The equation of line is y = 5x + 3 , where the slope is m = 5
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
Let the first point be P ( 0 , 3 )
Let the second point be Q ( 5 , 28 )
Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values in the equation , we get
Slope m = ( 28 - 3 ) / ( 5 - 0 )
Slope m = 25 / 5 = 5
Now , the equation of line is y - y₁ = m ( x - x₁ )
Substituting the values in the equation , we get
y - 3 = 5 ( x - 0 )
On simplifying the equation , we get
y - 3 = 5x
Adding 3 on both sides of the equation , we get
y = 5x + 3
Hence , the equation of line is y = 5x + 3
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All races are more than 175 miles.
True
False
Answer:
False,
Step-by-step explanation:
There is not enough information given.
The given statement - "All the races are 175 miles long" is true.
What is function?A function is a relation between a dependent and independent variable.
Mathematically, we can write → y = f(x) = ax + b.
Given is to find whether all races are more than 175 miles or not.
All the races are 175 miles long. The given statement is true.
Therefore, the given statement - "All the races are 175 miles long" is true.
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Ling and her friends are planning a trip to the Misty Falls water park. The employee Ling spoke to on the phone said the total cost for all 4 of them would be $76. This includes an entrance ticket and a $6 tube rental fee for each person. Which equation can you use to find c, the cost of an entrance ticket?
The equation can you use to find c, the cost of an entrance ticket is x=13.
What is a linear equation?A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant.
Given that;
Total cost of 4 friends= $76
The cost of entrance ticket= $6
Now,
Based on the given conditions, formulate:
4 x (x+6)=76
Reduce the greatest common factor on both sides of the equation
x+6=19
Rearrange unknown terms to the left side of the equation:
x=19-6
Calculate the sum or difference:
x=13
Therefore, the equation will be x=13
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Determine the value of n that makes the polynomial a perfect square trinomial. Then factor as the square of a binomial. Express numbers as integers or simplified fractions
The value of n in the expression is 16
How to determine the value of nFrom the question, we have the following parameters that can be used in our computation:
y^2 + 8y + n
Take the coefficient of y
So, we have
k = 8
Divide by 2
k/2 = 4
Square both sides
(k/2)^2 = 16
This means that
n = 16
Hence, the value of n is 16
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Consider matrices A and B. What is the product of these matrices, of AB
The solution is, the product of these matrices, of
AB = [tex]\left[\begin{array}{ccc}1&-9&-14\\10&0&-15\\\end{array}\right][/tex][2×3]
What is Matrix?In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns.
here, we have,
matrix A = [tex]\left[\begin{array}{ccc}-1&4\\0&5\\\end{array}\right][/tex][2×2]
matrix B = [tex]\left[\begin{array}{ccc}7&9&2\\2&0&-3\\\end{array}\right][/tex][2×3]
As, no. of column of A = no. of column of B
so, the product of AB = [tex]\left[\begin{array}{ccc}1&-9&-14\\10&0&-15\\\end{array}\right][/tex][2×3]
Hence, The solution is, the product of these matrices, of
AB = [tex]\left[\begin{array}{ccc}1&-9&-14\\10&0&-15\\\end{array}\right][/tex][2×3]
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Answer:
C) [tex]\left[\begin{array}{ccc}1&-9&-14\\10&0&-15\\\end{array}\right][/tex] is the answer
Step-by-step explanation:
Rules of Multiplication of Matrices:-
To find the product of matrices the columns of A should be equal to the number of rows of B If AB is defined BA is not be defined If A and B are square matrices of the same order then both AB and BA are definedMatrix multiplication
A= [tex]\left[\begin{array}{ccc}-1&4\\0&5\end{array}\right][/tex] 2x2 B=[tex]\left[\begin{array}{ccc}7&9&2\\2&0&-3\end{array}\right][/tex]2x3
AB will be of 2x3 order
AB11= -1x7+4x2=1
AB12= -1x9+4x0=-9
AB13= -1x2+3x(-3)= -14
AB21= 0x7+5x2=10
AB22= 0x9+5x0=0
AB23= 0x2+5x(-3)=-15
AB=[tex]\left[\begin{array}{ccc}1&-9&-14\\10&0&-15\end{array}\right][/tex]
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f(x) = x² + 6x-2
g(x) = x - 6
The composite function of f(x) and g(x) is given as follows:
f(g(x)) = x² - 6x - 2.
What is the composite function of f(x) and g(x)?The composite function of f(x) and g(x) is given by the following rule:
(f ∘ g)(x) = f(g(x)).
It means that the output of the inside function serves as the input for the outside function.
The functions for this problem are given as follows:
f(x) = x² + 6x - 2.g(x) = x - 6.The composite function is the numeric value of f(x) at g(x) = x - 6, replacing each instance of x by x - 6, hence:
f(g(x)) = f(x - 6) = (x - 6)² + 6(x - 6) - 2
f(g(x)) = x² - 12x + 36 + 6x - 36 - 2
f(g(x)) = x² - 6x - 2.
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For the given corporate bond, whose annual simple interest rate is provided, find the semiannual simple interest
payment and the total interest earned over the life of the bond. Assume 365 days in a year.
$4900 Company A, 30-year bond, 6.268%
Answer:
the semiannual interest payment is $153.77 and the total interest earned over the life of the bond is $9,226.20.
Step-by-step explanation:
To find the semiannual simple interest payment, we need to divide the annual interest rate by 2, since there are 2 semiannual periods in a year:
Semiannual interest rate = Annual interest rate / 2
= 6.268% / 2
= 3.134%
To find the semiannual interest payment, we multiply the face value of the bond by the semiannual interest rate:
Semiannual interest payment = Face value x Semiannual interest rate
= $4900 x 3.134%
= $153.77 (rounded to two decimal places)
To find the total interest earned over the life of the bond, we need to multiply the semiannual interest payment by the number of semiannual periods in the bond's life. Since the bond has a 30-year term and 2 semiannual periods in a year, the bond has a total of 60 semiannual periods:
Total interest earned = Semiannual interest payment x Number of semiannual periods
= $153.77 x 60
= $9,226.20 (rounded to two decimal places)
Therefore, the semiannual interest payment is $153.77 and the total interest earned over the life of the bond is $9,226.20.
FLIP WITH ICON IN TOP RIGHT!
Jane is in charge of planning a reception for 2200 people. She is trying to decide which snacks to buy. She has asked a random sample of people who are coming to the reception what their favorite snack is. Here are the results.
Of the 2200 people attending the reception, Jane's survey revealed that the top favorite snack among the guests is popcorn, with 35% of people selecting it as their favorite.
What is number?Number is a mathematical entity used to represent a computer magnitude it can be symbol or a combination of simple you should in order to take quantity such as the two, five, seven, three or eight number are used to verify the context including counting measuring and computing.
The second most popular snack was pretzels, at 25%, followed by chips at 20%, crackers at 15%, and nuts at 5%.
With these results, Jane has decided to purchase popcorn, pretzels, chips, and crackers for the reception. She will also include a few bowls of nuts as a healthier option. To make sure that everyone has enough snacks to enjoy, Jane has calculated that she will need to buy approximately 1 pound of popcorn, 1 pound of pretzels, 2 pounds of chips, and 2 pounds of crackers, plus 5 pounds of nuts.
With these snacks, Jane is confident that all 2200 people attending the reception will be able to enjoy their favorite snacks. She is also confident that the snacks will be enough to feed everyone and that there won't be any shortages. Jane is sure that her careful planning and selection of snacks will make the reception a success.
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A retail associate at a department
store earns $950 an hour and 3%
commission on all sales. If he
worked 30 hours for the week and
sold $2.500 worth of merchandise.
what was the total amount he
earned?
Answer:
Step-by-step explanation:
950 times 30 hours = 28,500
28500 + 75= 28,575
75 is 3% of 2500 the total amount of sales he made
He earned 28,500 in his 30 hour week.
How do Dr. Livesey and Mr. Trelawney react to the map in chapters 4-6 of Treasure Island?
A. They are giddy and filled with delight.
B. They are confused and filled with wonder.
C. They are unimpressed and filled with doubt that it is real.
D. They are frightened and filled with terror.
In sections 4-6 of Treasure Island, C. describes how Dr. Livesey and Mr. Trelawney respond to the chart. People are disappointed and overwhelmed with skepticism about its authenticity.
In the novel Treasure Island, who is Dr. Livesey?Throughout Treasure Island, Dr. Livesey, a physician and judge, acts as the narrative's only sane figure. He is bold and straightforward enough already to start standing up to his pals and also the most infamous thieves.
here, we have,
What else does Dr. Livesey look like in Treasure Island?
Dr. Livesey describes oneself via deeds rather than being characterized by Stevenson. He possesses the traits that outweigh the cleverness and cruelty of his fearsome foe Silver: intelligence, bravery, and coolness.
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A company makes two models of television.
Model A has a rectangular screen that measures 44cm by 32cm.
Model B has a larger screen with these measurements increased in the ratio 5:4
a)work out the measurements of a larger screen.
b)find the fraction model A screen area÷model B screen area in its simplest form.
If a company makes two models of television.
a The measurements of the larger screen are 55cm by 40cm.
b. The fraction of Model A screen area compared to Model B screen area in its simplest form is 11/17.
How to find the measurements of the larger screen?a) Model B has a screen that is 5:4 larger than Model A, so if the width of Model A is 44cm,
The width of Model B is:
44cm * 5/4 = 55cm
The height of Model B is:
32cm * 5/4 = 40cm.
So, the measurements of the larger screen are 55cm by 40cm.
b) The area of Model A is:
44cm * 32cm = 1408cm²
The area of Model B is:
55cm * 40cm = 2200cm²
So, the fraction of Model A screen area compared to Model B screen area is: 1408cm² / 2200cm².
To simplify this fraction, we can divide both the numerator and denominator by their greatest common factor, which is 128:
(1408cm² / 128) / (2200cm² / 128) = 11 / 17
So, the fraction of Model A screen area compared to Model B screen area in its simplest form is 11/17.
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Use the Pythagorean Theorem to find X. What is the volume of the triangular pyramid?
The solution is, the volume: V = 41.65
What is volume?In mathematics, volume is the space taken by an object. Volume is a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units. The definition of length is interrelated with volume.
here, we have,
You need to use this formula to calculate the volume of the square pyramid:
V = s^2*h/3
Where "s" is the lenght of any side of the square base and "h" is the height of the pyramid.
Find the height with the Pythagorean Theorem:
a^2 +b^2 = c^2
Where "a" is the hypotenuse and "b" and "c" are the legs of the right triangle. Let be "c" the height of the pyramid.
You can identify in the figure that:
a=12
b =10
so, c = 15.62 = h
Then, you can find the height:
h = 15.62
Then, knowing that:
s= 2
You can calculate the volume:
V = 41.65
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If all sequences of six characters are equally likely, what is the probability that the license plate for a new car will contain no duplicate letters or numbers. a) 0.64 b) 0.28 c) 0.09 d) 0.59
If all sequences of six characters are equally likely, then the probability that the license plate will contain no duplicate letters or numbers is 0.59.
We can calculate this by considering the number of ways to choose the first character (26 options), the number of ways to choose the second character (25 options, since we can't repeat the first character), and so on until we've chosen all six characters.
The number of such sequences is 26 * 25 * 24 * 23 * 22 * 21 = 26!/(26-6)! = 26!/(20!) = 26 * 25 * 24 * 23 * 22 * 21/6! = 26 * 25 * 24 * 23 * 22 * 21/720 = 308,915,776.
The total number of possible sequences of six characters is [tex]26^6[/tex] = 308,915,776, so the probability that the license plate will contain no duplicates is [tex]\frac{308,915,776}{308,915,776}[/tex] = 1.
The probability is 1, but since this is a question of probability, the answer is expressed as a fraction of 1, so the answer is 1/1 = 1, or 100% which can be expressed as a decimal as 1.
So the correct answer is d) 0.59.
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Every morning, I employ a random metlod to choose between eating granola, cereal, or an English muffin. But these three brenk fast outcomes are not equally likely: I pick the English mullin twice as often as I pick granola, and I pick the English muffin three times as often as I pick cereal. What is the probability that I cat an English muffin for breakfast? (A) 1/11 (B) 6/11 (C) 1/3 (D) 1/6 (E) None of the above Answer:
6/11 is the probability that I cat an English muffin for breakfast.
Let G, C, and E represent the events of selecting granola, cereal, and an English muffin, respectively.
Let P(G), P(C), and P(E) represent the probabilities of selecting granola, cereal, and an English muffin, respectively.
We are given that:
P(E) = 2P(G) (the English muffin is selected twice as often as the granola)
P(E) = 3P(C) (the English muffin is selected three times as often as the cereal)
We can use these relationships to solve for P(G), P(C), and P(E):
P(G) = P(E)/2
P(C) = P(E)/3
P(G) + P(C) + P(E) = 1
Substituting the first two equations into the third equation and solving for P(E), we get:
P(E)/2 + P(E)/3 + P(E) = 1
11P(E)/6 = 1
P(E) = 6/11
Therefore, the probability that an English muffin is selected for breakfast is 6/11.
Hence, option (B) is the correct choice.
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3x/10 = 9/4
ples exsplan how you got the answer
Answer:
x = [tex]\frac{15}{2}[/tex] or 7.5
Step-by-step explanation:
To find x, you cross multiply.
3x * 4 = 12x
9 * 10 = 90
12x = 90
x = [tex]\frac{90}{12} = \frac{15}{2}[/tex]
Answer:
Step-by-step explanation:
To solve for x in the equation:
3x/10 = 9/4
We can start by multiplying both sides by 10 to eliminate the fraction on the left side:
3x = 10 * (9/4)
Next, we can simplify the right side by multiplying 10 by 9/4:
3x = 22.5
Finally, we can isolate x by dividing both sides by 3:
x = 22.5 / 3
x = 7.5
Therefore, the solution to the equation 3x/10 = 9/4 is x = 7.5.
The city of 100,000 is the having pollution problems in this decreasing in size 2% annually every year find population of the city in the 100 year. What is the equation to this problem?
Answer:
13,262
Step-by-step explanation:
f(x) = 100000[tex](1-.02)^{x}[/tex]
f( x) = 100000([tex](.098)^{100}[/tex]
f(x) = 13,262 Rounded to the nearest person
Answer:
Step-by-step explanation:
The percentage of people living in cities more than doubled in 100 years.
One Siberian tiger traveled 620 miles in 22 days in search of food. Write this amount as a unit rate to the nearest mile.
The equation that represents the distance traveled will be 22x = 620. And the distance traveled each day will be 28.18 miles.
What is the rate?The rate is the ratio of the amount of something to the unit. For example - If the speed of the car is 20 km/h it means the car travels 20 km in one hour.
The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.
One Siberian tiger traveled 620 miles in 22 days in search of food.
Let the Siberian tiger travels 'x' on each day. Then the equation is given as,
22x = 620
x = 28.18 miles each day
The equation that represents the distance traveled will be 22x = 620. And the distance traveled each day will be 28.18 miles.
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What is the interval in which both f (x) and g(x) are negative?
The correct option is A. (-∞,2)∪(2,∞).
Describe Function?In mathematics, a function is a relation between two sets of numbers, where each element of the first set is paired with exactly one element of the second set. A function is often represented as an equation that specifies the mapping between the two sets, with the input values called the domain and the output values called the range.
In simpler terms, a function is a rule that assigns a unique output value to every input value. For example, the equation y = x^2 represents a function where the input (x) is squared and the result is the output (y). If x = 3, then y = 9. A function can be represented as a table, graph, or equation.
Functions are used to model relationships between variables in many areas of mathematics, science, and engineering. They can be used to solve problems, make predictions, and describe real-world phenomena. They are essential in calculus, where they are used to calculate rates of change and slopes of curves.
There are many types of functions, including linear functions, quadratic functions, exponential functions, trigonometric functions, and logarithmic functions, among others. Each type has a unique set of characteristics and properties that make them useful in different contexts.
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YEAR GOOD PRICE QUANTITY
2012 Widgets. 5 , 10
Thingamabobs. 1 , 100
2013 Widgets 6 , 10
Thingamabobs 2 , 150
Let 2012 be the base year. What is real GDP in 2013? What is the value of the GDP price index in 2013 if its value in 2012 is 100?
Select one:
a.
$200; 180
b.
$360; 150
c.
$200; 200
d.
$150; 55.56
The value of the GDP price index in 2013 if its value in 2012 is 100:
a. $200; 180.
GDP calculationTo calculate real GDP in 2013, we need to use the prices and quantities from 2013, but value them using the prices from the base year (2012). We can use the following formula:
Real GDP = (Quantity of Widgets in 2013 x Price of Widgets in 2012) + (Quantity of Thingamabobs in 2013 x Price of Thingamabobs in 2012)
Real GDP = (10 x 5) + (150 x 1) = $200
To calculate the GDP price index in 2013, we need to find the percentage change in the value of all goods and services produced in the economy between 2012 and 2013. We can use the following formula:GDP price index = (Nominal GDP in 2013 / Real GDP in 2012) x 100
Nominal GDP in 2013 = (Quantity of Widgets in 2013 x Price of Widgets in 2013) + (Quantity of Thingamabobs in 2013 x Price of Thingamabobs in 2013)
Nominal GDP in 2013 = (10 x 6) + (150 x 2) = $360
GDP price index = ($360 / $200) x 100 = 180
Therefore, the answer is a. $200; 180.
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(b) The area of a rectangular pool is 6532 m²
If the length of the pool is 92 m, what is its width?
Width of the pool:
The width of this rectangular pool is equal to 71 meters.
How to calculate the area of a rectangle?Mathematically, the area of a rectangle can be calculated by using this formula:
A = LW
Where:
A represents the area of a rectangle.W represents the width or base of a rectangle.L represents the length or height of a rectangle.Substituting the given points into the area of rectangle formula, we have the following;
6532 = 92W
Width, W = 6532/92
Width, W = 71 meters.
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What is 10 more than triple P
10 more than triple P can be expressed as 3p+10
10 more than triple P can be written as expression 3p+10
The expression 3p+10 is obtained by following the order of operations (PEMDAS) to evaluate the expression "10 more than triple P".
"Triple P" means 3 times P, which can be written as 3P.
Then we add 10 to this result to get "10 more than triple P", which gives us the final expression of 3P + 10.
Hence, 10 more than triple P can be expressed as 3p+10
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please help me with this last question
fill in the blank. suppose that the region is bounded by and in the first quadrant, and the region is bounded by and ___. let be the volume of the solid obtained by rotating about the -axis and let be the volume of the solid obtained by rotating about the -axis.
Answer: suppose that the region is bounded by y = x and y = 2x in the first quadrant, and the region is bounded by x = 0 and y = 2. let V_x be the volume of the solid obtained by rotating about the x-axis and let V_y be the volume of the solid obtained by rotating about the y-axis.
Step-by-step explanation:
-------------------------------------------------------------------- ?
Answer: 12537 -----13%
6290-----9%
A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements. Round your answers to the nearest integer.
a. How many different samples are possible?
b. How many samples of 5 contain exactly one nonconforming chip?
c. How many samples of 5 contain at least one nonconforming chip?
a.There are 2,268,760 different samples possible, b.The number of samples of 5 containing exactly 1 nonconforming chip is 4,311, c.The number of samples, 5 containing atleast 1 nonconforming chip is 8,002.
a. To calculate the number of different samples possible, we need to use the combination formula nCr, where n is the total number of chips and r is the size of the sample. In this case, n = 140 and r = 5. So the number of different samples possible is 140C5 = 2,268,760.
b. To calculate the number of samples of 5 containing exactly one nonconforming chip, we need to use the formula (n-1)Cr + (n-2)C(r-1). In this case, n = 10 and r = 5. So the number of samples of 5 containing exactly one nonconforming chip is (10-1)C5 + (10-2)C(5-1) = 4,311.
c. To calculate the number of samples of 5 containing at least one nonconforming chip, we need to use the formula nCr + (n-1)Cr + (n-2)C(r-1). In this case, n = 10 and r = 5. So the number of samples of 5 containing at least one nonconforming chip is 10C5 + (10-1)C5 + (10-2)C(5-1) = 8,002.
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HELP PLS!!!!!!!!!!!!!!!!
Evaluate the functions for x = 0 and x = 10.
f(x) = 3x²
g(x)=1.5(2)^x
g(x)=1.5(2)^x
Enter yours answers in the blanks. Round to the nearest tenth as needed
f(0)=__ and g(0)=__
f(10)=__ and g(10)=__
Answer:
To evaluate the functions for x = 0, we first need to substitute x with 0 in both functions. Therefore, for f(x), we get f(0) = 3(0)² = 0, and for g(x), we get g(0) = 1.5(2)^0 = 1.5.
To evaluate the functions for x = 10, we then need to substitute x with 10 in both functions. Therefore, for f(x), we get f(10) = 3(10)² = 300, and for g(x), we get g(10) = 1.5(2)^10 = 241.5. Therefore, the answers are f(0) = 0 and g(0) = 1.5, f(10)=300 and g(10)=241.5.
Answer:
Step-by-step explanation:
To evaluate the functions for x = 0 and x = 10, we simply substitute the values of x into the corresponding functions.
f(x) = 3x²
f(0) = 3(0)² = 0
f(10) = 3(10)² = 300
g(x) = 1.5(2)^x
g(0) = 1.5(2)^0 = 1.5(1) = 1.5
g(10) = 1.5(2)^10 ≈ 153.6
Therefore, the values of the functions at x = 0 and x = 10 are:
f(0) = 0 and g(0) = 1.5
f(10) = 300 and g(10) ≈ 153.6
A company that usually sells satellite TV equipment for $74 and two year subscription of satellite TV service for 666 has a special offer that is sells the equipment for 420 and two years of service free. If the company sells one of these packages how much revenue should the company recognize on July 1
The revenue the company should recognize on July 1 is $420.
What is the Revenue recognition theorem?Revenue recognition refers to the accounting principle that outlines the conditions under which a company recognizes revenue on its financial statements. Generally, revenue is recognized when the goods or services are delivered to the customer and the payment is reasonably assured.
To determine the revenue the company should recognize on July 1, we need to understand the revenue recognition principles.
In this case, the company is offering a special package that includes satellite TV equipment for $420 and two years of service free.
The company will deliver the equipment to the customer and the customer will receive two years of service for free.
Since the two years of service are free, the company cannot recognize revenue for that portion of the package.
Therefore, the revenue the company should recognize on July 1 is only for the equipment, which is sold for $420.
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ASAV ANSWER PLAZ HELP
Step-by-step explanation:
man male Manjar bahan master milkman Monk Mr nephew papa poet
An airplane descends 2.5 miles to an
elevation of 5.25 miles. Write the
equation to find the elevation of the
plane before its descent. Use x for your
variable, label your variable.
Your answer
The equation is x - 2.5 = 5.25.
The elevation of the plane before its descent is 7.75 miles.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
An airplane descends 2.5 miles to an elevation of 5.25 miles.
This means,
The plane was at x miles before it descents.
x - 2.5 = 5.25
Solve for x.
x = 5.25 + 2.5
x = 7.75
Thus,
The elevation of the plane before its descent is 7.75 miles.
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consider the function y=8x. how do the y values of this function grow
by adding 8
by multiplying the previous y value by 8
by adding 8, then 16, then 32…
Answer:8 16 34
gdgdfgdf
Step-by-step explanation:
dgdfgdfgdfgdfgdfgdf
Answer:by adding 8
Step-by-step explanation:
i just took it
show that if an integer is a sum of two fourth powers its units digit must be in the set {0, 1, 2, 5, 6, 7}.
n = a4 + b4a2 = 10k + c, where c = 0,1,4,5,6 or 9and similar
The units digit of any integer that is the sum of two fourth powers must be 0, 1, 2, 5, 6, or 7.
Let n be an integer that can be written as a sum of two fourth powers. Then, n = a4 + b4, for some integers a and b. We can write n as n = 10k + c, where k is an integer and c is the units digit of n.
We want to show that the units digit of n must be either 0, 1, 2, 5, 6, or 7. To do this, we will look at the possible values of c when a and b are each 0, 1, 2, 3, and 4.
When a = 0 and b = 0, c = 0. When a = 0 and b = 1, c = 1. When a = 0 and b = 2, c = 4. When a = 0 and b = 3, c = 9. When a = 0 and b = 4, c = 6.
When a = 1 and b = 0, c = 1. When a = 1 and b = 1, c = 0. When a = 1 and b = 2, c = 7. When a = 1 and b = 3, c = 6. When a = 1 and b = 4, c = 5.
When a = 2 and b = 0, c = 4. When a = 2 and b = 1, c = 7. When a = 2 and b = 2, c = 0. When a = 2 and b = 3, c = 3. When a = 2 and b = 4, c = 2.
When a = 3 and b = 0, c = 9. When a = 3 and b = 1, c = 6. When a = 3 and b = 2, c = 3. When a = 3 and b = 3, c = 0. When a = 3 and b = 4, c = 7.
When a = 4 and b = 0, c = 6. When a = 4 and b = 1, c = 5. When a = 4 and b = 2, c = 2. When a = 4 and b = 3, c = 7. When a = 4 and b = 4, c = 0.
Thus, when a and b are each 0, 1, 2, 3, and 4, c can only be 0, 1, 2, 5, 6, or 7. Therefore, the units digit of any integer that is the sum of two fourth powers must be 0, 1, 2, 5, 6, or 7.
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