than 212 ? express your answer as a simplified fraction or a decimal rounded to four decimal places.
The probability of reaching into the box and randomly drawing a chip number that is smaller than 212 is 0.9378
First, we should find the total number of chips in the box. The box contains 225 chips numbered from 1 to 225. Therefore, the probability of reaching into the box and randomly drawing a chip number that is smaller than 212 is 211/225.
The probability can be expressed as a simplified fraction or a decimal rounded to four decimal places. The probability is rounded to four decimal places is 0.9378.
The probability of drawing a chip number that is smaller than 212 from the box is 211/225 or 0.9378 (rounded to four decimal places).
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True or False, suppose a hypothesis test was performed with a level of significance of 0.05. then if the null hypothesis is actually true, then there is a 5% chance that the researcher will end up accepting the alternative hypothesis in error.
If the null hypothesis is actually true, then there is a 5% chance that the researcher will end up accepting the alternative hypothesis in error, the statement is true.
If a hypothesis test is performed with a level of significance of 0.05 and the null hypothesis is actually true, then there is a 5% chance (or 0.05 probability) that the researcher will reject the null hypothesis and accept the alternative hypothesis in error.
This is known as a Type I error. The Type I error rate is determined by the level of significance of the test.
In other words, if the null hypothesis is true, but the researcher concludes that it is false (i.e., accepts the alternative hypothesis), this is an incorrect decision that is made with a probability of 0.05 or 5%, assuming a significance level of 0.05.
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True or false (with a counterexample if false)?(a) The vectors that are not in the column space form a subspace.(b) If contains only the zero vector, then is the zero matrix.(c) The column space of equals the column space of .(d) The column space of equals the column space of .
(a) False; A subspace is formed by the set of vectors that do not belong to the column space.
(b) True; If the matrix contains solely the zero vector, then it is the zero matrix.
(c) True; The column space of a particular matrix is equivalent to the column space of another specified matrix.
(d) False; The column space of one matrix is identical to the column space of another matrix.
(a) False; if A = [1 0; 0 0], then the column space of A is { e1 }, where e1 is the standard unit vector in the plane. If v is not in the column space of A, but w is not in the column space of A, then v + w is not in the column space of A.
Therefore, the set of vectors that are not in the column space of A does not form a subspace.
(b) True; if every vector in Rn is in the null space of A, then in particular, every standard unit vector is in the null space of A. Thus, the ith column of A is zero for i = 1, . . . , n, so A is the zero matrix.
(c) True; the column space of A is generated by the columns of A, while the column space of AB is generated by linear combinations of the columns of AB. By definition of matrix multiplication, the columns of AB are linear combinations of the columns of A, so the column space of AB is a subspace of the column space of A. Conversely, let b be in the column space of A. Then there is an x in Rm such that Ax = b. Thus, ABx = A(Bx), so b is in the column space of AB. Therefore, the column space of A is a subspace of the column space of AB. Hence the two column spaces are equal.
(d) False; if A = [1 0; 0 0] and B = [0 0; 0 1], then the column space of A is { e1 }, while the column space of B is { e2 }. The column space of AB is { 0 }, so it is not equal to either column space.
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what is 1 half of 68 ?
Answer:
34
Step-by-step explanation:
1/2 multiplied by 68=34
34 is 1 half of 68 .
Here, we have,
given that,
what is 1 half of 68
we know that,
Half of a number can be found by dividing the number by 2.
i.e. we have to find half of 68,
for that, we need to divide 68 by 2
so, we get,
To find half of 68:
68 / 2 = 34
Therefore, half of 68 is 34.
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Each angle of a regular polygon is 1680. How
many sides has it? What is the name of this
polygon?
Answer: 2 solutions
Step-by-step explanation:
To find the angle of a regular polygon, use the formula 180(n-2)/n (where n is the amount of sides.)
Setting them equal, we get (180n-360)/n = 1680.
Multiplying by n on both sides, we get 180n-360 = 1680n.
Solving, we get 1500n = 360.
n = 0.24, which means it is not a shape, as you cannot have a shape with 0.24 sides.
The other way to look at it is to take full revolutions of 360 away from each angle, giving us 240 (the smallest remainder without it going negative). However, all the angles would be concave. If all the angles are concave, then it might connect backwards.
Subtracting 240 from 360 (to get the "exterior" angles, we get 120. Plugging it in to our equation 180(n-2)/n and solving, we get 180n-360 = 120n, and solving gives us 60n = 360, or n=6.
Since the amount of sides came together cleanly, we can classify this polygon as a normal hexagon, which has 6 sides.
Find the derivative of the function f(x), below. It may be to your advantage to simplify first. f(x)=x⋅5x
f′(x)=
The derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.
Using the product rule of differentiation, we can find the derivative of the function f(x) = x⋅5x as follows:
f'(x) = (x)'(5x) + x(5x)'
where (x)' and (5x)' are the derivatives of x and 5x with respect to x, respectively.
(x)' = 1 (the derivative of x with respect to x is 1)
(5x)' = 5 (the derivative of 5x with respect to x is 5)
Substituting these values, we get:
f'(x) = 1⋅5x + x⋅5
Simplifying further, we get:
f'(x) = 5x + 5x
Therefore, f'(x) = 10x.
In conclusion, the derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.
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the answer i need is Another pair it could be (_,10)
and every y value is _ every x value
Answer:
Another order pair could be (30,10).
Every y value is "one-third of" every x value.
1.3. The Dow Jones average (a stock market share index) dropped from 12 837 to 12 503 in one week in July 2012. 1.3.1. Calculate the drop in the share index. 1.3.2. If the price continued to drop at the same rate, calculate the Dow Jones average after 4 more weeks.
The Dow Jones average after 4 more weeks of the same rate of drop would be 11,167.
What is average?
Average, also known as mean, is a measure of central tendency that represents the typical or common value in a set of data. It is calculated by adding up all the values in a data set and then dividing the sum by the total number of values.
To calculate the drop in the Dow Jones average, we subtract the initial value from the final value:
Drop = Final Value - Initial Value
Drop = 12,503 - 12,837
Drop = -334
So the Dow Jones average dropped by 334 points in one week.
If the price continued to drop at the same rate for 4 more weeks, then the total drop after 5 weeks would be:
Total Drop = 5 x Drop
Total Drop = 5 x (-334)
Total Drop = -1670
To calculate the Dow Jones average after 4 more weeks, we need to subtract the total drop from the initial value:
New Dow Jones Average = Initial Value - Total Drop
New Dow Jones Average = 12,837 - 1,670
New Dow Jones Average = 11,167
Therefore, the Dow Jones average after 4 more weeks of the same rate of drop would be 11,167.
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simplest form please
hmmm what we do is firstly make the recurring part a variable, then we multiply it such that, the recurring digits move over from the decimal point to the left, so we'd multiply it by some power of 10, in this case power of 3, because we have three digits to move, 246, so let's do all that
[tex]0.\overline{246}\hspace{5em}x=0.\overline{246}\hspace{5em} \begin{array}{llll} 1000x&=&246.\overline{246}\\\\ &&246+0.\overline{246}\\\\ &&246+x \end{array} \\\\[-0.35em] ~\dotfill\\\\ 1000x=246+x\implies 999x=246\implies x=\cfrac{246}{999}\implies x=\cfrac{82}{333}[/tex]
If you run towards a faraway friend at 5 miles per hour and she bikes towards you at 15 miles per hour, how many miles closer are you to each other after 1 hour?
Using the unitary method we calculate that the friend would be 20 miles closer in an hour.
If you are running towards a faraway friend at a speed of 5 miles per hour and she is biking towards you at 15 miles per hour, According to relative motion's concept, the total speed at which you are approaching each other is:
5 miles / hour - (- 15 miles / hour) = 20 miles / hour
Also, we know that
speed= distance/time according to which, after 1 hour, you and your friend would have closed the distance by,
20 miles/hour × 1 hour = 20 miles
Therefore, you would be 20 miles closer to each other after 1 hour.
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gamboa, inc. sold 110 selfie sticks for $10 each. if producing the selfie sticks had an average cost of $3 , how much profit did the company make?
The company made a profit of $770
How much profit did the company make?Profit is the difference between revenue and costs. In this scenario, the revenue from selling 110 selfie sticks is $10 × 110 = $1100.
Therefore, the costs of producing the same number of selfie sticks are
110 × $3 = $330
So, the profit that Gamboa, Inc. made is
$1100 - $330 = $770.
As we can see, based on the number of selfie sticks together with their production, we managed to obtain a profit of $770.
Hence, the company made a profit of $770.
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3. The length of one leg of a 45-45-90 triangle is 7 m. What is the length of the other leg and the length of the hypotenuse?
The other leg is 7 m, and the hypotenuse is 7 m.
The other leg is 7 m, and the hypotenuse is 14 m.
O The other leg is 7√2 m, and the hypotenuse is 7 m.
The other leg is 7 m, and the hypotenuse is 7√2 m.
Answer: The other leg is 7 m, and the hypotenuse is 7√2 m.
Step-by-step explanation:
This is just a rule that in all cases, the two legs are equal and the hypotenuse is equal to the length of a leg times the square root of 2.
Hope this helps :)
A number is increased by 70% and the result is 42.5. What is the number?
A. 29.75
B. 27.5
C. 25
D. 17
E. 12.75
To compare the pain control offered by two different analgesics in pediatric patients, the authors selected the Wong-Baker FACES pain rating scale as the primary end point. Before beginning the clinical trial, the authors sought to validate this ordinal scale by showing a correlation with a previously validated visual analog scale. Which one of the following statistical test is most appropriate to assess whether a correlation exists between these two measurements?
A. Pearson correlation
B. Analysis of variance (ANOVA)
C. Spearman rank correlation
D. Regression analysis
The most appropriate statistical test to assess whether a correlation exists between the Wong-Baker FACES pain rating scale and a previously validated visual analog scale is the (C) Spearman rank correlation.
What is correlation?Correlation refers to the connection between two variables in which a modification in one variable is linked to a modification in the other variable. Correlation can be positive or negative.
Spearman rank correlation- A non-parametric approach to test the statistical correlation between two variables is Spearman rank correlation, also known as Spearman's rho or Spearman's rank correlation coefficient. This is based on the ranks of the values rather than the values themselves. The results are denoted by the letter "r".
The formula for Spearman's rank correlation coefficient:
Rs = 1 - {6Σd₂}/{n(n₂-1)}
Where, Σd₂ = the sum of the squared differences between ranks.
n = sample size
Thus, the most appropriate statistical test to assess whether a correlation exists between these two measurements is the (C) Spearman rank correlation.
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Evaluate the expression shown below and write your answer as a fraction in simplest form.
-0.25 + 0.3 - ( - 3/10 ) + 1/4
The evaluation of the expression -0.25 + 0.3 - ( - 3/10 ) + 1/4 is 3 / 5.
How to solve expression?An algebraic expression is made up of variables and constants, along with algebraic operations such as addition, subtraction, division, multiplication etc.
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.
Therefore, let's solve the expression as follows:
-0.25 + 0.3 - ( - 3/10 ) + 1/4
let's convert it to fraction
- 1 / 4 + 3 / 10 + 3 / 10 + 1 / 4
Hence,
3 / 10 + 3 / 10 + 1 / 4 - 1 / 4
3 + 3 / 10
6 / 10 = 3 / 5
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c) assume that 25% of the defendants in the state are innocent. in a certain year 200 people put on trial. what is the expected value and variance of the number of cases in which juries got the right decision?
The expected value of cases in which juries got the right decision is 150, and the variance is 375.
1. Since 25% of defendants in the state are innocent, that means that 75% of the defendants are guilty.
2. This means that in the given year, 150 out of the 200 people put on trial will be guilty.
3. Thus, the expected value of cases in which juries got the right decision is 150.
4. The variance of the number of cases in which juries got the right decision is calculated by taking the expected value and subtracting it from the total number of people put on trial, which is 200.
5. The result of the calculation is 375, which is the variance of cases in which juries got the right decision.
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parabola a and parabola b both have the x-axis as the directrix. parabola a has its focus at (3,2) and parabola b has its focus at (5,4). select all true statements.
a. parabola A is wider than parabola B
b. parabola B is wider than parabola A
c. the parabolas have the same line of symmetry
d. the line of symmetry of parabola A is to the right of that of parabola B
e. the line of symmetry of parabola B is to the right of that of parabola A
In the following question, among the given options, Option (b) "Parabola B is wider than Parabola A" and option (d) "The line of symmetry of Parabola A is to the left of that of Parabola B" are the true statements.
The following statements are true about the parabolas: c. the parabolas have the same line of symmetry, and d. the line of symmetry of parabola A is to the right of that of parabola B.
Parabola A and Parabola B have the x-axis as the directrix, with the focus of Parabola A at (3,2) and the focus of Parabola B at (5,4). As the focus of Parabola A is to the left of the focus of Parabola B, the line of symmetry for Parabola A is to the right of the line of symmetry of Parabola B.
Parabola A and Parabola B may have different widths, depending on their equation, but this cannot be determined from the information given.
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7. What is the greatest whole number that satisfies the inequality
3x - 1 < 8 ?
Answer:
2
Step-by-step explanation:
3(2)-1<8
6-1<8
5<8
if you go up to 3 as the whole number then the equation ends up 8<8, and the sign is less than (<) not less than or equal to.
So 2 would be the answer.
F(x)=-(x+3)(x+10) pls help
Answer:
Zeros: x = -10 and x = -3
Vertex: [tex](-\frac{13}{2} , \frac{49}{4} )[/tex]
Step-by-step explanation:
Pre-SolvingWe are given the following function:
f(x) = -(x+3)(x+10)
We want to find the zeros and the vertex of the parabola.
SolvingZerosThe zeros are the values of the function where f(x) = 0.
So, in order to find the zeros, we can set f(x) = 0.
0 = -(x+3)(x+10)
We can divide both sides by -1, to get:
0 = (x+3)(x+10)
To solve this, we will use zero product property.
Split and solve:
x+3 = 0
x = -3
x+10=0
x = -10
Vertex
Now, to find the vertex, we first get the average of the zeros.
Add the values of the zeros together, then divide by two:
[tex]\frac{-3-10}{2}[/tex] = [tex]\frac{-13}{2}[/tex]
Now, we plug this in for x to get the y value (found through f(x)) of the vertex.
[tex]f(-\frac{13}{2}) = -(-\frac{13}{2} + 3) (-\frac{13}{2} + 10)[/tex] = [tex]\frac{49}{9}[/tex]
So, the vertex is [tex](-\frac{13}{2} , \frac{49}{4} )[/tex]
the sides of a triangle have lengths 15, 20, 25. find the length of the shortest altitude of the triangles.
The length of the shortest altitude of the triangles is 15 units by using Heron’s formula.
We have, The sides of a triangle have lengths of 15, 20, and 25.
To find, The length of the shortest altitude of the triangle.
Steps to solve the problem:
Let us assume that the length of the b is h.According to the property of triangles, the area of the triangle can be calculated as:Area = 1/2 * base * height
We can choose any side as the base of the triangle, let us assume that 20 is the base of the triangle, and its corresponding height is h.Area of the triangle = 1/2 * 20 * h ⇒ 10h
Using Heron’s formula, the area of the triangle can be calculated as:A = √(s(s-a)(s-b)(s-c))
Where a, b, and c are the sides of the triangle, and s is the semi-perimeter of the triangle.
According to the given problem, the sides of the triangle are 15, 20, and 25.
s = (a + b + c)/2
= (15 + 20 + 25)/2
= 30
Therefore, the area of the triangle can be calculated as:
A = √(30(30-15)(30-20)(30-25))
= √(30*15*10*5)
= 150 sq. units
Therefore, we can write the formula for the area of the triangle as:
150 = 10 h
h = 15 units
Therefore, the length of the shortest altitude of the triangle is 15 units.
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Find the first 4 terms of the sequence represented by the expression 3n + 5
The first 4 terms of the sequence represented by the expression 3n + 5
is 8, 11, 14 and 17.
Sequence:
In mathematics, an array is an enumerated collection of objects in which repetition is allowed and in case order. Like a collection, it contains members (also called elements or items). The number of elements (possibly infinite) is called the length of the array. Unlike sets, the same element can appear multiple times at different positions in the sequence, and unlike sets, order matters. Formally, a sequence can be defined in terms of the natural numbers (positions of elements in the sequence) and the elements at each position. The concept of series can be generalized as a family of indices, defined in terms of any set of indices.
According to the Question:
Given, aₙ = (3n+5).
First four terms can be obtained by putting n=1,2,3,4
a 1=(3×1+5) = 8
a 2 =(3×2+5) = 11
a 3 =(3×3+5) = 14
a 4 =(3×4+5) = 17
First 4 terms in the sequence are 8, 11, 14, 17.
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Levi's investment account accrues interest biannually. The function below represents the amount of money in his account if the account is left untouched for
t years.
f(t) = 2000 (1.03)2t
The amount of money in the account ( increases or decreases )
by (2 , 3 or 103)
% (every six months, each year, or every two years)
Answer:
The amount of money in the account increases by 3% every six months, or biannually.
To see why, we can break down the function f(t) = 2000(1.03)^(2t):
The base amount in the account is $2000.The term (1.03)^(2t) represents the interest accrued over time.Since the interest is compounded biannually, the exponent of 2t indicates the number of six-month periods that have elapsed. For example, if t = 1, then 2t = 2, which means two six-month periods have elapsed (i.e., one year).
Each time 2t increases by 2, the base amount is multiplied by (1.03)^2, which represents the interest accrued over the two six-month periods.
Thus, the amount of money in the account increases by 3% every six months, or biannually.
As for the second part of the question, the amount of increase is not 2%, 3%, or 103%.
Construct triangle ABC, in which AB = 5 cm, angle BAC = 95° and
angle ABC = 34°.
Measure the length of BC.
Give your answer to 1 d.p.
The length of BC is approximately 3.5 cm
What is a triangle?A triangle is described as a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices.
We construct triangle ABC, through the following steps:
Draw a line segment AB of length 5 cm.At point A, draw a ray that makes an angle of 95 degrees with AB.At point B, draw a ray that makes an angle of 34 degrees with AB.The intersection point of the two rays is point C, which is the third vertex of the triangle.The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Mathematically shown as:
a/sin(A) = b/sin(B) = c/sin(C)
5/sin(95) = BC/sin(34)
BC = (5*sin(34))/sin(95)
BC ≈ 3.5cm
In conclusion, the length of BC is approximately 3.5 cm.
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Two teams, A and B, play in a series. Team A has a 60% chance of winning each game, independent of other games. The series ends and a winner is declared when one of the teams has won two more games than the other team. (a) What is the expected number of games played? (b) Given that Team B wins the first game, what is the probability that the series will last at least 8 games?
(a) The probability that the series will last exactly m games is:(1 - (pA + pB))^m(pA (1 - pB) + pB (1 - pA))where pA and pB are the probabilities of teams A and B, respectively. Therefore, the probability that the series will last less than 5 games is the probability that the series will last exactly 3 games or exactly 4 games:1 - (1 - 0.6 * 0.4)^3 - (1 - 0.6 * 0.4)^4 ≈ 0.684.
The probability that the series will last 5 games is the probability that the first four games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 4 ≈ 0.055.The probability that the series will last 6 games is the probability that the first five games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 5 ≈ 0.077The probability that the series will last 7 games is the probability that the first six games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 6 ≈ 0.091.
The expected number of games played is thus approximately:0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + ∑m=8^∞ (1 - (pA + pB))^m(m + 1)(pA (1 - pB) + pB (1 - pA))The sum above can be computed by expressing it as the product of three factors:1 - (pA + pB) is a common factor for all terms, (pA (1 - pB) + pB (1 - pA)) is a sum of two terms that can be replaced by 1 - (1 - pA)(1 - pB), and m + 1 is a sum of m and 1. After replacing the sum of two terms, we obtain:0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + (1 - (0.6 + 0.4))^8(8 + 1)(1 - (1 - 0.6 * 0.4)^2) / (0.6 + 0.4 - 0.6 * 0.4) ≈ 0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + 0.02525 / 0.34 ≈ 5.43.
Therefore, the expected number of games played is approximately 5.43.(b) Given that Team B wins the first game, the series can last 7 or 8 games. The probability that the series will last 8 games is the probability that the first seven games have three wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 ≈ 0.013824.The probability that the series will last at least 8 games is therefore approximately 0.091 + 0.013824 = 0.104824, or 10.48%.Answer: (a) The expected number of games played is approximately 5.43. (b) The probability that the series will last at least 8 games is approximately 0.104824 or 10.48%.
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