The number of selections that will contain no defective printers can be found using the nCr formula as well, but with n being the number of non-defective printers (34) and r being the number of items selected (2). So nCr = 34!/(2!(34-2)!) = 34!/(2!32!) = 5,832.
(a) There are 6,632 selections that can be made.
(b) There are 5,832 selections that will contain no defective printers.
(a)The number of selections can be found using the nCr formula: nCr = n!/(r!(n-r)!), where n is the total number of items (42) and r is the number of items selected (2). So nCr = 42!/(2!(42-2)!) = 42!/(2!40!) = 6,632.
(b) The number of selections that will contain no defective printers can be found using the nCr formula as well, but with n being the number of non-defective printers (34) and r being the number of items selected (2). So nCr = 34!/(2!(34-2)!) = 34!/(2!32!) = 5,832.
There are 6,632 selections that can be made when two printers are chosen from a shipment of 42, including 8 that are defective. Out of these selections, 5,832 will contain no defective printers.
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Given f(x) = x - 7 and g(x) = x2.
Find g(f(4)).
g(f(4)) means to first find f(4) and then use that result as the input for g(x).
So,
f(4) = 4 - 7 = -3
then we use the result, -3 as the input for g(x)
g(-3) = (-3)^2 = 9
Therefore, g(f(4)) = 9
tips for success read the assignment carefully and make sure you answer each part of the question or questions. check your work when you're done. your assignment jane is training for a triathlon. after swimming a few laps, she leaves the health club and bikes 16 miles south. she then runs 12 miles west. her trainer bikes from the health club to meet jane at the end of her run.
Jane continues her triathlon training by swimming, then biking 16 miles south and running 12 miles west. Her trainer meets her at the end of the run.
Jane has traveled a total of 28 miles - 16 miles biking and 12 miles running.
Jane is training for a triathlon and is putting in the work to reach her goal. After swimming a few laps in the health club, she bikes 16 miles south. She then runs 12 miles west, bringing her total distance to 28 miles. When she has finished her run, her trainer meets her to check on her progress and offer support for her continued success. Jane's hard work and dedication to her training is admirable and demonstrates her commitment to completing her triathlon. The support from her trainer will be invaluable in helping her reach her goal.
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Lisa is framing a rectangular painting. The length is three more than twice the width. She uses 30 inches of framing material. What is the length of the painting?
Answer:
11
Step-by-step explanation:
picture is too blurry, but here is the process to solve it
L = 3+2w
total length = 30 inches = 2L +2w
so 2w=30-2L
L= 3+2w = 3+30-2L
so 3L =33
L =11
5. Nicolet is fencing off a rectangular pen for her chickens. She has 80 feet of fencing and will use all of it. The area of the pen, A(x), is a function of its length, x, in feet.
The domain of the area function is given by the following interval:
(0, 40).
How to obtain the domain of the function?The domain of a function is the set composed by all the input values that the function accepts.
On the graph, the domain of the function is given by the values of x.
Considering the graph, the domain is given by the following interval:
(0, 40).
It is an open interval because if x = 0, then the length is of zero, and if x = 40, then the width is of zero, meaning that these dimensions are not possible.
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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
The postal service offers flat-rate shipping for priority mail in special boxes. Today, Jaden shipped 4 small boxes and 2 large boxes, which cost him $36 to ship. Meanwhile, Pamela shipped 3 small boxes and 9 large boxes, and paid $102. How much does it cost to ship these two sizes of box?
Shipping costs $
for a small box and $
for a large box.
The cost to ship the small boxes and large boxes are 4 and 10 dollars respectively.
How to represent system of equation?The postal service offers flat-rate shipping for priority mail in special boxes. Today, Jaden shipped 4 small boxes and 2 large boxes, which cost him $36 to ship. Meanwhile, Pamela shipped 3 small boxes and 9 large boxes, and paid $102.
The system of equation of represent the situation is as follows:
let
x = cost for small boxes
y = cost for large boxes
Hence,
4x + 2y = 36
3x + 9y = 102
Let's find how much used to ship the two sizes of box.
Multiply equation(i) by 4.5
18x + 9y = 162
3x + 9y = 102
Subtract the equations
15x = 60
x = 60 / 15
x = 4
Therefore,
2y = 36 - 4x
2y = 36 - 4(4)
2y = 36 - 16
2y = 20
y = 20 / 2
y = 10
Therefore,
cost for small boxes = 4 dollars
cost for large boxes = 10 dollars
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what is the y-intercept of the line passing through the point (9, -7) with a slope of -1/8
NO LINKS!!!
60. If a bond earns 5% interest per year compounded continuously, how many years will it take for an initial investment of $100 to $1000?
Answer:
46.05 years
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
Given:
A = $1000P = $100r = 5% = 0.05Substitute the given values into the continuous compounding formula and solve for t.
[tex]\implies 1000=100e^{0.05t}[/tex]
[tex]\implies 10=e^{0.05t}[/tex]
[tex]\implies \ln 10=\ln e^{0.05t}[/tex]
[tex]\implies \ln 10=0.05t\ln e[/tex]
[tex]\implies \ln 10=0.05t[/tex]
[tex]\implies t=\dfrac{1}{0.05}\ln 10[/tex]
[tex]\implies t=20\ln 10[/tex]
[tex]\implies t=46.0517018... \rm years[/tex]
Therefore, it will take 46.05 years for the initial investment of $100 to reach $1000.
The population (in millions) of Calcedonia as a function of time t (years) is P(1) = 55 . 20^005t. Determine the number of years it takes the population to double. (Give an exact answer. Use symbolic notation and fractions where needed.) years: Let g(t) = a2^kt. Determine the expression for g(t + 1/ķ) (for any positive constants a and k). (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the expression in terms of a, k, and t.
g(t+1/k)
Interpret the obtained results. - The population does not change significantly over time. - The population increases by 1/k after one year. - The population doubles after 1/k years - The population halves after 1/k years
Give the expression in terms of a, k, and t is g(t+1/k) = a2^(k(t+1/k)) = a2^kt * 2^(1/k)
The population doubles when P(t) = 2 * P(0), so we can set up the equation:
2 * 55 = 20^(5t)
Solving for t:
t = log(2)/(5 * log(20)) = log(2)/log(400)
The population doubles after approximately 0.173 years or 63.1 days.
g(t+1/k) = a2^(k(t+1/k)) = a2^kt * 2^(1/k)
Interpretation: g(t+1/k) is the population at time t+1/k, where a and k are positive constants. The population at t+1/k is equal to the population at time t multiplied by 2^(1/k). This means that if k is a large number, the population will change very little over time, if k is a small number, the population will change more quickly.
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A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 47 and 57 minutes. Find the probability that a given class period runs between 50.5 and 51 minutes?
The probability would be length of the interval [50.75, 51.25] divided by the length of the interval [47.0, 57.0] which is 0.05
Probability :Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Now we have given :-the lengths of her classes are uniformly distributed between 47 and 57 minutes.
so probability hat a given class period runs between 50.5 and 51 minutes=
(51-50.5)/(57-47)
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The probability would be length of the interval [50.75, 51.25] divided by the length of the interval [47.0, 57.0] which is 0.05
Probability :Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Now we have given :-the lengths of her classes are uniformly distributed between 47 and 57 minutes.
so probability hat a given class period runs between 50.5 and 51 minutes= (51-50.5)/(57-47) = 0.05
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Larry leaves home at 9:06 and runs at constant speed to the lamppost 200 m away. He reaches the lamppost at 9:09, immediately turns, and runs to the tree (another 1,000 m away). Larry arrives at the tree at 9:13.
What is Larry's average velocity, in m/min, during each of these two intervals. Find v1 and v2.
Larry ran at an average velocity of 66.7 m/min to the lamppost 200 m away and an average velocity of 250 m/min to the tree 1000 m away.
v1 = 200 m / 3 min = 66.7 m/min
v2 = 1000 m / 4 min = 250 m/min
Larry left his home at 9:06 and ran at a constant speed to the lamppost 200 m away. He arrived at the lamppost at 9:09. Immediately after arriving, he turned around and ran to a tree 1000 m away, arriving at 9:13. To find his average velocity during each interval, we can divide the distance traveled by the time it took to travel the distance. For the first interval, Larry traveled 200 m in 3 minutes, so his average velocity was 200 m divided by 3 min, or 66.7 m/min. For the second interval, Larry traveled 1000 m in 4 minutes, so his average velocity was 1000 m divided by 4 min, or 250 m/min.
Larry ran at an average velocity of 66.7 m/min to the lamppost 200 m away and an average velocity of 250 m/min to the tree 1000 m away.
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Find the area of the composite figure.
Is this correct?
When the indicated sides are multiplied by the figure's area, 44 is the result.
How can I find my location?An expression for the size of a planar layer contained within a two-dimensional figure, form, or plane is called an area. 2D shapes that can be drawn on a plane are included in layered shapes. B. A parallelogram, square, triangle, rectangle, trapezoid, and circle.
polygon:
A two-dimensional object made of only straight lines is a polygon. Triangles, hexagons, and pentagons are a few examples of polygons. The number of sides is indicated in the shape's name. For instance, a rectangle has four sides and a triangle has three.
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consider the following balls-and-bin game. we start with one black ball and one white ball in a bin. we repeatedly do the following: choose one ball from the bin uniformly at random, and then put the ball back in the bin with another ball of the same color. we repeat until there are n balls in the bin. let x
The number of white balls is equally likely to be any number between 1 and n − 1.
This problem can be solved by considering the number of ways to get a certain number of white balls in the bin. Since each ball can either be white or black, there are 2^n possible sequences of n balls in the bin.
Now consider the number of ways to get exactly k white balls in the bin. It can be done in the following way:
Choose k white balls out of n balls, which can be done in C(n, k) ways, and then arrange the k white balls and n-k black balls in (n-k + k)!/(n-k)!k! ways.
Therefore, the number of ways to get exactly k white balls in the bin is C(n, k)(n-k + k)!/(n-k)!k! = C(n, k)(n!)/(n-k)!k!.
Since each sequence of balls is equally likely, it follows that the number of white balls is equally likely to be any number between 1 and n - 1.
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--The given question is incomplete; the complete question is
"Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following: choose one ball from the bin uniformly at random, and then put the ball back in the bin with another ball of the same colour. We repeat until there are n balls in the bin. Show that the number of white balls is equally likely to be any number between 1 and n − 1."--
How many unique permutations are in “HAPPINESS”
Answer: There are 120 ways to arrange the letters.
Step-by-step explanation:
The number of unique permutations will be 90720.
To find unique permutations for any word, one needs to break the word into letters,
In the present case the word has 9 letters, out of which 'P' and 'H' are the repeating letters.
The basic method of finding unique permutations for a word is calculating the total number of words and doing its factorial.
for e.g. in the word "WORD"
the number of unique permutations is 4! = 24 unique combinations.
But for the given question the number of unique permutations will be as follows:-
9!/2!*2!= 90720
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the functions f and g are differntiable for all real numbers, explain why there must be a value c for 1 < c < 3
The reason behind the the functions f and g are differentiable for all real numbers for the limit 1 < c < 3 is the limitation of constant term.
Function in the math is known as a kind of rule that, for one input, it gives you one output.
Here we the functions f and g are differentiable for all real numbers
And here we need to explain why there must be a value c for 1 < c < 3.
As we all know that in the differentiable function that is defined as a function that can be approximated locally by a linear function.
Where the general form of the function is written as,
=> [f(c + h) − f(c) h ] = f (c).
Here we have also know that the domain of f is the set of points c ∈ (a, b) for which this limit exists.
Here we know that the value of a = 1 and b = 3.
So due to the limitation the function will only work on the given range.
Complete Question:
The function f and g are differentiable for all real numbers and the value of g'(3)=k, where k is a constant. The function h is given by h(x) = f(g(x))- g(f(x)). then explain the reason why there must be a value c for 1 < c < 3.
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in this exercise we will write code to rotate the cube. you will need to implement a loop asking the user what direction to rotate in roobikrunner, and implement left(), right(), up() and down() methods in roobikcube that will rotate the cube in the given direction. for example, if the cube looks like: w b o g y r and you rotated left, the cube would now look like: w o g r y b
Here is an example of how the code to rotate the cube could be implemented in Python:
class RooCube:
def __init__(self):
self.colors = ["w", "b", "o", "g", "y", "r"]
def left(self):
self.colors = [self.colors[1], self.colors[5], self.colors[2], self.colors[3], self.colors[0], self.colors[4]]
def right(self):
self.colors = [self.colors[4], self.colors[0], self.colors[2], self.colors[3], self.colors[5], self.colors[1]]
def up(self):
self.colors = [self.colors[3], self.colors[1], self.colors[0], self.colors[5], self.colors[4], self.colors[2]]
def down(self):
self.colors = [self.colors[2], self.colors[1], self.colors[5], self.colors[0], self.colors[4], self.colors[3]]
class RooRunner:
def __init__(self):
self.cube = RooCube()
def rotate(self):
direction = input("What direction would you like to rotate (left, right, up, down)? ")
if direction == "left":
self.cube.left()
elif direction == "right":
self.cube.right()
elif direction == "up":
self.cube
This would give us the required result.
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There are 56 trees growing in the park, 4 of which are giant oaks. How many times are there fewer big oaks than other trees?
The percentage of oak trees in park is 7.14% and there are 14 times more trees in the park than giant oak trees.
What is percentage?
A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion therefore refers to a component per hundred.
The total number of trees in a park is = 56
The number of giant oak trees in the park is = 4
To find the percentage of oak trees, use the formula -
(Number of Oak Trees/Total number of trees) × 100
Substitute the values into the formula -
(4/56) × 100
(1/14) × 100
7.14%
The number of times (x) that bigger oaks is lesser in comparison to other trees can be found using the arithmetic operation of division.
The equation is -
4x = 56
x = 56/4
x = 14
Therefore, there are 14 times more trees.
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6[tex]6\sqrt{24} \alphax^{4}[/tex]
Write in standard form
The equation y-4 = 5/3 ( x+3). can be written in standard form as 3y-5x = 15
How do we write equation in standard form?A linear equation is the equation in which the highest power of it's variable is 1. A linear equation can be with a variable or more. y-4 = 5/3 ( x+3) is an example of linear equation with two variables.
The standard form of writing the linear equation is in the form Ax+By = C
Where C Is the constant and A and B are the coefficient of the variable x and y.
y -4 = 5/3 (x+3)
opening the parentheses
y = (5/3)x +5
y - (5/3)× = 5
multiply both sides by 3
3y -5x = 15
therefore the standard form of writing y-4 = 5/3 ( x+3) is 3y -5x = 15
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The figure below represents a rectangular prism net. The rectangles labeled B represent both bases of the prism. What is the difference between the lateral and total surface area of the prism? (Hint: What are the only rectangles you are finding the area of?)
A : 156in
B : 128in
C : 30in
D: 48in
The total surface area and lateral area have a difference of 48 square inches. (Correct choice: D)
How to difference lateral area and total surface area in a right prism
In this problem we find a right prism with a rectangular base, of which we must determine its lateral area and its total surface area, both in cubic inches.
Lateral area is the sum of all faces A and C (four rectangles), whereas total surface is the sum of lateral area and all faces B (two rectangles). Area formula for rectangles is shown below:
A = w · l
Where:
w - Width, in inches.l - Length, in inches.A - Area, in square inches.Now we proceed to determine the lateral area and the total surface area of the prism:
Lateral area
A' = 2 · (3 in) · (5 in) + 2 · (8 in) · (5 in)
A' = 30 in² + 80 in²
A' = 110 in²
Total surface area
A = A' + 2 · (8 in) · (3 in)
A = 110 in² + 48 in²
A = 158 in²
And the difference between total surface area and lateral area is equal to 48 in².
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-
The gas mileages (in milles per gallon) for 35 cars are shown in the frequency distribution. Approximate the mean of the frequency
distribution
The approximate mean of the frequency distribution is
(Round to one decimal place as needed.)
Gas Mileage
(in miles per gallon)
27-30
31-34
35-38
39-42
Frequency
12
14
3
6
Answer:
The gas mileages (in miles per gallon) for 35 cars are shown in the frequency distribution. This means that the gas mileage of 35 cars has been recorded and grouped into categories. The categories are 27-30, 31-34, 35-38, and 39-42. The frequency is the number of cars in each category. For example, there are 12 cars that have a gas mileage between 27 and 30 miles per gallon. Approximating the mean of the frequency distribution means finding the average gas mileage of all the 35 cars. To do this, you need to add the gas mileages of all the 35 cars and divide the sum by 35. The approximate mean of the frequency distribution is 33.4 miles per gallon. An analogy that can help you understand this concept is that of a classroom full of students. If you want to calculate the average height of the students, you would need to add up the heights of all the students and divide the sum by the total number of students. The same concept applies to the frequency distribution.
GIVE ME BRAINIESTSix years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ?
A. 11/P + 6
B. P/11 +6
C. 17 - P/6
D. 17/P
E. 11.5P
option A is correct-Ben is 11/P + 6 old now in terms of P .
Six years ago,
Anita was P times as old as Ben was.
Let Ben's age now be B.
let Anita's age now be A.
linear equation in two variables is as follows,
(A-6) = P(B-6)
we know A = 17,put the value of A,
17 - 6 = P(B - 6)
11 = P(B - 6)
11/P = B- 6
B = 11/P + 6
therefore,
Ben is 11/P + 6 old now in terms of P .
linear equation
A linear equation in two variables is of the form Ax + By + C = 0, in which A and B are the coefficients, C is a constant term, and x and y are the two variables, each with a degree of 1.
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Solve the inequality, then identify the graph of the solution -3x-3 less than or equal to 6
The solution of the inequality -3x - 3 ≤ 6 is x ≥ -3 represented by (D).
What is an equation?An equation is an expression that shows the relationship between numbers and variables. Equations are either linear, quadratic, cubic and so on
Inequality is an expression that shows the non equal comparison between two or more numbers and variables.
Number line contains number placed at regular intervals on a straight line.
Given the inequality:
-3x - 3 ≤ 6
adding 3 to both sides:
-3x - 3 + 3 ≤ 6 + 3
-3x ≤ 9
Dividing by -3:
x ≥ -3
The solution is D. It a solid dark circle beginning from -3 to the right.
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April borrows $21000 at an interest rate of 3% to purchase a new automobile. At what rate (in dollars per year) must she pay back the loan, if the loan must be paid off in 5 years?
Answer:
$18,000
rate of 5 %
Step-by-step explanation:
Answer:
7000 per year. 3000 interest
Step-by-step explanation:
If x equals 44, find the measures of angles 1 and 2
The measurement of ∠1 = 136° and ∠2 = x = 44°, using Vertically opposed angles.
What are Vertically opposed angles?Vertical angles are produced when two lines intersect at a point. They are always on an even playing field. Or, to put it another way, whenever two lines cross or intersect, four angles are formed. As we can see, two opposed angles are equal and are referred to as vertical angles.
They are also referred to as "vertically opposed angles" because of their position directly opposite one another.
∠x = 44
∠2 = x = 44
∠1 + ∠2 = 180
∠1 = 180 - 44 = 136°
Therefore, the angles are 36° and 144°.
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List the sample space for rolling a fair eight-sided die.
A. S = {1}
B. S = {8}
C. S = {1, 2, 3, 4, 5, 6}
D. S = {1, 2, 3, 4, 5, 6, 7, 8}
The sample space for rolling a fair eight-sided die is S = {1, 2, 3, 4, 5, 6, 7, 8}; the correct option is D
What is probability?Probability is a way to gauge how likely something is to happen. According to the probability formula, the likelihood that an event will occur is equal to the proportion of positive outcomes to all outcomes. The probability that an event will occur P(E) is equal to the ratio of favorable outcomes to total outcomes. The likelihood of an event occurring might range from 0 to 1. A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results.
Given;
An eight-sided die,
The sample space=?
Now, when eight sides die is tossed there will be 8 sample spaces that are;
1, 2, 3, 4, 5, 6, 7, 8,
Hence, S = {1, 2, 3, 4, 5, 6, 7, 8}
Therefore, the probability sample space will be S = {1, 2, 3, 4, 5, 6, 7, 8}
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what is an explicit formula for 9, 3, 1, 1/3
An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
What is Geometric Progression ?
Geometric Progression can be defined as the sequence of numbers in which the common ratio between any consecutive numbers is same.
When you say, "each term in the sequence is obtained by multiplying -1/3 to the previous term," that is the recursive definition.
An explicit definition can give a value for the nth item in a sequence by simply using the value of n:
an = -27 * (-1/3)n-1
(more generally written as: an = a1 * (-1/3)^n-1 )
if n = 1
we get,
an = -27 *1 = -27
if n=2
we get,
an = 9
and if n=3
we get,
an = -3
similarly if n=4
than an= 1
Therefore, An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
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Which expression is equivalent to 20y² + 5y?
O 5y(4y - 1)
O-5y(4y + 1)
O 5y(4y + 1)
O-5y(4y - 1)
Answer:
[tex]5y\left(4y+1\right)[/tex]Step-by-step explanation:
[tex]20y^2+5y[/tex]
[tex]20yy+5y[/tex]
[tex]5\cdot \:4yy+5\cdot \:1\cdot \:y[/tex]
[tex]5y\left(4y+1\right)[/tex]
Step-by-step explanation:
5y(4y + 1)
= 20y × y + 5y × 1
= 20y² + 5y
NB: BODMAS
What is the mean with the outlier of the following data set 20 17 19 22 18 17 5
According to the given information in the question the mean is 16.71.
What does the statistical definition of mean mean?A data set's mean (average) is calculated by summing all of the numbers in the set, then reducing by the total number of values within the set. The median is the middle when a data set is sorted from least to greatest. The value that pops up most frequently in a data set is its mode.
How are mean and median determined?The data set's values must first be summed together before being divided by the total number of values to determine the mean. List the data set's values in numerical order to get the median, then note the value that sits in the middle of a list.
mean = (20 + 17 + 19 + 22 + 18 + 17 + 5) / 7
mean = 16.71.
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which of the following are correct converstions between a teragram (tg) and a gram? select any that apply. group of answer choices 1 tg
3 teragrams is equal to 3 million grams.
The formula to convert teragrams (tg) to grams (g) is: grams = teragrams x 1000000000000. For example, to convert 15 teragrams to grams, we multiply 15 by 1000000000000, which equals 1.5e+13 or 150000000000000 grams. To convert grams to teragrams, we reverse the formula by dividing grams by 1000000000000. For example, to convert 1500000000000 grams to teragrams, we divide 1500000000000 by 1000000000000, which equals 1.5 teragrams.1 tg (teragram) is equal to 1000000 g (grams).
To calculate how many grams are in a teragram, you can use the following formula:
Grams = Teragrams x 1,000,000
For example, let's say you have 3 teragrams. To find out how many grams this is, you would use the formula above:
Grams = 3 tg x 1,000,000
Grams = 3,000,000 g
In other words, 3 teragrams is equal to 3 million grams.
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consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. given that the volume of this set is $\displaystyle {{m n\pi}\over p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m n p$.
The solution of m+n+p = 462+40+3 = 505.
The set is divided into several parts: the large 3x4x5 parallelepiped, 6 external parallelepipeds that share a face with the large parallelepiped and have a height of 1, 1/8 spheres (one at each vertex of the large parallelepiped), and 1/4 cylinders connecting each adjacent pair of spheres.
The parallelepiped has a volume of $3 times 4 times 5 = 60 cubic units.
The external parallelepipeds have a volume of $2(3 times 4 times 1)+2(3 times 5 times 1)+2(4 times 5 times 1)=94$.
Each of the 1/8 spheres has a radius of 1. Their combined volume is [tex]\frac{4}{3} \pi[/tex].
There are 12 of the 1/4 cylinders, allowing for the formation of 3 complete cylinders. Their volumes are 3[tex]\pi[/tex], 4[tex]\pi[/tex], and 5[tex]\pi[/tex], totaling 12[tex]\pi[/tex].
The total volume of these parts is 60+94+ [tex]\frac{4}{3} \pi[/tex] +12[tex]\pi[/tex] = [tex]\frac{462 + 40\pi }{3}[/tex]. Thus, the solution is m+n+p = 462+40+3 = 505.
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