The horizontal distance from the point where it was launched is; Option C: 7.14 feet
How to solve quadratic model of projectile functions?
Horizontal distance is defined as the distance between two points measured at a zero percent slope.
We are told that the path of the marshmallow can be modeled by the quadratic equation;
y = −0.07x² + x + 2.2
The maximum point will occur when dy/dx = 0
Thus;
dy/dx = -0.14x + 1
At dy/dx = 0,
-0.14x + 1 = 0
1 = 0.14x
x = 7.14
This value of x is the x-coordinate of the vertex and it represents the horizontal distance from the point where it was launched.
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given triangle ABC find the values of x and y. In your final answer, include all of your calculations.
Answer:
Step-by-step explanation:
X = ?
y = ?
The big triangle ABC is a right triangle, we can use Pythagoras theorem to find side y, then use SOH CAH TOA to find angles :)
also, there is a special rule for a triangle set up like this one
2CD = BD
for this triangle then,
2CD = 9
CD = 4.5
X = 13.5
now use Pythagoras to find y
[tex]13.5^{2}[/tex] = [tex]6^{2}[/tex] + [tex]y^{2}[/tex]
182.25 -36 = [tex]y^{2}[/tex]
[tex]\sqrt{146.25}[/tex] = y
12.093386 = y
now that you have all 3 sides of the big triangle, use which ever trig function you are most comfy with to find angle B or C, i'll find C
CAH = Cos(C) = Adj / Hyp
Cos(C) = 6 / 13.5
arcCos(Cos(C)) = arcCos(6/13.5)
C = 63.6122°
now find B
180 = 90 + 63.6122 + B
26.3877° = B
got it?
The diagram shows 3 points, A, B and C, on a circle, centre O.
AC is a diameter of the circle.
B
Angle BAO is aº and angle BCO is yº
What is the size of angle ABC?
Answer:
90 degrees
Step-by-step explanation:
the centre angle of the triangle is 180°.
ABC is its circumference angle, that is half the centre one
36) The table below shows the growth rate of a vine over several weeks:
Week
1
3
6
9
11
Length (inches)
5
6
9
11
14
What is the average rate of change from weeks 3 to week 9?
A) 0.83
B) 0.9
C) 1.2
D) 1.1
0.83 is the average rate change of length of vine from weeks 3 to 9.
What is an average?In mathematics, the average value of a set of integers is the middle value, which is determined by dividing the sum of all values by the number of values.
When we need to get the average of a set of data, we add all the values and then divide this sum by the number of values.
Calculation:The length of the plant at the end of the third week is 6 inches.
At the conclusion of week nine, the length of the vine is 11 inches.
So the change in the length of the plant is 11-6 = 5 inches.
The change in weeks is 9-3 = 6 weeks.
We know that average growth = [tex]\frac{change in x}{ change in y}[/tex] = Δx÷Δy
∴ change in length/ change in time = [tex]\frac{5}{6}[/tex] = 0.833
0.83 is the average rate change of length of vine from weeks 3 to 9.
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(b) If X = {2,3,5) and R is a relation on X where "a R b" means "a = b" for each a EX ,b Ex 1 Write R and represent it by an arrow diagram. 2 Show that R is a function, then find its range. m
The arrow diagram is
2 3 5
| | |
v v v
2 3 5
To prove that R is a function, we must show that "a R b" exists for each element an in X. R is a function since the only element b in X that fulfills the relation is a.
The range of R is X = {2, 3, 5}.
What is the range?Generally, The relation R on X is defined as "a R b" if and only if "a = b" for all a,b in X. This means that for any element a in X, the only element b in X that satisfies the relation is a itself. We can represent this relation using an arrow diagram as follows:
2 3 5
| | |
v v v
2 3 5
To show that R is a function, we need to demonstrate that for each element a in X, there is exactly one element b in X such that "a R b". In this case, we have already shown that the only element b in X that satisfies the relation is a itself, so R is a function.
The range of the function R is the set of all possible values of b for which "a R b" is true. In this case, the range of R is the set X itself, because for any element a in X, the only element b in X that satisfies the relation is a itself. Therefore, the range of R is X = {2, 3, 5}.
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Can we use AAA congruence criteria to make two congruent triangle give reason?
The sides of a triangle with all respective angles equal, i.e. AAA condition, may or may not be equal. Only if two triangles with the same respective angles are identical would the congruence stay true.
Knowing only angle-angle-angle (AAA) does not work because it can produce similar but not congruent triangles. When you're trying to determine if two triangles are congruent,
there are 4 shortcuts that will work. Because there are 6 corresponding parts 3 angles and 3 sides, you don't need to know all of them. If the three angles (AAA) between two triangles are congruent, that does not imply that the triangles must be congruent. They have the same shape (and can be considered comparable), but we don't know how big they are.
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17. (4 points) On the set of axes below, draw the graph of the equation y = -3/4x +3.
(2 points) is the point (3,2) a solution to the equation?
Explain your answer based on the graph drawn.
(3,2) is not a solution because it does not satisfy the equation and the line on the graph does not pass through it.
How to find the solution to an equation?
We perform the same operations on both sides to ensure that the equality of both expressions is maintained.
Solving equations generally entails determining the values of the variables used in the equation for which the equation under consideration is true.
Given that:
The graph of the equation is y = -3/4x + 3.
To find if the point (3, 2) a solution to the equation.
substitute the point to to satisfy the equation;
y = -3/4x + 3
y = -3/4(3) + 3
y = -9 / 4 + 3
y = 3/3 = 1
Hence, (3,2) isn’t a solution because it doesn't satisfy the equation and also the line on the graph doesn’t go through it .
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C. P. =Rs 50,s.p.=Rs 65
Answer: Profit = Rs 15, Profit %= 30%
Step-by-step explanation: When S.P.( Selling Price) is greater than C.P.(Cost Price), there is a Profit.
Profit= S.P-C.P
Hence Rs 15 is the profit.
To calculate Profit %, we use the formula Profit ÷ C.P. × 100%
Hence, the Profit % is 30%.
Can someone help me with this question?
Consider the mathematical sentence 2 = 2.
Is there any variable or constant that you could add to or subtract from both sides, or
multiply or divide both sides by using the Properties of Equality to make this true sentence false? Choose variables and constants to create new mathematical sentences to justify your conclusion.
If a statement is true, then it cannot be falsified no matter what.
A true statement will yield a true statement.
A false statement can yield a true or false statement.
The answer to your question is no, we cannot do anything to make the statement "2=2" false
Eight people of various ages were polled and asked to estimate the number of cds they had bought in the previous year. The following table contains the collected data. The variable x represents the age. The variable y represents the number of cds. Age number of cds 18 12 20 17 24 10 25 6 28 4 30 4 32 2 34 3 using the data in the table, choose which set of axes and which set of ordered pairs should be used in making a scatter plot of the data.
The ordered pairs of the data that should be used to plot the points on the scatter plot are (18, 12), (20, 17), (24, 10), (25, 6), (28, 4), (30, 4), (32, 2), and (34, 3).
The appropriate set of axes to make a scatter plot of the data is a horizontal x-axis and a vertical y-axis. The horizontal x-axis would represent the age of the people while the vertical y-axis would represent the number of CDs bought. The ordered pairs of the data that should be used to plot the points on the scatter plot are (18, 12), (20, 17), (24, 10), (25, 6), (28, 4), (30, 4), (32, 2), and (34, 3).
A scatter plot is a graph that is used to represent the relationship between two variables. It is a graphical representation of data that shows the relationship between two variables by plotting a set of ordered pairs on a coordinate plane. The ordered pairs of the data are represented by a mark, usually a dot, on the graph. The horizontal x-axis is usually used to represent the independent variable and the vertical y-axis is used to represent the dependent variable. In this case, the x-axis is the age of the people and the y-axis is the number of CDs purchased.
A scatter plot can be used to find patterns in data and to identify relationships between two variables. By plotting the ordered pairs of the data on the coordinate plane, the scatter plot will show any correlations or trends that are present in the data. For example, if there is a positive correlation between age and the number of CDs purchased, the scatter plot will show a line of points that goes up and to the right. On the other hand, if there is a negative correlation between the two variables, the scatter plot will show a line of points that goes down and to the right. By plotting the data points on a scatter plot, it is possible to identify any patterns or trends that are present in the data.
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a ferry will safely accommodate 66 tons of passenger cars. assume that the mean weight of a passenger car is 1.7 tons with standard deviation 0.5 tons. if a random sample of 36 cars are loaded onto the ferry, what is the probability that the maximum safe weight will be exceeded? hint! think about the average safe weight g
There is a 4.5% (or 0.045) probability that the maximum safe weight will be surpassed.
What is probability?The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true.
The probability is computed by dividing the total number of possible outcomes by the number of possible ways the event could occur.
Probability and odds are two distinct ideas.
Odds are calculated by dividing the likelihood of an event by the likelihood that it won't.
So, we know that;
Average u = 1.8 tons
r = 0.5 for the standard deviation
n = 35 data points
68 tons or more would be the combined weight of the 35 automobiles. Each car's average weight should be greater than:
X = 68/35 = 1.943
After that, we can utilize that to calculate the z value.
Z = (X - u)/(r/√n)
Z = (1.943 - 1.8)/(0.5/√35)
Z = 1.692
We must determine the p-value at Z > 1.692 in order to evaluate the likelihood:
P(Z > 1.692) = 0.0453
Therefore, there is a 4.5% (or 0.045) probability that the maximum safe weight will be surpassed.
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What is the sum of roots of 3x² 10x 8 0?
The sum of the roots of 3x² 10x 8 0 is -2.
The given equation, 3x² 10x 8 0, is a quadratic equation with two roots. To find the sum of the roots of this equation, we must first solve the equation. To do this, we use the quadratic formula, which states that the roots of the equation are x = [-b ± √(b² - 4ac)]/2a. Here, a = 3, b = 10 and c = 8.
Therefore, the roots of the equation are x = [-10 ± √(10² - 4(3)(8))]/2(3). This simplifies to x = [-10 ± √(-4)]/6. Since the square root of a negative number is not a real number, the roots of the equation are not real numbers, and the sum of the roots is undefined. Therefore, the sum of the roots of 3x² 10x 8 0 is -2.
x = [-10 ± √(10² - 4(3)(8))]/2(3)
x = [-10 ± √(-4)]/6
x = [-10 ± √(-4)]/6
Since the square root of a negative number is not a real number, the roots of the equation are not real numbers, and the sum of the roots is undefined.
Therefore, the sum of the roots of 3x² 10x 8 0 is -2.
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pls help me with this problem
Answer:
Step-by-step explanation:
Hi
Answer:
The mean price for this sample is $1,385.00
Step-by-step explanation:
⭐What is the "mean" of a sample?
The mean of a sample is another word for the average of said sampleYou add up all the values and divide that sum by the number of values presentTo find the mean for this data, we need to divide the sum of the prices for the computers bought by the total number of computers bought.
Sum of the prices for the computers:
[tex]14(1400)+12(1050)+5(2150)[/tex]
[tex]= 19,600 + 12,600 + 10,750[/tex]
[tex]= 42,950[/tex]
Total number of computers bought:
[tex]14+12+5[/tex]
[tex]= 31[/tex]
Mean price:
[tex]\frac{42,950}{31}[/tex]
[tex]= 1,385.00[/tex]
Annie scored 29 out of 35 marks in a test.
What percentage of the test did Annie get correct?
Give your answer to 1 decimal place.
HELP!!!!
Drag numbers to complete the table for missing values of x, x2, and x3
Here, the value of x = 20, x² = 400 and x³ = 8000.
What is Factorization ?
The breaking or breakdown of one entity into a product of another entity, or factors, which when multiplied together yield the original number, is known as factorization or factoring in mathematics.
We've x³ = 8000 ...............equation 1
we can factor it as follows :
8000 = 2×2×2×10×10×10
8000 = 2³ × 10³
8000 = 20³ .............................equation 2
Now, By comparing equation 1 and 2,
We get, x = 20.
So, x² = 20 × 20 = 400.
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What is a 5th root of unity?
The fifth root of unity is given by Z = 1, e^(2πi/5) ,e^(-2πi/5), e^(4πi/5), e^(-4πi/5).
As given in the question,
Using the Euler Identity ,
e^(πi) = -1
Calculate the nth root of Unity so that it can be used to calculate for any value of n:
Zⁿ = 1
⇒Z = (1)^(1/n)
Apply Euler's Formula we get,
Z = cos (2kπ + 0)/n + i sin(2kπ + 0 )/n
⇒Z = cos ( 2kπ / n) + isin( 2kπ / n)
Using Euler' identity we can write this as :
Z = e^(2kπ/n)
For k = 0 , 1 , 2 , 3..... (n-1)
Now substitute n = 5 to get the value of 5th root of unity we have
Z = e^(2kπ/5)
When k = 0 , 1, 2, 3, 4
k = 0 ⇒ Z = 1
k= 1 ⇒ Z = e^(2πi/5)
k = 2 ⇒ Z = e^(4πi/5)
k= 3 ⇒ Z = e^(-2πi/5)
k = 4 ⇒ Z = e^(-4πi/5)
Therefore, the value of 5th root of unity is equal to Z = 1, e^(2πi/5) ,e^(-2πi/5), e^(4πi/5), e^(-4πi/5).
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What is the coefficient of the variable in the expression 2 − 5x − 4 8?
The coefficient of the variable x in the expression 2-5x-4+8 is -5
Given that,
The expression is 2-5x-4+8 which can be simplified as -5x+6
To find : The coefficient of the variable x in the expression
The coefficient of x in 2-5x-4+8 is -5
A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6
Coefficient is the number that is beside the variable in the equation.
As we can see that the term that is in multiplication with x is -5.
Hence, the coefficient of x in the equation is -5.
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How do you calculate maximum value?
Therefore , the solution of the problem of equation is you can calculate maximum value of a equation by max = c - (b2 / 4a).
What is the equation?
Equations are created by connecting two expressions with the same sign to form a mathematical statement. An example is the equation
3x - 5 = 17. We may ascertain that the variable x has a value of 4 by resolving this equation.
According to the given question:
You can use the formula max = c - (b2 / 4a) to determine the maximum value if you are given the formula y = ax2 + bx + c.
The greatest value is k in the equation y = a(x-h)2 + k when the a component is negative.
Therefore , the solution of the problem is you can calculate maximum value is by max = c - (b2 / 4a).
Complete question:
How do you calculate maximum value of a equation?
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find a and b if the point p (6,0) and Q 3 2 lie on the graph of ax+ by
Answer:
a=b=0
Step-by-step explanation:
y=(ax)/b so now plug in the points
first 0=6a/b
next 2=3a/b
0=6a
2b=3a
a,b =0
You probably put the wrong problem but this is what I got.
5x - 4y = -12 I really need help please
Answer:
To solve the equation 5x - 4y = -12, we can first get all of the terms containing x on one side of the equation and all of the terms containing y on the other side. To do this, we need to add 4y to both sides of the equation. This gives us 5x = -12 + 4y. Then, we can divide both sides of the equation by 5 to get x = (-12 + 4y)/5. This is the value of x that makes the equation true.
Step-by-step explanation:
Answer: I= 3÷y-5x÷12
solve for y
Step-by-step explanation: 5x−4y=−12I
Swap sides so that all variable terms are on the left-hand side.
−12I=5x−4y
Divide both sides by −12.
-12I/-12= 5x−4y/-12
Dividing by −12 undoes the multiplication by −12.
I=-12I/-12= 5x−4y/-12
Divide 5x−4y by −12
I=3÷y-5x÷12
What is a word expression example?
A word expression example is "as busy as a bee." This phrase is used to describe someone who is very busy and active.
The Hard Work of a Busy Bee: Exploring the Benefits of an Active LifestyleThe phrase “as busy as a bee” is often used to describe someone who is very active and busy. This phrase is a testament to the hard work that bees have to do every day in order to ensure the survival of their hive. Similarly, people who live an active lifestyle can reap the same benefits of a hard-working bee. From physical health benefits to improved mental health, living an active lifestyle can have a profound impact on one’s overall well-being.
When it comes to physical health, engaging in regular physical activity can reduce the risk of chronic diseases, such as heart disease, stroke, and diabetes. Additionally, regular physical activity can help to improve physical fitness and boost energy levels. Not only does this make it easier for individuals to perform everyday tasks, but it can also help to improve one’s mood and reduce feelings of stress and anxiety. In addition to physical health benefits, an active lifestyle can also promote improved mental health. Exercise can help to reduce symptoms of depression, as well as improve one’s self-esteem and overall sense of well-being.
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Which of the following is not a quadratic equation * x²+ 4x 11 X² X² 4x 5x² 90 2x X² X² 5?
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. So , x + 4x 11 cannot be solved as a quadratic equation.
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The expression x + 4x 11 does not fit this form, and therefore is not a quadratic equation. In the equation, x is an unknown variable, and 4x 11 is a linear expression. Since this expression does not contain the variable squared, it is not a quadratic equation.
x + 4x 11 cannot be solved as a quadratic equation.
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Mrs. Jain went for lunch with her friends at a restaurant which offers her a 20% discount
The total amount she needs to pay is ₹6396
What is discount?A reduction made from the gross amount or value of something.
Given that, Mrs. Jain goes to lunch with her friends at a restaurant which offers her a 20% discount. The cost of the food that they order is 6,480. Given that there is a service charge of 5% and GST is at 18%
Discount Offered is 20%, so price after discount will be-
= ₹6480-₹(20% of 6480)
= ₹5200
Service Charge she has to pay:
5% of ₹5200
= ₹260-Answer
Total Amount of Bill:
= ₹5200+Service Charge+GST
= ₹5200+₹260+₹(18% of ₹5200)
= ₹(5200+260+936)
= ₹6396
Hence, The total amount she needs to pay is ₹6396
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find sin2theta and cos2theta and tan2theta if cos theta is 1/3 and theta is less than 360 but greater than 270
someone help!!
Answer:
sin2theta
= (4root2)/9
cos2theta = -7/9
tan2theta
= (4root2)/-7
Step-by-step explanation:
see image.
Draw a picture to help see your angles and the triangle trigonometry. Use Pythagorean theorem and triangle trig to find sintheta and tantheta. Use Double Angle Formulas to find the answers to the question. See image.
The value of the sin 2θ, cos 2θ, and tan 2θ will be - 0.628, -0.778, and 0.808, respectively.
What is trigonometry?Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.
The trigonometric equation is given below.
cos θ = 1/3
cos θ = 0.3333
θ = 289.47°
The value of the sine of angle 2θ will be given as,
sin 2θ = sin (2 × 289.47°)
sin 2θ = sin 578.94°
sin 2θ = -0.628
The value of the cosine of angle 2θ will be given as,
cos 2θ = cos (2 × 289.47°)
cos 2θ = cos 578.94°
cos 2θ = -0.778
The value of the tangent of angle 2θ will be given as,
tan 2θ = tan (2 × 289.47°)
tan 2θ = tan 578.94°
tan 2θ = 0.808
The value of the sin 2θ, cos 2θ, and tan 2θ will be - 0.628, -0.778, and 0.808, respectively.
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Determine whether the following equence i an arithmetic or geometric progreion. Give a reaon for your anwer. 100p,50p,25p,
Answer:
Geometric.
Step-by-step explanation:
It is Geometric because there is a common ratio between the terms.
50p/100p = 1/2
25p/50p = 1/2
The common ratio is 1/2.
Each term is obtained by multiplying by 1/2, so the next term in this progression is 25p * 1/2 = 12.5p.
a group of seven people are attending the movies together. (a) two of the seven insist on sitting side-by-side. in how many ways can the seven be seated together in a row? (b) 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?
The probability of 2 people insists on sitting side by side is 2/7. In the next, the probability of 2 people insists on sitting side by side is 2/5
What is probability?The possibility of occurring a particular events is called probability.
it is expressed in ratio of two data. The probability has no unit to measure.
how to calculate probability?Number of students =7, among them 2 are insists on sitting side by side
now probability to sit in such a way = 2/7
again, 2 students will never sit side by side so the number became 5.
probability to sit side by side = 2/5
hence, the result is 2/7 and 2/5
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A rectangular lawn has an area of 126 square meters. Surrounding the entire lawn is a flower border that is 4 meters wide. The area of the flower border alone (not including the lawn) is 264 square meters. A sprinkler is installed in the middle of the lawn that sprays water in a circle. Rounded to the nearest 10th of a meter, what is the minimum spraying radius necessary in order for the sprinkler to water both the lawn AND the entire flower border?
[10 points]
Answer:
To solve this problem, we can first use the area of the flower border to calculate the perimeter of the flower border. Since the flower border is 4 meters wide and has an area of 264 square meters, we know that it has a perimeter of 264 / 4 = 66 meters.
Next, we can use the perimeter of the flower border to calculate the dimensions of the lawn. Since the total perimeter of the lawn and flower border is 66 meters and the width of the flower border is 4 meters, we know that the length and width of the lawn must be 31 meters and 5 meters, respectively.
Finally, we can use the dimensions of the lawn to calculate the radius of the sprinkler. Since the lawn is a rectangle with a length of 31 meters and a width of 5 meters, we know that the center of the lawn is located 15.5 meters from each of the longer sides and 2.5 meters from each of the shorter sides. Therefore, the minimum radius of the sprinkler must be 15.5 meters in order to reach the entire flower border.
Rounded to the nearest tenth of a meter, the minimum spraying radius necessary is 15.5 meters.
What is median in 7th grade math?
The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.
The data set's median is the value in the center. Put all the numbers in ascending or descending order, and then choose the middle number to determine the median. There will only be one middle integer if the number of data points is odd. If the number of data points is even, you must choose the middle two numbers, add them together, and divide by two. Your median will be that value.
Example: The median of 4, 1, and 7 is 4 because when the numbers are put in order (1,4,7)the number 4 is in the middle.
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How do I solve 8(4x+3) ≥ −1+7x
If all possible rearrangements of the word REVISIONS are sorted alphabetically, the n-th word is VISIONERS. What is the remainder when n is divided by 100?
Answer:
75
Step-by-step explanation:
You want the remainder when n is divided by 100, where VISIONERS is the n-th word in the alphabetically-ordered list of the permutations of REVISIONS.
Lexicographical orderThe letters of the word REVISIONS are, in alphabetic order, are EIINORSSV. We notice there are two each of the letters I and S.
When the permutations of the letters of REVISIONS are written in lexicographical order, the list will first have all of the permutations beginning with E, then those beginning with I, and so on until the permutations beginning with S are exhausted.
Counting permutationsTo find the rank (number n) of the first permutation beginning with V, we must count the number of permutations ahead of it in the list. This will be the sum of the numbers of permutations beginning with E, I, N, and so on through S.
The straightforward way to do this is to consider the letters individually, and the number of permutations associated with each one. For example, the letters following E will be the 8 letters IINORSSV, which have two repeat pairs. The number of permutations of these 8 letters is 8!/(2!·2!) = 10080.
The next leading letter in the list is I, which can be followed by any of the letters EINORSSV, for a total of 8!/2! = 20160 permutations.
The letter N will be the next leading letter, and it will have a number of permutations equal to the number for the letter E, 10080.
At this point, we have exhausted 4 of the 8 letters that must precede V in the ordered list, and have counted 10080 +20160 +10080 = 40320 permultations. We notice this is, in effect, 10080 permutations per letter, the number of permutations of the letters not including V.
If we were to work through the remaining letters (including the repeated S), we would find the total to be 8·10080 = 80640 permutations before any starting with V.
Effectively, our formula for the number of added permutations preceding words matching up to the k-th letter of our target word is ...
p(9-k)!/q
where p is the number of letters in the (remaining) ordered letter list preceding the k-th letter of VISIONERS, and q is the factor that compensates for repeated letters among those remaining. "q" is the product of the factorials of the number of times a letter is repeated. For 2 letters, each repeated once, q = 2!·2! = 4.
Application of the formulaV is the 9th letter of EIINORSSV; there are 8 letters preceding, including 2 repeated twice. k=1, so there are 8·(9-1)!/(2!·2!) = 80640 permutations preceding ones starting with V.
I is the 2nd letter of EIINORSS; there is 1 letter preceding, including 2 repeated twice. k=2, so there are 1·(9-2)!/(2!·2!) = 1260 permutations preceding ones starting with VI.
S is the 6th letter of EINORSS; there are 5 letters preceding, including 1 repeated twice. k=3, so there are 5·(9-3)!/2! = 1800 permutations preceding ones starting with VIS.
I is the 2nd letter of EINORS; there is 1 letter preceding, with no repeats. k=4, so there are 1·(9-4)! = 120 permutations preceding ones starting with VISI.
O is the 3rd letter of ENORS; there are 2 letters preceding. k=5, so there are 2·(9-5)! = 48 permutations preceding ones starting with VISIO.
N is the 2nd letter of ENRS; there is 1 letter preceding. k=6, so there are 1·(9-6)! = 6 permutations preceding ones starting with VISION.
We note that the remaining letters, ERS, are in alphabetical order, so there are no additional permutations preceding VISIONERS.
RankThen the number of permutations preceding VISIONERS is ...
80640 +1260 +1800 +120 +48 +6 = 83874
and the rank of VISIONERS is 83875.
RemainderThe remainder when this is divided by 100 is 75.
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Additional comment
The usual counting method goes into more detail than the one described here. For problems of reasonable size, it is often easier to count the permutations generated ahead of the one of interest using a "next permutation" function.
In the attachment, we generated them all, and searched for the one of interest.
b. a new ground-approach radar system approved by the federal aviation administration is being considered as a means of reducing congestion. under this system, planes can be processed at a constant rate of 15 per hour (i.e., the variance is zero). what would be the expected number of airplanes circling the airport, waiting in queue for clearance to land, if this system were to be used?
The expected number of airplanes circling the airport, waiting in queue for clearance to land, if this new radar system were to be used is zero.
The new radar system is being considered as a means of reducing congestion, which implies that there is currently an issue with congestion at the airport. One way to reduce congestion is to process planes at a faster rate, which is exactly what the new radar system aims to do.
The statement "planes can be processed at a constant rate of 15 per hour (i.e., the variance is zero)" indicates that the system is able to process planes at a consistent rate of 15 per hour, with no variation in this rate. This means that there should be no planes waiting in queue for clearance to land, as they can all be processed at the constant rate of 15 per hour. Therefore, the expected number of airplanes circling the airport and waiting in queue for clearance to land is zero.
Learn more about Constant Rate here:
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