The probability of Bo, Colleen, Jeff, and Rohini winning the first, second, third, and fourth prizes, respectively, is 1 in 6,497,400.
The probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in the contest with 52 entrants and no one winning more than one prize can be calculated as follows:
1. Probability of Bo winning the first prize: 1/52 (since there are 52 people, and Bo is one of them)
2. Probability of Colleen winning the second prize after Bo has won the first prize: 1/51 (since there are now 51 people left)
3. Probability of Jeff winning the third prize after Bo and Colleen have won: 1/50 (50 people remaining)
4. Probability of Rohini winning the fourth prize after Bo, Colleen, and Jeff have won: 1/49 (49 people left)
To find the overall probability, multiply all individual probabilities together:
(1/52) x (1/51) x (1/50) x (1/49) = 1/6,497,400
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The probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in the drawing is approximately 1.0625 × 10^-9.
To find the probability of Bo, Colleen, Jeff, and Rohini winning the first, second, third, and fourth prizes, respectively, follow these steps:
Calculate the total number of ways to choose 4 winners from 52 people (order matters, since the prizes are distinct). This is a permutation, so use the formula nPr = n! / (n - r)!,
where n = 52 (total number of people) and r = 4 (total number of prizes). In this case, 52P4 = 52! / (52 - 4)! = 52! / 48!.
Calculate the probability of Bo winning the first prize:
There is 1 favorable outcome (Bo winning) out of 52 possible outcomes, so the probability is 1/52.
Calculate the probability of Colleen winning the second prize, given that Bo has already won:
There is 1 favorable outcome (Colleen winning) out of the remaining 51 possible outcomes, so the probability is 1/51.
Calculate the probability of Jeff winning the third prize, given that Bo and Colleen have already won:
There is 1 favorable outcome (Jeff winning) out of the remaining 50 possible outcomes, so the probability is 1/50.
Calculate the probability of Rohini winning the fourth prize, given that Bo, Colleen, and Jeff have already won: There is 1 favorable outcome (Rohini winning) out of the remaining 49 possible outcomes, so the probability is 1/49.
To find the overall probability of this specific order of winners (Bo, Colleen, Jeff, Rohini), multiply the individual probabilities: [tex](1/52) × (1/51) × (1/50) × (1/49) ≈ 1.0625 × 10^-9.[/tex].
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in conditional statements, the part of the statement following ‘if’ is called ___antecedent or consequent
The part of the statement following "if" is called the antecedent, and the part of the statement following "then" is called the consequent in conditional statements.
The if statement evaluates the test expression inside the parenthesis ().
If the test expression is evaluated to true, statements inside the body of if are executed.
If the test expression is evaluated to false, statements inside the body of if are not executed.
The part of the statement following "if" is called the antecedent, and the part of the statement following "then" is called the consequent in conditional statements.
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In conditional statements, the part of the statement following 'if' is called the antecedent.
The antecedent is the condition that needs to be true for the consequent to occur.
The consequent is the part of the statement that follows 'then.'
An antecedent is a noun or pronoun that denotes a specific being, place, object, or clause.
It's also referred to as a referent. Without an antecedent, a sentence may be insufficient or nonsensical since it is
required to establish what or to whom a pronoun in a sentence is referring.
In summary, a conditional statement is structured as "if (antecedent) then (consequent)."
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three six-sided, fair dice are rolled. what is the probability that none of the outcomes is an even number
the probability that none of the outcomes is an even number is 1/8 or approximately 0.125
The probability of rolling an even number on a fair six-sided die is 3/6 or 1/2, and the probability of rolling an odd number is also 1/2.
To find the probability that none of the outcomes is an even number, we need to calculate the probability of rolling an odd number on all three dice. Since the dice are rolled independently, we can multiply the probabilities together to get the overall probability:
P(rolling odd on one die) = [tex]1/2[/tex]
P(rolling odd on two dice) = [tex](1/2)^2 = 1/4[/tex]
P(rolling odd on three dice) =[tex](1/2)^3 = 1/8[/tex]
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Help me please….
it’s due tomorrow!!!
What is the value of g(x)=x^2+4x
The value of g(x) = x^2 + 4x depends on the value of x.
If, for example, x = 2, then:
g(2) = 2^2 + 4(2) = 4 + 8 = 12
If x = -3, then:
g(-3) = (-3)^2 + 4(-3) = 9 - 12 = -3
In general, we can evaluate g(x) for any value of x by substituting that value into the expression x^2 + 4x.
What is x if cos(15%) = x/5?
A. cos(15%)/5
B. cos(75°)
C. 5cos(15%)
D. cos(3⁰)
There are 410 calories in five ounces of a certain ice cream. How many calories are there in three pounds?
Hence, three pounds of ice cream contain 1968 calories as we can apply a ratio to determine how many calories are contained in 3 pounds .
what is unitary method ?Researchers utilise a method called the unitary approach to tackle proportional issues. It entails identifying a single product value that can be utilized to calculate the values of connected units. The method is predicated on the notion that if the value of one unit of even a quantity is known, then the value of any further unit of the same quantity can be determined. It is frequently employed in the computation of costs, rates, and dimensions.
given
Let's begin by converting 5 ounces to pounds:
A pound weighs 16 ounces.
5 ounces equals 5/16 pounds.
So, we can apply a ratio to determine how many calories are contained in 3 pounds:
(3 pounds/x calories)/5/16 pounds/410 calories
After finding x, we obtain:
3 pounds * 410 calories divided by 5/16 pounds equals 1968 calories.
Hence, three pounds of ice cream contain 1968 calories as we can apply a ratio to determine how many calories are contained in 3 pounds.
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Question # 9
Multiple Choice
Estimate the difference.
10{1}{9} - 5{14}{15}
5
4{1}{2}
4
5{1}{2}
To estimate the difference 10 1/9 - 5 14/15, we can round both numbers to the nearest whole number and it gives (c) 4
What is the Estimated difference?From the question, we have the following parameters that can be used in our computation:
10 1/9 - 5 14/15
10 1/9 is between 10 and 11, but closer to 10 so we can round it to 10.Similarly, 5 14/15 is between 5 and 6, but closer to 6 so we can round it to 6.Using the above as a guide, we have the following:
The estimated difference is: 10 - 6
Evaluate
The estimated difference is: 4
Therefore, the Estimated difference is 5
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4. Select Yes or No to indicate whether each ordered pair is a point of intersection
between the line y = x + 1 and the parabola y = x² +1
Ordered Pair
(0,1)
(1,2)
(2,0)
The point of intersections between y = x + 1 and y = x² +1 are (0, 1) and (1, 2)
Indicating the point of intersectionsFrom the question, we have the following parameters that can be used in our computation:
y = x + 1
y = x² +1
(0,1): Yes, (0,1) is a point of intersection between the line y = x + 1 and the parabola y = x² +1, since 1 = 0 + 1 = 0² + 1.
(1,2): No, (1,2) is a point of intersection between the line y = x + 1 and the parabola y = x² + 1, since 2 = 1 + 1 = 1² + 1.
(2,0): Yes, (2,0) is not a point of intersection between the line y = x + 1 and the parabola y = x² +1, since 0 = 2 + 1 ≠ 2² + 1.
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Question #5 I really don’t get it I would appreciate an answer it’s been an hour and no response
The value of x is 11 and the measure of the supplementary angle m∠N is equal to 143°. That is:
a. equation:37° + 3x = 180°
b. x = 11, m∠M = 37°, m∠N = 143°
What are supplementary anglesSupplementary angles are a pair of angles that add up to 180 degrees. In other words, if you have two angles that are adjacent (share a common vertex and side) and their measures add up to 180 degrees, then they are supplementary angles.
Given that m∠M and m∠N are supplementary then:
37° + 3x = 180°
3x = 180° - 37° {collect like terms}
3x = 143°
x = 143/3 {divide through by 3}
x = 11
m∠N = 3(11) = 143°.
Therefore, the value of x is 11 and the measure of the supplementary angle m∠N is equal to 143°.
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I need help with this. please
The only exponential growth equation is the third one:
y = 2*(4)^x
Which equation will produce a growth?A general exponential equation is:
y = A*(b)^x
if 0 < b < 1, then it is a decay.
if 1 < b, then it is a growth.
So any equation where the base, b, is larger than 1, represents a growth.
Then the correct option is the third one:
y = 2*(4)^x
All the other options are decays.
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rosanne makes 112 liters of strawberry lemonade. she pours 14 of a liter of lemonade into a thermos to take to the park. her brother drinks 25 of the remaining lemonade. how much lemonade does rosanne's brother drink?write your answer as a whole number, fraction, or mixed number. simplify any fractions. liters
If rosanne makes 112 liters of strawberry lemonade, she pours 14 of a liter of lemonade into a thermos to take to the park, rosanne's brother drink 25 liters of lemonade
To solve this problem, we can start by subtracting the 14 liters of lemonade that Rosanne poured into the thermos from the total 112 liters to get the amount of lemonade that remained:
112 - 14 = 98 liters
Next, we need to subtract the amount of lemonade that Rosanne's brother drank, which is 25 liters, from the remaining amount:
98 - 25 = 73 liters
Therefore, Rosanne's brother drank 25 liters of lemonade.
To check our answer, we can add the amount of lemonade in the thermos, the amount that Rosanne's brother drank, and the remaining amount to see if it adds up to the original 112 liters:
14 + 25 + 73 = 112
Since the sum is indeed equal to 112, we can be confident that our answer of 25 liters for the amount of lemonade that Rosanne's brother drank is correct.
In summary, to find the amount of lemonade that Rosanne's brother drank, we subtracted the amount that Rosanne poured into the thermos and the amount that her brother drank from the total amount of lemonade that Rosanne made. The answer is 25 liters.
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help Here is your graph of the points on the previous screen.
Connect the points in order to create polygon `ABCDEF`.
1.2 Enter the length of the segment betwee
The length of the given line AB is 6 units and the polygon ABCDEF has been created.
What is a graph?A graph is a mathematical structure made up of a collection of points called VERTICES and a set of lines connecting some pair of VERTICES that may or may not be empty.
There is a chance that the edges will be directed, or orientated.
If the lines are directed or undirected, respectively, they are referred to as ARCS or EDGES.
Make a sequence of bars on graph paper as an example of a graph.
So, in the given situation the polygon ABCDEF has been created:
(Refer to the graph attached below)
Now, the length of the side AB:
Count the units as follows which comes to 6 units.
Therefore, the length of the given line AB is 6 units and the polygon ABCDEF has been created.
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Correct question:
Help Here is your graph of the points on the previous screen.
Connect the points in order to create polygon `ABCDEF`.
1.2 Enter the length of the segment between A and B.
for a normal distribution, a positive value of z indicates that group of answer choices all the observations must have had positive values. the area corresponding to the z is either positive or negative. the sample mean is smaller than the population mean. the sample mean is larger than the population mean.
For a normal distribution ,a positive value of z simply means that the observation or sample mean is above the population mean this implies
none of the options provided is completely accurate.
All the observations must have had positive values,
It is not necessarily true for a positive value of z.
The value of z indicates how many standard deviations away from the mean a particular observation or sample mean is.
A positive value of z simply means that the observation or sample mean is above the population mean.
The area corresponding to the z is either positive or negative,
It is also not accurate.
The area under the normal curve corresponds to probabilities, not positive or negative values.
The area to the right of the mean corresponds to positive z-values, and the area to the left of the mean corresponds to negative z-values.
The sample mean is smaller than the population mean,
It is a possibility when the z-value is negative, indicating that the sample mean is below the population mean.
The sample mean is larger than the population mean,
It is a possibility when the z-value is positive, indicating that the sample mean is above the population mean.
Therefore, the sample mean can be either larger or smaller than the population mean depending on the direction of the z-value.
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two of the people do not like each other and do not want to sit side-by-side. now how many ways can the seven be seated together in a row? incorrect: your answer is incorrect.
In a group of seven people are attending the movies together. The number of possible ways for the seven be seated together in a row with two do not want to sit side-by-side are equal to the 4320.
Combinations : it is defined as the total number of possible arrangements of objects without think about the order. We have a group of seven people are attending the movies together. We have to determine the number of ways can the seven be seated together in a row, with conditions. Now, total number of people in group = 7 , so total number of possible arrangements = 7!
If two of seven insists on side-by-side, then total possible arrangements to seat in a row = 6! ( possible pairs = 3 in 7 people)
If two of the people do not like each other and want to sit side-by-side, then the possible arrangements to seat in a row = 6!
So, number of ways of arrengement seven persons in a row when two do not like each other and do not sit side-by-side = total possible arrangements - arrangements when sear together = 7! - 6!
= 6!( 7 - 1) = 6( 6!)
= 4320
Hence, required value is 4320.
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Complete question:
A group of seven people are attending the movies together. Refer to this solution to answer the following questions. Two of the people do not like each other and do not want to sit side-by-side. Now how many ways can the seven be seated together in a row?
I need some help bro
Answer:
first answer is correct
y = -1/2 x^2 + x + 1
Step-by-step explanation:
based on the graph, you can find two points
(2,1) and (0,1)
I input (2, 1) to each equation to check
y = -1/2 x^2 + x + 1 is correct
An employee's current annual gross wage is $63,900.
Part A: Calculate how much will be needed in retirement if the employee wants to have enough saved to live off 80% of the current annual gross wage and withdraw 4% the first year. Show all steps. (3 points)
Part B: The employee determines that they can contribute $600 per month to a retirement account with a 5.5% monthly compounded interest rate. Calculate the account balance if the employee plans to retire in 40 years. Show all steps. (3 points)
Part C: Using your values from Part A and Part B, calculate the difference between the employee's goal and the actual retirement account balance. Explain whether the employee will meet their retirement goal. (4 points)
Part A:The employee need to have $1,278,000 in retirement savings.
Part B:The retirement account balance after 40 years be $1,007,610.40.
Part C: The employee will not meet their retirement goal as the actual retirement account balance falls short by $270,389.60.
What is compounded interest?Compounded interest refers to the interest earned on both the principal amount and any accumulated interest over time.
It can lead to exponential growth in the value of an investment or debt.
Part A: To calculate how much the employee will need in retirement to live off 80% of their current annual gross wage and withdraw 4% the first year, we can use the following formula:
Retirement Savings Needed = (Annual Gross Wage × 0.8) / 0.04
Retirement Savings Needed = (63,900 × 0.8) / 0.04 = $1,278,000
Part B: To calculate the retirement account balance after 40 years of monthly contributions of $600 with a 5.5% monthly compounded interest rate, we can use the following formula:
Retirement Account Balance = $600 × (((1 + 0.055/12)⁴⁰ˣ¹²)-1) / (0.055/12)
Retirement Account Balance = $600 × (292.827 - 1) / (0.055/12) = $1,007,610.40
Part C: The difference between the employee's retirement goal and the actual retirement account balance is:
Retirement Savings Needed - Retirement Account Balance = $1,278,000 - $1,007,610.40 = $270,389.60
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factor completely show your work 9x^3 +36x^2 -x -4
Answer:
To factor 9x^3 + 36x^2 - x - 4, we can use the grouping method:
9x^3 + 36x^2 - x - 4
= (9x^3 - x) + (36x^2 - 4)
= x(9x^2 - 1) + 4(9x^2 - 1)
= (9x^2 - 1)(x + 4)
Therefore, the factored form of 9x^3 + 36x^2 - x - 4 is (9x^2 - 1)(x + 4).
In morning Emily studied 40 minutes for a math exam. Later that evening Emily studied for x more minutes. Write an equation that represents the total number of minutes y emily studied for the math exam
Equation that represents the total number of minutes y Emily studied for the math exam is y = 40 +x.
We can represent the total number of minutes Emily studied for the math exam using the equation,
y = 40 + x
Here, 'y' represents the total number of minutes Emily studied for the math exam, '40' represents the number of minutes she studied in the morning, and 'x' represents the number of minutes she studied later that evening.
By adding the number of minutes studied in the morning to the number of minutes studied later that evening, we can calculate the total number of minutes Emily studied for the math exam.
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a prescription for ear drops specifies that the patient put 2 drops in each ear twice a day for 9 days. approx how many millilitres of medication would be needed for this prescription? ( dosage×frequency×day supply )
approximately 3.6 milliliters of medication would be needed for this prescription.
To calculate the total milliliters of medication needed for this prescription, we will use the formula dosage × frequency × day supply.
In this case, the dosage is 2 drops per ear, the frequency is twice a day, and the day supply is 9 days.
First, let's find out how many drops are needed per day:
2 drops/ear × 2 ears = 4 drops
Since the frequency is twice a day, multiply this number by 2:
4 drops × 2 = 8 drops per day
Now, multiply the number of drops per day by the day supply (9 days):
8 drops/day × 9 days = 72 drops
Typically, 1 drop is approximately 0.05 millilitres. To find out how many millilitres of medication are needed, multiply the total number of drops by the volume of one drop:
72 drops × 0.05 mL/drop = 3.6 mL
So, approximately 3.6 millilitres of medication would be needed for this prescription.
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Approximately 1.8 milliliters of medication would be needed for this prescription.
To calculate how many milliliters of medication would be needed for this prescription, we need to use the following formula:
dosage × frequency × day supply.
The dosage is 2 drops, the frequency is twice a day, and the day supply is 9 days.
First, we need to convert drops to milliliters.
One drop is equal to approximately 0.05 milliliters.
2 drops would be equal to 0.1 milliliters.
Next, we can plug in the values into the formula:
[tex]0.1 milliliters/drop \times 2 drops/twice a day \times 9 days = 1.8 milliliters. [/tex]
The patient will be using exactly 2 drops in each ear for each dose, and that there will be no waste or spillage of the medication.
To determine how many milliliters of medicine would be required for this prescription:
dose frequency day supply.
The recommended dosage is 2 drops, twice day, with a 9-day supply.
We must first convert droplets to milliliters.
A drop is around 0.05 milliliters in volume. As a result, 2 drops are equivalent to 0.1 milliliters.
The values may then be entered into the formula as follows:
0.1 milliliters each drop times twice daily for nine days equals 1.8 milliliters.
As a result, this prescription would require about 1.8 milliliters of medicine.
It's crucial to note that this estimate is predicated on the patient applying precisely 2 drops to each ear for each dose.
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triple the difference of c and b, then subtract the result from a
Therefore, if you triple the difference of c and b (which is 2), and then subtract the result from a (which is 10), you get the answer of 4.
The mathematical expression for "triple the difference of c and b, then subtract the result from a" is:
a - 3(c - b)
In this expression, "c" and "b" are variables representing numerical values, and "a" is another variable representing a numerical value.
To evaluate this expression, you would need to substitute specific numerical values for "a", "b", and "c". For example, if a = 10, b = 5, and c = 7, then the expression would become:
10 - 3(7 - 5) = 10 - 6 = 4
Therefore, if you triple the difference of c and b (which is 2), and then subtract the result from a (which is 10), you get the answer of 4.
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Jaxon made 5% of his free throws over the season. If he shot 220 free throws, how many did he make?
Answer:
Jaxon made 11 free throws over the season.
Step-by-step explanation:
If he made 5% of his free throws, we know that he will make 5% of the total number of free throws he took, which is 220:
We multiply 220 by 0.05, or 5% to find out how many free throws he made:
220*0.05 = 11
After that, we now know that Jaxon made 11 of his free throws out of 220 over the course of the season, or 5%.
For each natural number , let ≔ {4, 4 + 1, … ,5}. Find ⋃[infinity] =1 and
⋂[infinity] =1 and prove your answers
The only possible elements that can appear in all the sets are 4 and 5. Since both 4 and 5 appear in each set, they must be in the intersection.
Based on the given set definition, we have:
A₁ = {4, 5}
A₂ = {4, 5, 6}
A₃ = {4, 5, 6, 7}
A₄ = {4, 5, 6, 7, 8}
and so on.
To find the union of all the sets, we need to find all the distinct elements that appear in at least one of the sets. We can do this by listing the elements in each set and removing duplicates:
⋃[infinity] =1 Aᵢ = {4, 5, 6, 7, 8, ...}
We can see that the union includes all the natural numbers greater than or equal to 4. We can prove this by showing that any natural number n greater than or equal to 4 is included in at least one of the sets. Indeed, for any such n, we can choose i = n - 3, which is a natural number since n is at least 4, and then we have n in Aᵢ.
To find the intersection of all the sets, we need to find the elements that appear in all the sets. We can do this by listing the elements in each set and finding the common elements:
⋂[infinity] =1 Aᵢ = {4, 5}
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patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. assume the time that patients wait is exponentially distributed. what is the probability a randomly selected patient will see the doctor within eleven minutes of the scheduled time?
The probability is approximately 0.446 or 44.6%.
Waiting time is exponentially distributed, we can use the cumulative distribution function (CDF) of the exponential distribution to solve this problem.
Let X be the waiting time in minutes.
X is exponentially distributed with a mean of 8 minutes.
The rate parameter λ is equal to 1/8.
The CDF of the exponential distribution is given by:
[tex]F(x) = 1 - e^(-\lambda x)[/tex]
The probability that a patient will see the doctor within eleven minutes of the scheduled time.
The value of F(11) - F(0).
[tex]F(11) = 1 - e^(-\lambda \times 11) = 1 - e^(-11/8)[/tex] ≈ 0.554
[tex]F(0) = 1 - e^(-\lambda \times 0) = 1 - e^0 = 1[/tex]
Therefore, the probability that a randomly selected patient will see the doctor within eleven minutes of the scheduled time is:
F(11) - F(0) ≈ 0.554 - 1 ≈ 0.446
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please help! and explain and show work of how you got the answer. I will mark you brainliest.
Answer: i dont know if this is right but i put a^4+b^4=c^4
Step-by-step explanation:
I have no idea tho
Chris is making a tabletop from some leftover tiles. He has 9 tiles that measure 3 1/8 inches long and 2 3/4 inches wide. What is the greatest area he can cover with these tiles?
Answer: The length of each tile is 3 1/8 inches and the width is 2 3/4 inches. To find the area of one tile, we need to multiply the length and width:
Area of one tile = (3 1/8) x (2 3/4)
= (25/8) x (11/4)
= 275/32
Therefore, the area of 9 tiles is:
Area of 9 tiles = 9 x (275/32)
= 2475/32
= 77.34 (rounded to two decimal places)
Since Chris cannot use a fraction of a tile, he can cover the greatest area by arranging the tiles in a rectangle with as close to equal length and width as possible. Let x be the length of the rectangle and y be the width. Then:
x = (number of tiles in length) x (length of one tile)
= 3 x 3 1/8
= 9 3/8
y = (number of tiles in width) x (width of one tile)
= 3 x 2 3/4
= 8 1/4
The area of the rectangle is:
Area of rectangle = x * y
= (9 3/8) * (8 1/4)
= 77.34 (rounded to two decimal places)
Therefore, the greatest area Chris can cover with these tiles is 77.34 square inches, using all 9 tiles arranged in a rectangle.
Step-by-step explanation:
PLEASE HELP DUE TODAY!!!!!!!
Consider the functions g(x) = 2x + 1 and h(x) = 2x + 2 for the domain 0 < x < 5
a. Without evaluating or graphing the functions, how do the ranges compare?
b. graph the 2 functions and describe each range over the given interval
Answer:
see the images and explanation
Step-by-step explanation:
for the graph:
the domain 0 < x < 5
the range for each functions:
g(x) = 2x + 1
g(x) = y , 1 < y < 11
h(x) = 2x + 2 , 2 < y < 12
A classmate of yours stated that a solid line is not a good representation of an arithmetic sequences. What logical assumption is your classmate using?
The classmate is not correct. A line is a good representation of an arithmetic sequence.
A line is a series of dots that represent each value of the sequence.
A line has the same slope as the common difference in the sequence.
An arithmetic sequence is a set of discrete values, whereas a line is a continuous set of values.
The logical assumption used is: An arithmetic sequence is a set of discrete values, whereas a line is a continuous set of values.
What is arithmetic sequence?An arithmetic sequence is a set of numbers where, with the exception of the first term, each term is obtained by adding a fixed constant to the term before it. Every pair of following terms in the sequence has the same fixed constant, which is known as the common difference. A1 stands for the first term in an arithmetic sequence, while an is used to represent the nth term.
A solid line symbolises continuous numbers, whereas the classmate's logical presumption is that an arithmetic series comprises of discrete values. This presumption is untrue, though, as a line can effectively represent an arithmetic series.
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Show your work please
Answer:
Perimeter:
[tex]2( \frac{2}{3} + \frac{1}{2} ) = 2( \frac{4}{6} + \frac{3}{6} ) = 2( \frac{7}{6} ) = \frac{7}{3} = 2 \frac{1}{3} [/tex]
The perimeter is 2 1/3 inches.
Area:
[tex] \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} [/tex]
The area is 1/3 of a square inch.
a frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13.$ a move sequence is a sequence of coordinates which correspond to valid moves, beginning with $0,$ and ending with $39.$ for example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. how many move sequences are possible for the frog?
There are 16 possible move sequences for the frog.
The possible moves that the frog can make.
The frog can move to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13.
Let's consider the first type of move, where the frog moves to the closest point with a greater integer coordinate that is a multiple of 3.
If the frog is at the point [tex]$n$[/tex], then the closest point with a greater integer coordinate that is a multiple of 3 is either [tex]$n+3$ or $n+6$[/tex].
Similarly, if the frog is at the point [tex]$n$[/tex] and wants to move to a multiple of 13, then the closest point with a greater integer coordinate that is a multiple of 13 is either[tex]$n+13$[/tex] or [tex]$n+26$[/tex].
Constructing move sequences.
The sequence starts with 0 and ends with 39.
The frog has two choices for its first move: either move to 3 or move to 13.
Let's assume that the frog moves to 3.
From there, the frog has two choices again:
move to 6 or move to 13. If the frog moves to 6, then it has two choices again: move to 9 or move to 13.
If it moves to 13, then it has two choices: move to 26 or move to 16.
We can continue this process until we reach 39.
We can create a tree diagram to help us keep track of the possible move sequences:
0
|
3 or 13
/ \
6 or 13 13 or 26
/ \ / \
9 13 16 26
/ \ / \ / \ / \
... ... ... ... ... ... ... ...
Each branch represents a different move sequence.
We can see that there are 16 possible move sequences in total.
We can also see that some sequences are longer than others.
The longest sequence has 7 moves, while the shortest sequence has only 2 moves.
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A person has a bag containing quarters and dimes. There are a total of 52 coins in the bag, and the total value of the coins is $8.35.
There are 23 quarters and 29 dimes in the bag.
How many quarters and dimes are in the bag?Let's denote the number of quarters as "q" and the number of dimes as "d".
We can set up a system of two equations based on the given information:
The total number of coins is 52:
q + d = 52
The total value of the coins is $8.35, where each quarter is worth $0.25 and each dime is worth $0.10:
0.25q + 0.10d = 8.35
To solve this system of equations, we can use substitution, elimination, or any other appropriate method.
Let's use elimination to solve this system of equations.
0.10(q + d) = 0.10(52)
25q + 10d = 835
25q + 10d = 835
0.25q + 0.10d = 8.35 (multiply by 100)
Now, let's multiply both sides of the first equation by 2 and the second equation by 8 to make the coefficients of "q" the same:
50q + 20d = 1670
200q + 80d = 6680
Now, let's subtract the first equation from the second equation to eliminate "q":
200q + 80d - (50q + 20d) = 6680 - 1670
150q + 60d = 5010
Now, let's divide both sides of the resulting equation by 30 to solve for "q":
150q/30 + 60d/30 = 5010/30
5q + 2d = 167
After trying different values, we find that when "q" = 23 and "d" = 29, the equation is satisfied and the condition "q + d = 52" is also satisfied. Therefore, there are 23 quarters and 29 dimes in the bag.
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