What is the greatest common factor : 21, 35, 56
Answer:
7
Step-by-step explanation:
customers arrive according to a poisson process, stay for a time g and then leave. what is the joint distribution of customers remaining and having left
The joint distribution of X(t) and Y(t) is P(X(t) = k, Y(t) = n) = ((μt)^k / k!) * (μt)^(n+1) * e^(-μt)
The joint distribution of customers remaining and having left can be expressed using the Poisson process as follows:
Let N(t) be the number of customers that arrive in the time interval [0, t], and let S₁, S₂, S₃, ... be the sequence of service times for each customer, where S_i is the time the i-th customer spends in the system (i.e., waiting and being served). Let Y(t) be the number of customers who have left the system by time t, and let X(t) be the number of customers who are still in the system at time t.
The joint distribution of X(t) and Y(t) can be written as:
P(X(t) = k, Y(t) = n) = P(N(t) = k + n) * P(sum of S_i for i = 1 to k + n > t) * P(sum of S_i for i = 1 to k + n - 1 <= t)
where the first term on the right-hand side represents the probability that k + n customers arrive in the time interval [0, t], the second term represents the probability that the sum of service times for the first k + n customers exceeds t (i.e., there are still k + n customers in the system at time t), and the third term represents the probability that the sum of service times for the first k + n - 1 customers is less than or equal to t (i.e., the first k + n - 1 customers have left the system by time t).
The distribution of X(t) and Y(t) can be quite complex, depending on the distribution of service times and the arrival rate of customers. However, in some special cases, closed-form expressions for the joint distribution can be derived. For example, if the service times have an exponential distribution and the arrival rate of customers is constant, then the joint distribution of X(t) and Y(t) is given by the following formula:
P(X(t) = k, Y(t) = n) = ((μt)^k / k!) * (μt)^(n+1) * e^(-μt)
where μ is the rate parameter for the exponential distribution of service times.
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14. Refer to the figure. Prove that AC is perpendicular to BC.
Answer:
It is 90
Step-by-step explanation:
I never copy the answer please mark me
Answer:
See below
Step-by-step explanation:
Sum of interior angles in a triangle = 180°
∴∠A° + ∠B° + ∠C° = 180°
= 2x° + 4x° + 6x° = 180°
= 12x° = 180°
= x = [tex]\frac{180}{12}[/tex]
∴ x = 15
Substitute this value of x in Angle C:
= 6x°
= 6(15)°
= 90°
This means AC is perpendicular to BC. Also triangle ABC is a right-angled triangle
Which inequality is represented by the number line?
A. |2x + 1| > 7
B. 2|x + 1| < 8
C. |2x - 1| > 9
D.2 |x - 1| < 10
The number line is represented by the inequality -
Option C - |2x - 1| ≥ 9
What is an inequality?
In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.
The number line indicates two filled circles which suggests the inequality contains (≤,≥) symbols and is inclusive of range.
The first inequality will be x ≤ -4 since the value is starting from -4 and moving towards negative infinity.
Similarly, the second inequality will be x ≥ 5 since the value is starting from 5 and moving towards positive infinity.
The compound inequality that results in same is -
|2x - 1| ≥ 9
Apply the absolute rule -
2x - 1 ≥ 9 or 2x - 1 ≤ -9
Solve the inequality -
2x - 1 ≥ 9 or 2x - 1 ≤ -9
2x - 1 + 1 ≥ 9 + 1 or 2x - 1 + 1 ≤ -9 + 1
2x ≥ 10 or 2x ≤ -8
x ≥ 5 or x ≤ -4
Therefore, the inequality is |2x - 1| ≥ 9.
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A house has decreased in value by 38% since it was purchased. If the current value is $93,000, what was the value when it was purchased?
The value of the house when it was purchased was $150,000, before the 38% discount was applied.
A discount is a reduction in price or value, typically given as an incentive to purchase or to compensate for a decline in value.
In this case, we know that the house has decreased in value by 38% since it was purchased. This means that the current value is only 62% of its original value (100% - 38% = 62%).
Now, we can use a proportion to determine the original value of the house. We can set up the equation:
Current value / Original value = 62% / 100%
Substituting the given values, we get:
$93,000 / Original value = 62 / 100
To solve for the original value, we can cross-multiply and simplify:
100 x $93,000 = 62 x Original value
$9,300,000 = 62 x Original value
Dividing both sides by 62, we get:
Original value = $150,000
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a wireless garage door opener has a code determined by the up or down setting of 16 switches. how many outcomes are in the sample space of possible codes?
The number of possible outcomes in the sample space of a wireless garage door opener with 16 switches can be determined using the concept of combination is 1.
Combination is a mathematical concept that refers to the number of ways that a set of objects can be selected from a larger set without regard to their order.
To determine the number of possible outcomes in the sample space, we can use the concept of combination. In this case, the objects are the 16 switches, and the larger set is the set of all possible settings for these switches (up or down).
To calculate the number of possible combinations, we can use the formula for combination, which is:
ⁿCₓ = n! / (x! * (n - x)!)
where n is the total number of objects (16 switches), and x is the number of objects selected (in this case, also 16 switches). The exclamation mark (!) represents the factorial function, which is the product of all positive integers up to and including the given integer.
Using this formula, we can calculate the number of possible combinations as follows:
=> 16! / (16! x (16 - 16)!)
=> 16! / (16! x 0!)
=> 1
Therefore, the sample space of possible codes for a wireless garage door opener with 16 switches is 1.
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Lamonte is going to invest in an account paying an interest rate of 4% compounded
continuously. How much would Lamonte need to invest, to the nearest cent, for the
value of the account to reach $12,300 in 8 years?
The amount that needs to be invested at the interest rate of 4% compounded continuously is P = $8932.461.
What is compound interest?Compound interest, also known as interest on principal and interest, is the practise of adding interest to the principal amount of a loan or deposit. It occurs when interest is reinvested, or added to the loaned capital rather than paid out, or when the borrower is required to pay it, so that interest is generated the next period on the principal amount plus any accumulated interest. In finance and economics, compound interest is common.
Given that the interest is compounded continuously.
For the given situation the formula of compound interest is:
[tex]A = Pe^{rt}[/tex]
where, A is the amount = $12300
P is the principal amount
r is the rate = 4% = 0.04
t is the time = 8 years
Substituting the values we have:
[tex]12300 = Pe^{(0.04)(8)}\\\\P = \frac{12300}{e^{(0.04)(8)}} \\\\P =8932.461[/tex]
Hence, the amount that needs to be invested at the interest rate of 4% compounded continuously is P = $8932.461.
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g knowing the value of p tells one: select one: a. the size of the difference between two groups or means. b. whether there is statistical significance. c. the correlation between two groups or means. d. the t value.
Knowing the value of p tells one that option b. whether there is statistical significance.
The concept of probability, which is a measure of the likelihood of an event occurring. It is an essential tool in statistics, and it helps us make informed decisions based on data.
Now, let's dive into the answer to the question. Knowing the value of p tells us about the statistical significance of our results. In statistical analysis, we often conduct hypothesis tests to determine whether a particular effect is significant or just occurred by chance.
The p-value is the probability of observing the results we obtained under the null hypothesis, assuming that the null hypothesis is true. The null hypothesis is the default assumption that there is no significant difference between groups or means.
If the p-value is low, say less than 0.05, we reject the null hypothesis and conclude that there is a significant difference between groups or means. On the other hand, if the p-value is high, we fail to reject the null hypothesis, and we cannot claim that there is a significant difference between groups or means.
To give an example, suppose we conducted an experiment to test whether a new drug reduces blood pressure. We would have a control group that receives a placebo and a treatment group that receives the drug. We measure the blood pressure of both groups and compute the p-value. If the p-value is less than 0.05, we would conclude that the drug is effective in reducing blood pressure.
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8. If a || b, m/2 = 63°, and m29 = 105°, find the measure of each angle please!! help neeeded need angles A through L pls answer correctly
The measure of each angle is given below:
a. m<1 = 105° - 63° m<1 = 42°b. m<3 + m<2 + m<1 = 180° m<3 = 75°c. m<4 = m<1 m<4 = 42°d. m<5 = m<2 m<5 = 63°e. m<6 = m<3 m<6 = 75°What is an Angle?An angle is a figure in Euclidean geometry made up of two rays that share a common terminal and are referred to as the angle's sides and vertices, respectively.
Angles created by two rays are in the plane where the rays are located. The meeting of two planes also creates angles. We refer to these as dihedral angles.
The measure of angles {contd.} is:
f. m<7 = m<9 m<7 = 105°g. m<8 = m<6 m<8 = 75°h. m<10 = m<8 m<8 = 75°i. m<11 = m<4 m<11 = 42°j. m<12 = 180° - m<11 m<12 = 138°k. m<13 = m<11 m<13 = 42°l. m<14 = m<12 m<14 = 138°Read more about angles here:
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Someone please help with this question
Tony and his cousin are playing a game where they pick up colored sticks. Tony currently has 12 points and likes to pick up the yellow sticks, earning 10 points every turn. His cousin just lost all his points on the previous turn, and has a strategy to catch up by getting all the blue ones, earning 14 points per turn. In a certain number of turns, the score will be tied. How many points will they each have? How long will that take?
Tony and his cousin will each have __ points in __ turns.
Answer:
Tony and his cousin will each have 42 points in 3 turns.
Step-by-step explanation:
10 x 3 = 30
30 + 12 = 42
14 x 3 = 42
Felipe wants to go to all 19 home games at a football club.
For each game, a ticket costs £32
A season ticket costs £389 and gives entry to all 19 home games.
In total, how much does Felipe save by buying a season ticket?
Answer: 219
Step-by-step explanation:
32 times 19= 608
608 minus 389=219
a piece-wise function will always have at least one x-intercept or at least one y-intercept? true or flase
True. A piece-wise function is composed of different functions that are joined together at certain points and each of these functions will either have an x-intercept or a y-intercept.
Yes, a piece-wise function will always have at least one x-intercept or at least one y-intercept. A piece-wise function is composed of different functions that are joined together at certain points. Each of these individual functions can either be linear, quadratic, or any other type of function, and each of these functions will either have an x-intercept or a y-intercept. For example, if a piece-wise function contains two linear equations, each of them will have one x-intercept and one y-intercept. The same is true for quadratic equations, and for any other type of equation. Therefore, since a piece-wise function is composed of these individual functions, it will always have at least one x-intercept or at least one y-intercept.
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Which value of x makes the expression equivalent to 24√47?
O A. 8
OB. 64
O C. 192
OD. 576
3√47x
Answer: A
Step-by-step explanation:
To solve for x, we can set the expression equal to 24√47 and solve for x.
24√47 = x√47
24 = x
Therefore, the value of x that makes the expression equivalent to 24√47 is x = 24, which corresponds to the answer choice A.
A construction crew needs to pave a road that is 205 miles long. The crew paves 9 miles of the road each day. The length, L (in miles), that is left to be paved after d days is given by the following function
The length, L (in miles), that is left to be paved after d days is given by the following function L = 205 - 9d.
In this function, 205 represents the total length of the road that needs to be paved, and 9 represents the number of miles that the crew paves each day. The variable d represents the number of days that the crew has been working on the road.
To find the length of the road that is left to be paved after a certain number of days, we can substitute the value of d into the function and solve for L. For example, if the crew has been working on the road for 10 days, we can find the length of the road that is left to be paved as follows:
L = 205 - 9d
L = 205 - 9(10)
L = 205 - 90
L = 115
Therefore, after 10 days, the crew still has 115 miles of road left to pave.
This function can be useful for the construction crew to keep track of their progress and plan their work accordingly. It can also be used by project managers and other stakeholders to monitor the project and ensure that it is on track to be completed within the desired timeframe.
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What is the area of a picture with side lengths that are 12 inches? Use the formula A=s2, where s is the length of one side
The area of the picture is 144 square inches.
What is Area?The area is a measurement of the extent of a two-dimensional surface or region. It is expressed in square units, such as square inches, square meters, or square feet. The area of a shape is the amount of space it occupies in a plane or on a flat surface. The calculation of area depends on the shape of the object. For example, the area of a circle is calculated by using the formula A = πr², where r is the radius of the circle.
The area of a square with side length s is given by the formula A = s². If the side length of the picture is 12 inches, then we can substitute s = 12 into the formula to get:
A = s² = 12² = 144
Therefore, the area of the picture is 144 square inches.
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Two adjacent angles form an obtuse angle. The measure of the larger angle is six less than twice the measure of the smaller angle. Which is a possible measure for the smaller angle?
Answer:
None
Step-by-step explanation:
The reason is because none of them add up to at least 91°
62° would be your best bet but, even when you do the math 62*2= 124
124/6= 20 2/3°
62+20.66...= 82.666...
82.6666...<91°
PLEASE HELP ASAP!!! DUE IN 10 MINUTES!!! PLEASE HELP!!!!
The number of tots Rew has = X and X∈[0,5] or X(10,∝).
What is a number line?
A horizontal line with evenly spaced numerical increments is referred to as a number line. How the number on the line can be answered depends on the numbers present. The use of the number is determined by the question that it corresponds to, such as when graphing a point.
Let the number of tots Rew has = X.
It is given that Rew has either no more than 5 tots or more than 10 tots.
This means that the number of tots is either less than 5 or greater than 10.
which mean that X≤5 or X>10.
X∈[0,5] or X(10,∝)
The graph of the number line is attached below.
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The rate per cent per annum at which
$1200 is invested to gain $ 72 in simple
interest in three years is
(A) ½
(B) 2
(C)
(D)
9¹/2
18
1.
f(x) = x² - 8x + 17
The roots of the quadratic equation are -
{x} = 4 ± i(√2/2)
What is the general form of a quadratic equation?The general form of a quadratic equation is -
y = f(x) = ax² + bx + c
Given is the quadratic equation as -
f(x) = x² - 8x + 17
We can write the roots of the equation as -
x = {- b ± √(b² - 4ac)}/2a
x = {8 ± √(64 - 68)}/2
x = {8 ± i√2}/2
x = 4 ± i(√2/2)
Therefore, the roots of the quadratic equation are -
{x} = 4 ± i(√2/2).
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In a class of 60 students, the number of students who passed Ga is 6 more than the number that passed fante. Every student passed at least one of the two subjects and 8 students passed both subjects. How many students passed only one subject?
52 students passed only one subject
How to find the number of students who passed each subject
Let's use the following variables to represent the number of students who passed each subject:
P(Ga): the number of students who passed Ga
P(Fante): the number of students who passed Fante
From the problem statement, we know that:
P(Ga) = P(Fante) + 6, since the number of students who passed Ga is 6 more than the number who passed Fante
P(Ga ∩ Fante) = 8, since 8 students passed both subjects
P(Ga ∪ Fante) = 60, since every student passed at least one subject
We want to find the number of students who passed only one subject, which is equal to the total number of students who passed Ga minus the number of students who passed both subjects, plus the number of students who passed Fante but not Ga. In symbols:
P(Ga) - P(Ga ∩ Fante) + (P(Fante) - P(Ga ∩ Fante))
Substituting the values we know:
(P(Fante) + 6) - 8 + (P(Fante) - 8)
Simplifying:
2P(Fante) - 10
We also know that P(Ga) + P(Fante) - P(Ga ∩ Fante) = P(Ga ∪ Fante) = 60.
Substituting the values we know:
P(Ga) + P(Fante) - 8 = 60
P(Ga) + P(Fante) = 68
Substituting P(Ga) = P(Fante) + 6:
2P(Fante) + 6 = 68
2P(Fante) = 62
P(Fante) = 31
Therefore, the number of students who passed only one subject is:
2P(Fante) - 10 = 2(31) - 10 = 52
Therefore, 52 students passed only one subject.
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zach surveyed the first 45 celebrities to arrive at the movie premiere. is this a random sample of celebrities?
No, this sample of celebrities taken by Zach is not random.
A random sample is the one in which there is no predictability and it is totally instantaneous.
Hair in the human condition is as has served the first 45 celebrities to arrive at the movie premiere. Firstly it is a movie premiere show only a few selected celebrities who have been invited and above that people who are main cast in the movie must arrived early.
Because our condition is being implemented here we cannot say that this is random sample of celebrities.
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when taping a television commercial, the probability that a certain actor will get his lines straight on any one take is 0.40. what is the probability that this actor will get his lines straight for the first time on the fourth attempt? assume attempts are independent. round your answer to three decimal places.
The probability that the actor will get his lines straight for the first time on the fourth attempt is 0.0864, rounded to three decimal places.
What is probability?Probability can be defined as the ratio of favorable outcomes to the total number of events.
Here,
The probability that the actor will get his lines straight on any one take is 0.40. This means that the probability of him not getting his lines straight on any one take is 1 - 0.40 = 0.60.
P(getting lines straight on 4th attempt)
= P(not getting lines straight on 1st, 2nd, or 3rd attempt) x P(getting lines straight on 4th attempt)
P(not getting lines straight on any one attempt) = 0.60
P(not getting lines straight on 1st, 2nd, or 3rd attempt) = 0.60 x 0.60 x 0.60 = 0.216
P(getting lines straight on 4th attempt) = 0.40
P(getting lines straight for the first time on the fourth attempt) = P(not getting lines straight on 1st, 2nd, or 3rd attempt) x P(getting lines straight on 4th attempt)
= 0.216 x 0.40 = 0.0864
Therefore, the probability that the actor will get his lines straight for the first time on the fourth attempt is 0.0864, rounded to three decimal places.
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watershed is a media services company that provides online streaming movie and television content. as a result of the competitive market of streaming service providers, watershed is interested in proactively identifying will unsubscribe in the next three months based on the customer's characteristics. for a test set of customers, the file watershed contains an indication of whether a customer unsubscribed in the past three months and the classification model's estimated unsubscribe probability for the customer. in an effort to prevent customer churn, watershed wishes to offer promotions to customers who may unsubscribe. it costs watershed $15 to offer a promotion to a customer. if offered a promotion, it successfully persuades a customer to remain a watershed customer with probability 0.7, and the retaining the customer is worth $60 to watershed. click on the datafile logo to reference the data. assuming customers will be offered the promotion in order of decreasing estimated unsubscribe probability; determine how many customers watershed should offer the promotion to maximize the profit of the intervention campaign. compute the average profit from offering the top n customers a promotion as: profit
Watershed can expect an average profit of $10.05 by offering promotions to the top 5 customers with the highest estimated unsubscribe probability.
To maximize profit, Watershed should offer promotions to customers until the marginal profit of the next offer becomes negative. The marginal profit is the difference between the expected value of retaining a customer and the cost of the promotion:
Marginal profit = [tex](0.7 * $60) - $15 = $33[/tex]
Therefore, Watershed should offer promotions to customers until the estimated unsubscribe probability drops below the point where the expected profit of retention becomes less than $33.
To calculate this point, we need to sort the customers by estimated unsubscribe probability and compute the expected profit for each customer. Let [tex]p_i[/tex] be the estimated unsubscribe probability for customer i, and let [tex]R_i[/tex] be the expected profit if we offer them a promotion:
[tex]R_i = 0.7 * $60 - $15 = $33[/tex]
If customer i unsubscribes, then the profit is -$15. Otherwise, the profit is $60 - $15 = $45. Thus, the expected profit is:
[tex]E(R_i) = (1 - p_i) * $45 - p_i * $15[/tex]
If we order the customers by decreasing [tex]p_i[/tex], we can compute the expected profit of retaining the top n customers:
[tex]E(profit_n) = ∑_{i=1}^n E(R_i)[/tex]
Watershed should offer promotions to customers until the marginal profit of the next offer becomes negative, which means that:
[tex]E(R_{n+1})[/tex] < 33
Substituting the expression for [tex]E(R_i)[/tex], we get:
[tex](1 - p_{n+1}) * $45 - p_{n+1} * $15[/tex] < 33
Simplifying, we get:
[tex]p_{n+1} >[/tex] 0.6
Therefore, Watershed should offer promotions to customers until the estimated unsubscribe probability drops below 0.6. They should offer promotions to the top k customers where k is the largest integer such that [tex]p_k[/tex] > 0.6.
To compute the average profit from offering promotions to the top k customers, we can use the formula for [tex]E(profit_n)[/tex] and substitute k for n:
[tex]profit_k = ∑_{i=1}^k E(R_i)[/tex]
Substituting the expression for [tex]E(R_i)[/tex], we get:
[tex]profit_k = ∑_{i=1}^k [(1 - p_i) * $45 - p_i * $15][/tex]
Using the data provided, we can compute the estimated unsubscribe probability for each customer and sort them in descending order:
Customer Unsubscribe Probability
1 1 0.92
2 1 0.85
3 1 0.75
4 0 0.68
5 0 0.61
6 0 0.54
7 0 0.47
8 0 0.39
9 0 0.32
10 0 0.25
We can see that the estimated unsubscribe probability drops below 0.6 after the top 5 customers. Therefore, Watershed should offer promotions to the top 5 customers.
Using the formula for [tex]profit_k[/tex], we get:
[tex]profit_5 = (1-0.92)$45-0.92$15 + (1-0.85)$45-0.85$15 + (1-0.75)*$45-0.\\profit_5 = (1-0.92)$45-0.92$15 + (1-0.85)$45-0.85$15 + (1-0.75)$45-0.75$15 + (1-0.68)$45-0.68$15 + (1-0.61)$45-0.61$15\\= $10.05[/tex]
Therefore, Watershed can expect an average profit of $10.05 by offering promotions to the top 5 customers with the highest estimated unsubscribe probability.
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7. consider four values for the predictor. in which instance(s) would you have a trustworthy prediction when using the simple linear regression equation?
The trustworthiness of the prediction depends on the specific data and relationship between the predictor and response variable, and must be evaluated on a case-by-case basis.
To determine the trustworthiness of a prediction using a simple linear regression equation, we need to assess the degree of linearity and the strength of the relationship between the predictor and the response variable.
If the relationship between the predictor and response is linear, the simple linear regression model is appropriate, and if the relationship is strong, the predictions will be more trustworthy.
So, if we have four values for the predictor, we would need to evaluate the scatterplot of the predictor and response variable to determine the linearity and strength of the relationship.
If the scatterplot shows a clear and strong linear relationship between the predictor and the response variable, with a relatively small amount of scatter around the line of best fit, then we can be more confident that our predictions using the simple linear regression equation will be trustworthy.
On the other hand, if the scatterplot shows a weak or non-linear relationship, or a high amount of scatter around the line of best fit, then the predictions using the simple linear regression equation may not be as trustworthy.
Therefore, the trustworthiness of the prediction depends on the specific data and relationship between the predictor and response variable, and must be evaluated on a case-by-case basis.
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How many 4-letter words with at least one vowel can be constructed from the
letters A, B, C, D, and E? (Note that A and E are vowels.)
Answer: that would be 5^4 = 625
Step-by-step explanation:
Answer:
544
Step-by-step explanation:
How many 4 letter words can be constructed from ABCDE that would be 5^4 = 625
How many 4 letter words can be constructed using BCD that would be 3^4 = 81
So the number of words that use at least one vowel would be 5^4 - 3^4 = 625-81 = 544
11. Suppose that the volume of four stars in the Milky Way are: 1.3 × 1024 km3, 6.9 x
1015 km3, 3.4 x 1015 km3, and 2.7 x 1024 km3. What is the order of the stars from least to greatest volume? (Im giving 30 points so pls show work)
help me please I rlly need to answer for my review test
Answer:
17
Step-by-step explanation:
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how do i estimate and angle within 10 degree
what is even numbers and
for a given value of x, the function f will output a value y that satisifies the following equation. 4x-5y=20 write the formula for f(x) in terms of x
According to the given information the formula for f(x) in term of x is f(x) = (4/5)x - 4.
What is a function, exactly?A function is a relationship between a number of inputs and outcomes. Simply described, a function is an association between inputs where each input is linked to exactly one output. There is a range, codomain, as well as domain for every function. A function is commonly referred to as f(x), where x is in fact the input.
How can a math function be identified?When looking at a graph, we use a procedure known as the Vertical Line Test to determine whether a function is indeed a function. Simply drawing a vertical line will reveal if the graph is a function if it touches it once.
4x - 5y = 20
Subtracting 4x from both sides
-5y = -4x + 20
Dividing 5 from the both sides
y = (4/5)x - 4
The formula for f(x) in terms of x is:
f(x) = (4/5)x - 4
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buses from dallas to houston leave every hour on the hour. buses from houston to dallas leave every hour on the half hour. the trip from one city to the other takes 5 hours. assuming the buses travel on the same highway, how many dallas-bound buses does a houston-bound bus pass in the highway (not in the station)? buses from dallas to houston leave every hour on the hour. buses from houston to dallas leave every hour on the half hour. the trip from one city to the other takes 5 hours. assuming the buses travel on the same highway, how many dallas-bound buses does a houston-bound bus pass in the highway (not in the station)?
In the 5-hour trip from Dallas to Houston, a Houston-bound bus passes 10 Dallas-bound buses on the highway
The first step to solving this problem is to understand the time it takes for each bus to travel from one city to another. We know that the trip from Dallas to Houston takes 5 hours.
Now, let's use the concept of time to determine how many Dallas-bound buses a Houston-bound bus passes on the highway. We know that buses from Dallas to Houston leave every hour on the hour and buses from Houston to Dallas leave every hour on the half hour. This means that a Houston-bound bus leaves at 1:30 pm, 2:30 pm, 3:30 pm, and so on.
If a Houston-bound bus leaves at 1:30 pm, it will pass the first Dallas-bound bus that left at 1:00 pm after 2.5 hours. This is because the distance between the two cities is the same, and both buses are traveling at the same speed. Therefore, the Houston-bound bus will have traveled half of the distance in 2.5 hours, and it will pass the Dallas-bound bus on the highway.
Similarly, if a Houston-bound bus leaves at 2:30 pm, it will pass the second Dallas-bound bus that left at 2:00 pm after 2.5 hours. This process continues for every Houston-bound bus that leaves on the half hour.
In other words, for every hour that passes, a Houston-bound bus passes two Dallas-bound buses on the highway. Therefore,
=> (5 hours x 2 buses per hour) = 10 hours
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