The reaccepts 10 integers from the user using an array and a loop is explained below.
To start, you need to declare an array variable that can store 10 integers. For example, you can declare an array variable called "numbers" as follows:
int numbers[10];
Next, you will use a loop to repeatedly ask the user to input an integer. The loop will run 10 times to receive 10 integers. You can use the for loop for this purpose.
Here is an example:
for (int i = 0; i < 10; i++) {
printf("Enter integer %d: ", i + 1);
scanf("%d", &numbers[i]);
}
In this code, the for loop starts with the value of i being 0. It runs until i is less than 10. After each iteration, i is incremented by 1.
After the loop finishes, all 10 integers received from the user are stored in the numbers array. You can access them using the index of the array.
In conclusion, to receive 10 integers from the user using an array and a loop, you need to declare an array variable, use a for loop to repeatedly ask the user for input, and store the input in the array using the scanf function.
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Pam’s Pet Supply Palooza buys bags of dog food (m) directly from the manufacturer at a 50% discount off the original market price. Pam then sells each bag of dog food at a price that is 3 times the amount that she paid for it.
Which expression correctly represents the cost of a bag of dog food to the customer in terms of (m), the original market price?
The expression that correctly represents the cost of a bag of dog food to the customer in terms of (m), the original market price is 1.5m
As per the given data Pam’s Pet Supply Palooza buys bags of dog food directly from the manufacturer.
Market Price = m
Got 50% discount off the original market price.
Discount = 50%
Pam then sells each bag of dog food at a price that is 3 times the amount that she paid for it.
Selling Price =3 × (Amount she paid)
If she gets a discount of 50 %, then she pays 50 % (100 - 50% ).
So, the amount she paid is:
Amount = 50 % × Market Price
Amount = [tex]$\frac{50}{100}[/tex] × m
Amount = 0.5 × m
Amount = 0.5 × m
She sells at three times that amount.
So, the selling price is:
Selling Price = 3 × 0.5 m
Selling Price = 1.5 m
Therefore the expression is 1.5 m.
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Use the drop-down menus to compare −2. 25 and −3.
please i need help!!!
-2.25 is larger when it compares to -3, because when you order negative numbers, the larger the number after the negative sign, the smaller the number actually is. This means -2.25 > -3
What is negative numbers?
On a number line, numbers always move from right to left and grow (become "more positive") in that direction. The numbers to the right are bigger than the numbers to the left, while the numbers to the left are smaller. Numbers with a minus sign before them are considered negative numbers. They could be fractions, decimals, or integers. Negative numbers include, for instance, -4, -15, -4/5, and -0.5. A negative number in mathematics represents the reverse. A negative number is one that is less than zero in the real number system. Negative numbers are frequently used to indicate how great a loss or a shortage is.
-2.5>-3 because the bigger a negative number is like -15,-18, etc the smaller it’s value actually gets so -2.5 is greater because it is closer to 0 on a number line.
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I need it quick pls almost done with home work
E = 18.1 h will be a proportional equation for e in terms of h that matches the context.
What is proportionality?the property of having suitable proportions in terms of size, number, degree, harshness, etc.: If a defensive action against an unfair attack results in the destruction that contravenes the proportionality criterion, it may even go far beyond a justifiable defense.
Given a table of data about total earnings and working hours,
Since its given that the data is proportional thus we can make an equation in terms of earning and working hours variable.
Let "e" be the earnings of Penelope and " h" be the total working hours
Thus,
e/h = 362/20
e = 18.1h
Therefore, a proportional equation for e in terms of h that matches the context will be E = 18.1h.
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Write an equation in point-slope form of the line that passes through the point (1.0) with slope -
An equation is
Answer:
Step-by-step explanation:
The equation of the line in point-slope form can be written as:
y - 0 = -1/4 (x - 1)
where (1, 0) is the point and -1/4 is the slope of the line.
A party planner organized a dinner party. The party planner recorded the height of the candlesticks over time and graphed the relationship.
graph with the x axis labeled time in hours and the y axis labeled height of candlestick in inches and a line going from the point 0 comma 9 through the point 2 comma 6
Find and interpret the slope and y-intercept in this real-world situation.
The slope is negative two thirds, and the y-intercept is 9. The candle starts at a height of 9 inches and decreases two thirds of an inch every hour.
The slope is negative three halves, and the y-intercept is 9. The candle starts at a height of 9 inches and decreases three halves of an inch every hour.
The slope is 9, and the y-intercept is negative two thirds. The candle starts at a height of two thirds of an inch and decreases 9 inches every hour.
The slope is 9, and the y-intercept is negative three halves. The candle starts at a height of three halves of an inch and decreases 9 inches every hour.
The meaning of the slope and of the intercept of the linear function are given as follows:
The slope is negative -3/2, and the y-intercept is 9. The candle starts at a height of 9 inches and decreases 3/2 of an inch every hour.
How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
The slope m represents the rate of change of the output variable relative to the input variable.The intercept b represents the value of y when the graph of the function touches of crosses the y-axis.The two points for the function in this problem are given as follows:
(0,9) and (2,6).
When x = 0, y = 9, hence the intercept b is given as follows:
b = 9.
Meaning that the initial value of the problem is of 9.
The slope is given as the change in y divided by the change in x, hence:
m = (6 - 9)/(2 - 0)
m = -3/2.
Meaning that the rate of change is a decrease of 3/2 units for each unit of the input.
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A container ha the following dimenion,120cm×300cm×430cm. How many litre of liquid can the box hold
The container that has the following dimensions 120cm×300cm×430cm can hold 15480 liters of liquid
What is volume?It is the space occupied by a body, it is calculated by multiplying its dimensions, for example: length, height and width.
Calculating the volume of the container we get the liter of liquid it can hold in cubic centimeters :
v = h*l*w
v = 120cm×300cm×430cm
v = 15480000 cubic centimeters
Converting the volume from cubic centimeters to liter we have:
1 liter is equal to 1000 cubic centimeters
x = 15480000 cubic centimeters * (1 liter / 1000 cubic centimeters)
x = 15480 liters
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Identify the x and y intercept for -4x+y=8
The x and y-intercept for -4x+y = 8 is (-2,0) and (0, 8)
What is the intercept?
The points where a line crosses an axis are known as the x-intercept and the y-intercept, respectively. The slope intercept formula y = mx + b is used when you know the slope of the line to be examined and the point given is also the y-intercept (0, b).
Here, we have
Given: -4x+y = 8
Let's start with the interception point with the x-axis. Here we know that y = 0, so we can put that in our original expression:
-4x = 8
x = -2
The function intercepts the x-axis at (-2, 0)
Let's continue with the y-axis. Here we know that x = 0, so let's put that into our expression/function:
y = 8
This gives us an interception point with a y-axis at (0, 8)
Hence, the x and y-intercept for -4x+y = 8 is (-2,0) and (0, 8)
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Factor the expression to fill in the blank.
48 + 96 = ? (1 + 2)
Answer:
48
Step-by-step explanation:
48+96 =144
x * (1+2) = 3x = 144
so x = 48
Mariano delivers newspaper. He always puts 3/4 of his weekly earnings in his savings account and then divides the rest equally into 3 piggy banks for spending at the snack shop, the arcade and the subway
a. Fraction of earnings that Mariano put into each piggy bank is 1/12
b. Mariano earns $28.8
Mariano divides his earnings into a saving account and 3 piggy banks.
First, we have to know the fraction that we divide in the 3 piggy banks.
Piggy banks =Total earning-Saving account
Saving account 3/4,
Total earnings is 1
to rest both terms we have to get the same denominator, so we can say that the total earning is 4/4
Piggy banks =4/4-3/4=( 4-3)/4=1/4
Piggy banks =1/4
Each piggy bank will be 1/3 of that 1/4
1/3 x 1/4=
For fraction multiplication, multiply the numerators and then multiply the denominators to get
1×1/3×4=1/12
This fraction cannot be reduced.
b. Mariano adds $2.40
1/12 is 2.40
We use cross-multiplication
a/b=c/d replacing (1/12)/2.40 = 1/ ?
So, the total earning (?) is 1 x 2.40/ (1/12)
= 2.40 x 12=$28.8
Complete question:
Mariano delivers newspapers. He always puts 3/4 of his weekly earnings in his savings account and then divides the rest equally into 3 piggy banks for spending at the snack shop, the arcade, and the subway.
a. What fraction of his earnings does Mariano put into each piggy bank?
b. If Mariano adds $2.40 to each piggy bank every week, how much does Mariano earn per week delivering papers?
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Mariano always puts 3/4 of his weekly earnings in his savings account and then 1/4 part divides into 3 piggy banks and in each it has 1/12 .
Let the total income of Mariano who delivers newspaper be x.
He puts 3/4(x) of his total income of his weekly earnings in his savings account.
The remaining income be x/4 and it is divided into 3 piggy banks.
so the income divides into each piggy bank will be 1/3(x/4) = x/12
so, therefore the amount that Mariano puts in each piggy bank is 1/12 part of his weekly income.
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Researchers are investigating the effectiveness of using a fungus to control the spread of an insect that destroys trees. The researchers will create four different concentrations of fungus mixtures: O milliliters per liter (ml/L), 1.25 ml/L, 2,5 ml/L, and 3.75 ml/L, An equal number of the insects will be placed into 20 individual containers. The group of insects in cach container will be sprayed with one of the four mixtures, and the researchers will record the number of insects that are still alive in each container one week after spraying (a) Identify the treatments, experimental units, and response variable of the experiment Treatments Experimental units: Response variable: (b) Does the experiment have a control group? Explain your answer, (c) Describe how the treatments can be randomly assigned to the experimental units so that each treatment has the same number of units.
(b) Yes, the experiment has a control group.
(c) The treatments can be randomly assigned to the experimental units by randomly selecting 20 containers and assigning each of the four treatments to 5 containers.
a) Treatments: Four different concentrations of fungus mixtures (0 ml/L, 1.25 ml/L, 2.5 ml/L, and 3.75 ml/L).
Experimental units: 20 individual containers with a group of insects.
Response variable: Number of insects that are still alive in each container one week after spraying.
(b) A control group is used to compare the results of the treatment group to a group that is not exposed to the treatment.
In this experiment, the control group can be the group of insects that are not exposed to any of the fungus mixtures.
c) The treatments can be randomly assigned to the experimental units by randomly selecting 20 containers and assigning each of the four treatments to 5 containers.
This way, each treatment has the same number of units and the assignment of treatments to units is random, which reduces the risk of bias in the experiment.
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The lateral height of a cone is 8 inches and the area of the base of the cone is 49π in². It requires 2.5 minutes to paint the cone. The area of the base is doubled. How long will it take to paint this cone if it can be painted at the same rate? Use π≈3.14. Enter your answer, rounded to the nearest tenth, in the box.
___ Min
(GIVEING BRAINLEST)
The time it take to paint this cone if it can be painted at the same rate is 4.21 minutes.
What is Total Surface area?The region that includes the base(s) and the curved portion is referred to as the total surface area. It is the overall area that the object's surface occupies. The total area of a shape with a curved base and surface is equal to the sum of the two areas.
Given:
Base area = 49π in²
h= 8 inches
So, area of a circle, A = πr^2
r = √(A/π).
r = √(49π/π)
r = 7 in.
Now, they doubled cone has a base area
= 2 x 49 in²
= 98 π in²
Then, r = √(98π/π)
r = 7√2 inch
Thus, the time taken
= TSA (R= 7) / TSA (r= 7√2) x2.5
= πR(l+ R) / πr(l+r)
= 14(4√2+7)π/105π x 2.5
= 4.21 min
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While playing a game of chance, Tarik rolled a regular six-sided die 100 times and noticed that it landed on the number 1 exactly 52 times. What could Tarik conclude about the die?
A: We are 90% confident that the true proportion of times the die would land on the number 1 is between 0.422 and 0.618. Since this includes 0.5, the die appears to be fair.
B: We are 90% confident that the true proportion of times the die would land on the number 1 is between 0.422 and 0.618. Since this is much greater than 1 out of 6, the die appears to be unfair.
C: We are 90% confident that the true proportion of times the die would land on the number 1 is between 0.438 and 0.602. Since this includes 0.5, the die appears to be fair.
D: We are 90% confident that the true proportion of times the die would land on the number 1 is between 0.438 and 0.602. Since this is much greater than 1 out of 6, the die appears to be unfair.
We are 90% confident that the true proportion of times the die would land on the number 1 is between 0.438 and 0.602. Since this includes 0.5, the die appears to be fair.
The correct option is C.
What is the probability?The Probability in mathematics is the possibility of an event in time. In simple words, how many times that incident is happening in any given time interval.
Given:
While playing a game of chance,
Tarik rolled a regular six-sided die 100 times and noticed that it landed on the number 1 exactly 52 times.
The probability P is = 52/100 = 0.52.
The lower limit = P - z√(P(1 - P)/n
The lower limit = 0.52 - 1.645√(0.52 x 0.48)/100
The lower limit ≈ 0.438.
The upper limit = P + z√(P(1 - P)/n
The upper limit = 0.52 + 1.645√(0.52 x 0.48)/100
The upper limit ≈ 0.602.
Therefore, the die would land on the number 1 is between 0.438 and 0.602.
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when u, v are nonzero vectors, then span{u, v} contains the line through u and the origin, as well as the line through v and the origin. true or fal
When u, v are nonzero vectors, then span{u, v} contains the line through u and the origin, as well as the line through v and the origin, this statement is false.
The span of two nonzero vectors, u and v, is the set of all linear combinations of u and v. This can be a line, a plane, or even the entire space, depending on the vectors u and v. In general, the span of two nonzero vectors does not contain the line through each vector and the origin, unless the two vectors are collinear (i.e. they lie on the same line).
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Fathi ha \$1. 10$1. 10dollar ign, 1, point, 10 in hi printing account. Each heet of paper he ue reduce hi printing account balance by \$0. 25$0. 25dollar ign, 0, point, 25. Fathi want to print out a PDF document that i 474747 page long. To ave paper, he decide to print on both ide of each heet and to print two page on each ide of the heet
Since Fathi has $1.10 in his printing account, he will have -$59342.28 after printing the 474747-page PDF document.
Let's calculate how much Fathi will spend on printing the 474747-page PDF document.
Since Fathi is printing 2 pages on each side of a sheet, he uses 1/2 sheet of paper per 2 pages. And since each sheet of paper reduces his printing account balance by $0.25, he will spend $0.25 per 2 pages.
So, the cost per page is $0.25 / 2 = $0.125.
Now, we can calculate the total cost of printing the 474747 pages by multiplying the cost per page by the number of pages:
$0.125 * 474747 = $59343.38
Since Fathi has $1.10 in his printing account, he will have $1.10 - $59343.38 = -$59342.28 after printing the 474747-page PDF document.
So, Fathi will not have enough funds in his printing account to print the document and will have to add more money to his account or find another way to print the document.
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Since Fathi has $1.10 in his printing account, he will have -$59342.28 after printing the 474747-page PDF document.
Let's calculate how much Fathi will spend on printing the 474747-page PDF document.
Since Fathi is printing 2 pages on each side of a sheet, he uses 1/2 sheet of paper per 2 pages. And since each sheet of paper reduces his printing account balance by $0.25, he will spend $0.25 per 2 pages.
So, the cost per page is $0.25 / 2 = $0.125.
Now, we can calculate the total cost of printing the 474747 pages by multiplying the cost per page by the number of pages:
$0.125 * 474747 = $59343.38
Since Fathi has $1.10 in his printing account, he will have $1.10 - $59343.38 = -$59342.28 after printing the 474747-page PDF document.
So, Fathi will not have enough funds in his printing account to print the document and will have to add more money to his account or find another way to print the document.
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22 miles in 20 minutes A < B > C = 38 miles in 30 minutes
It is found that 22 miles in 20 minutes is LESS
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
We have to compare the two rates:
=> 22 miles in 20 minutes
=> 38 miles in 30 minutes
Let's simplify each rate by finding the speed, which is distance divided by time.
That is:
Speed = distance/time
For the first one:
s = 22/20 = 1.1 milesl/min
For the second one:
s = 38/30 = 1.27 milesl/min
Since 1.1 is less than 1.27, so, 22 miles in 20 minutes is less than 38 miles in 30 minutes.
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The complete question is;
Is 22 miles in 20 minutes greater than, equal to or less than 38 miles in 30 minutes?
b. What is the value of the camper after 5 years? Round to the nearest dollar.
$13320 is the value of the camper after 5 years.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
m=23680-29600/3-2
=-5920
Now let us find the y intercept
37000=-5920(1)+b
b=37000+5920
b=42920
y=-5920(5)+42920
=-29600+42920
y=13320
Hence, $13320 is the value of the camper after 5 years.
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Question 1 (2 points)
Kala is financing a house for $123,500. She has to pay $300 plus 1. 05% for a brokerage fee.
How much are the mortgage brokerage fees? (2 points)
a$429. 68
b$1,296. 75
c$1,596. 75
d$2,152. 50
Total mortgage brokerage fees is $1,596.75. Thus, option c is correct.
What is the math formula for mortgage?These factors include the total amount you're borrowing from a bank, the interest rate for the loan, and the amount of time you have to pay back your mortgage in full. For your mortgage calc, you'll use the following equation: M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1].
Calculate H = P*J, this is your current monthly interest.
Calculate C = M - H, this is your monthly payment minus your monthly interest, so it is the amount of principal you pay for the month.
Calculate Q = P - C, this is the new balance of your principal of your loan.
P = the payment.
L = the loan value.
c = the period interest rate, which consits of dividing the APR as a decimal by the frequency of payments. ...
n = the total number of payments in the life of the loan (for monthly loan payments this is the loan term in years times twelve)
With given data:
123,500 x 1.05% =$1,296.75 percentage of the fee.
Total mortgage brokerage fees
$1,296.75 + $300 =$1,596.75
Thus, Total mortgage brokerage fees is $1,596.75. Thus, option c is correct.
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Which sequence of transformations produces A’B’C’ from ABC? a translation up 3 and then a 90 degree counterclockwise rotation about the origin a 90 degree counterclockwise rotation about the origin and then a translation up 3 a reflection over the x-axis and then a translation left 2 and down 1 a translation left 5 and down 2 and then a reflection over the x-axis
A reflection over the x-axis changes the signs of the y-coordinates of the points, and a translation left 2 and down 1 changes the position of the figure, but it does not change the orientation of the figure.
A 90 degree counterclockwise rotation about the origin followed by a translation up 3 produces A’B’C’ from ABC. A 90 degree counterclockwise rotation about the origin changes the coordinates of the points, but it does not change the relative positions of the points. The translation then moves the entire figure up 3 units along the y-axis. A reflection over the x-axis changes the signs of the y-coordinates of the points, and a translation left 2 and down 1 changes the position of the figure, but it does not change the orientation of the figure.
A translation left 5 and down 2 followed by a reflection over the x-axis changes the position and orientation of the figure, but it does not produce the same result as a 90 degree counterclockwise rotation about the origin followed by a translation up 3.
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how to find the speed of a roller coaster math ia
To find the speed of a roller coaster, measure the distance it travels in a certain amount of time, then divide that distance by the amount of time it took to travel.
To find the speed of a roller coaster, start by measuring the distance it travels in a certain amount of time. This can be done by using a measuring tape to measure the total length of the track. Then, use a stopwatch to measure the amount of time it takes for the coaster to complete one full lap of the track. Once you have both of these measurements, divide the total distance traveled by the amount of time it took to travel. This will give you the speed of the roller coaster, given in units such as meters per second.
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What is Rational Root Theorem Definition?
The Rational Root Theorem states that for a polynomial equation with integer coefficients to have a rational solution, that solution must be represented by the a factor of the constant (denoted by the variable p) divided by a factor of the leading coefficient (denoted by the variable q).
Define polynomial equation?An algebraic equation, also known as a polynomial equation, is a mathematical equation of the form P=0, where P is a polynomial with coefficients in some field, most commonly the field of rational numbers.Polynomial equations are equations formed with variables, exponents, and coefficients. It can have different exponents, with the higher one referred to as the equation's degree.A polynomial equation is a mathematical expression made up of variables, coefficients, and exponents. A polynomial function has only one independent variable.A polynomial equation is essentially a polynomial expression that has been equated to 0. 3x2 - 5 = 0 is a polynomial equation, just as 3x2 - 5 is a polynomial expression.To learn more about polynomial equation refer to:
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The Johnsons set aside money in their budget for food for their anniversary party. They spent $320 but had set aside 25% more than that. How much money did the Johnsons set aside in their budget for food for the party?
Answer:
$400
Step-by-step explanation:
25% = 0.25
320 times 0.25 = $80
$320 + $80 = $400
So, the Johnsons set aside $400 in their budget for food for the party.
I need the second answer please.
Answer: 40.75
Step-by-step explanation:
PLSSSS HELP IF YOU TURLY KNOW THISSS
Answer:
the Answer is x=5
Step-by-step explanation:
so all you do is 2+5 is 7 example say 2+x=4 what is 2+2=4
What is the length of the missing side? *NOTE: side lengths of 10 and 12.
Answer:
i think its 2 but i am not sure
help me please :(.......
Answer:
a) $2125
b) $2000
Step-by-step explanation:
In this problem, we are asked to solve for the manufacturing price of different printers if the printer company sells their printers at a price that makes them 25% profit, assuming there are no extra expenses.
We can solve this type of problem by manipulating ratios, like for example:
manufacturing price : sale price
100 : 125
This ratio represents the the sale price being 25% more than the manufacturing price.
a) We can equate the above ratio to the ratio of the manufacturing price (in this problem, $1700) to the sale price of the printer (represented by x).
100 : 125 = 1700 : x
This can be solved for x when the ratios are represented as fractions.
[tex]\dfrac{100}{125} = \dfrac{1700}{x}[/tex]
↓ simplify the fraction on the left side
[tex]\dfrac{4}{5} = \dfrac{1700}{x}[/tex]
↓ cross multiply
[tex]4x = 1700 \cdot 5[/tex]
↓ divide both sides by 4
[tex]x=\$ \, 2125[/tex]
So, the price that the customer would pay for a printer that the company bought for $1700 is $2125.
b) We can solve in the same way that we did for section a.
However, this time, the variable (y) represents the manufacturing price, NOT the sale price.
100 : 125 = y : 2500
↓ rewrite with fractions
[tex]\dfrac{100}{125} = \dfrac{y}{2500}[/tex]
↓ simplify the fraction on the left side
[tex]\dfrac{4}{5} = \dfrac{y}{2500}[/tex]
↓ cross multiply
[tex]5y = 2500 \cdot 4[/tex]
↓ divide both sides by 5
[tex]y = \$ \, 2000[/tex]
So, the price that the company would pay for a printer that a customer buys for $2500 is $2000.
Find the slope and the -intercept of the line y=5/2x-9
Answer:
The slope is 5/2 and the y-intercept is -9.
Step-by-step explanation:
You do not have to graph anything for this problem, as the slope and y-intercept are already in the problem.
In this problem it will be useful to recall the following properties of logarithms: log(xy)=log(x)+log(y) and log(xa)=alog(x).
Find the least integer k such that f(n) is O(nk) for each of the following functions:
f(n) = 18 + 28 + � + n8
f(n) = log(n!)
f(n) = log(nn) / (n9 + 1)
The least integer k such that f(n) is O(nk) for each of the following functions is k = 2.
log (n!) = ln(n!)
First we know ,
[tex]\lim_{n \to \infty} \frac{ln(n!)}{n^{k} } = L[/tex]
where L is a limited number.
then ,
[tex]\lim_{n \to \infty} \frac{ln(n!)}{n^{k} } = L \\\\\lim_{n \to \infty} \frac{ln(n+1)!-ln(n!)}{(n+1)^{k}- n^{k} } = L[/tex] Stolz's Formula
By solving the upper limit we get k =2
It can make
[tex]\lim_{n \to \infty} \frac{1}{k} \frac{ln(n+1)}{n^{(k-1)} } _[/tex]
[tex]\lim_{n \to \infty} \frac{1}{2} \frac{ln(n+1)}{n^{} } = 0[/tex]
The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain.
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Six Republicans and six Democrats orally cast their vote in the U.S. Senate. In how many orders can they vote they must alternate Republican, Democrat, Republican, Democrat, and so on, and if a Republican must always be the first to vote?
If Six Republicans and six Democrats orally cast their vote in the U.S. Senate , if they are alternate Republican, Democrat, Republican, Democrat, and so on, and if a Republican must always be the first to vote then the number of ways in which they can vote is 518400 ways .
the number of Republicans for voting is = 6 ;
the number of Democrats for voting is = 6 ;
so , total number of voters is = 12 ,
they have to vote alternately starting with a Republicans .
the first republican has to choose from = ⁶C₁ = 6 ;
the second republicans has to choose from the remaining 5 republicans , = ⁵C₁ = 5 ;
Similarly following the pattern ,
we get ;
Total chances of required way of casting votes ;
= 6×6×5×5×4×4×3×3×2×2×1×1
= (1×2×3×4×5×6)²
= (6!)² = (720)² = 518400 ways .
Therefore , the number of ways of voting is 518400 ways .
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Evaluate the integral. (5x cos(6x) dx To use the integration-by-parts formula ſu du = wv - fu du, we must choose one part of 5x cos(6x) dx to be u, with the rest becoming du.
The required integration-by-parts formula of given function is 5x (1/6) sin(6x) - (5/36) cos(6x) + C.
How to integrate a function by part formula?To evaluate the integral using integration-by-parts, we need to find two functions u and dv such that their derivative is equal to the integrated, and such that the anti-derivative of dv can be easily found. The integration-by-parts formula states that:
According to question:∫ uv = u(v) - ∫ v du
Let's choose u = 5x and dv = cos(6x) dx. Then, du = 5 dx and v = (1/6) sin(6x).
Using the integration-by-parts formula, we get:
∫ 5x cos(6x) dx = 5x (1/6) sin(6x) - ∫ (1/6) sin(6x) 5 dx
The antiderivative of (1/6) sin(6x) 5 is (5/36) cos(6x), so:
∫ 5x cos(6x) dx = 5x (1/6) sin(6x) - (5/36) cos(6x) + C,
where C is the constant of integration.
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determine the new limits of integration after the given substitution is made.
[tex]$\int_{-2}^{4} f(x)dx$[/tex] with the substitution [tex]$u = x^2$[/tex] . The new limits of integration are 0 and 8.
The substitution [tex]$u = x^2$[/tex] implies that [tex]$du = 2x dx$[/tex], so when [tex]x = -2$, $u = (-2)^2 = 4$[/tex]. Similarly, when [tex]x = 4$, $u = 4^2 = 16$[/tex]. Thus, the new limits of integration are 0 and 8.
To determine the new limits of integration, we must first find the values of u for the original limits of integration. Since we have [tex]$u = x^2$[/tex], when x = -2, we have [tex]$u = (-2)^2 = 4$[/tex]. Similarly, when x = 4, we have [tex]u = 4^2 = 16$.[/tex]Therefore, the new limits of integration are 0 and 8.
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