1) the rate change of function f(x) in given interval =4(b+1)
2) the rate change of function g(x) in given interval =1
3) the rate change of function j(x) in given interval =[tex]3h^2+9h+9[/tex]
To find out the average rate change of function on given interval is -
Use the graph of f to determine f (a) and f (b) for the provided range [a,b]. Remember that the graph of f's point with the y-coordinate of f (a) and the x-coordinate of an is called f. Similar to that, f ( b ) is the y-coordinate of the point on the f graph whose x-coordinate is b.
Step 2: Calculate the average rate of change of f over the range [a, b] using the average rate of change formula.
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
1)
[tex]f(x)=4x^2-7 \ on\ [1,b]\\\\f(1)=4-7=-3\\\\f(b)=4b^2-7\\\\the \ rate \ change =\frac{4b^2-7+3}{b-1}\\\\=4(b+1)[/tex]
2)
[tex]p(x)=3x+4 \ on\ [2,2+h]\\\\p(2)=10\\\\p(2+h)=10+h\\\\The\ average\ rate \ change= \frac{10+h-10}{2+h-2}\\\\=1[/tex]
3)
[tex]j(x)=3x^3 \ on \ [1,1+h]\\\\j(1)=3\\\\j(1+h)= 3(1+h)^3\\\\\\so, the\ rate\ change = 3h^2+9h+9[/tex]
The complete Question is: -
For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.
1. f(x)= 4x^2 - 7 on [1, b]
2. p(x)= 3x+4 on [2,2+h]
3. j(x)= 3x3 on [1, 1+h]
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Mason has 2/3 of a foot of ribbon. He needs to divide the ribbon into 1/6 -foot pieces.
How many pieces can he cut from the ribbon?
Based on the information provided, Mason can cut four pieces of ribbon.
How much ribbon does Mason have?Based on the information provided, Mason has 2/3 of a foot ribbon, which is equivalent to 0.66.
How many pieces can he cut out of the total ribbon he has?To know the answer, use a division with the following formula:
Total ribbon/ size of the desired pieces
0.66 / 2/3 or 0.166 (2/3 = 0.166)
0.66 / 0.166
4 pieces
Based on this, it can be concluded that Mason will be able to get 6 1/6 feet long pieces.
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2. The table shown below shows some values for functions f(x) and g(x). What is/are the solution(s)
when f(x) = g(x)? Explain your answer(s).
The solutions when the function f(x) = g(x) is x = -3 and x = -1, since the output of the function is similar.
What are functions?With the condition that each input is associated to exactly one output, a function is a relationship between a collection of inputs and a set of allowed outputs. A mapping from A to B will only be a function if every element in set A has one end and only one image in set B. Let A & B be any two non-empty sets.
As we can observe from the given table, the two values for which the two function produce a similar output are at x = -3 and x = -1.
Hence, the solutions when the function f(x) = g(x) is x = -3 and x = -1, since the output of the function is similar.
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3x + 5y = - 9 + -6x - 5y = 18?
Answer: The given equations are not valid because there is no equal sign between the first and second equation. If you are trying to solve a system of equations it should be in the form of
3x + 5y = - 9
-6x - 5y = 18
You can solve for x and y by using different methods such as substitution, elimination or graphing.
In this case, you can add the first equation to the second equation:
3x + 5y = -9
-6x - 5y = 18
-3x = 27
x = -9
Now you can substitute this value of x into one of the equations to find the value of y.
3x + 5y = -9
3(-9) + 5y = -9
-27 + 5y = -9
5y = 18
y = 3.6
So, the solution is x = -9 and y = 3.6
It's important to check your solution by substituting the values of x and y in the original equations to make sure that the solution is consistent.
Step-by-step explanation:
A teacher in a summer school program notices that her students seem less able to pay attention
when the temperature in the classroom gets warmer. The teacher decides to record the
temperature every day (utilizing Fahrenheit degrees) to track classroom heat conditions. The
measurement of the temperature used here used here is
1.
2.
3.
C
Nominal
r
Ordinal
Interval
Save
Ratio
The level of measurement used for the temperature is given as follows:
Interval.
How to obtain the levels of measurement of the data?There are four levels of measurement for the data, given as follows:
Nominal.Ordinal.Interval.Ratios.The measures of central tendency for each level are given as follows:
Nominal: ModeOrdinal: Mode and median.Interval: Arithmetic mean, mode and median.Ratios: Arithmetic mean, geometric mean, mode and median.For the temperature, we obtain the arithmetic mean, hence it has the level of measurement of interval.
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what of the following statements is not true about samples? group of answer choices wording and ordering of questions can affect results samples should reflect the population being studied samples picked randomly are more representative of a population a variety of ways can be used to contract the samples larger samples are more accurate
The statement that is not true about samples is: "samples picked randomly are more representative of a population."
To determine this:
While random sampling is one way to ensure that a sample is representative of a population, it is not the only way. There are other methods of sampling such as stratified sampling, cluster sampling, and systematic sampling that can also be used to obtain a representative sample.
It is important to note that, even with random sampling, a sample may not be representative of the population if the sample size is too small or if there is a significant amount of bias in the sampling process.
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let x be a numeric vector. create a new vector y which consists of all entries of x between -3 and 3 (including -3 and 3), in order of appearance as in the vector x. for example, if
For the creation of y, we first need to know the placement of the numbers and then write an R code to create a subset for the same.
To create another vector y which contains all the numbers between 3 and -3 in order of appearance, we first need to know exactly what is the order of the appearance of 3 and -3.
Hence we will write the R code
which(x == 3)
which(x == -3)
These commands will return us the specific position of the numbers 3 and -3
Let us assume that 3 is the fourth number and -3 is the 31st number. Then we will write y as a function that contains the subset of x from the fourth to the 31st number as follows.
y <- x[4:31]
Typing this command will give us the required subset.
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Complete Question
Let x be a numeric vector and suppose you want to extract all the entries of x that are between -3 and 3 (including both boundaries -3 and 3). Create a new vector y which consists of all entries of x between -3 and 3 (including -3 and 3), in order of appearance as in vector x. For example, if X = c(-5, 7, 9, 3, 14, 9, 5, 10, -2, 0, 12) then, printing y should read у [1] 0 3 -2 0
what is the solution set of y= x^2 + 2x + 7 and y = x + 7?
Answer:
Step-by-step explanation:
(0, 7) and (-1, 6)
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m = 6
point = (12, -8)
b = ?
solve it
Step-by-step explanation:
you mean the slope-intercept form of a line equation ?
y = mx + b
or what is the problem ?
if my assumption is correct, then m is the slope of the line.
and we got a point on the line.
so, we could start with the point-slope form and then transform.
the point-slope form is
y - y1 = m(x - x1)
again, m is the slope, (x1, y1) is a given point on the line.
so, we have
y - -8 = 6(x - 12)
y + 8 = 6x - 72
y = 6x - 80
that means b = -80
You are inflating a balloon in a room. The volume of the balloon is
given by V(m, θ) = sqrtθ [ln(m)],
where m is the mass of the air in the balloon, and θ is the temperature of the room.
Suppose that at time t, the mass of air in the balloon is m(t) = t^2+3 and the temperature of the room is θ(t) = e^t.
Use the chain rule to find the rate at which the volume is
changing at time t.
According to the question rate at which the volume is dV/dt = dV/dm * dm/dt + dV/dθ * dθ/dt.
What is the straightforward meaning of volume?The space covered within an object's borders in three dimensions is referred to as its volume. It is refereed as the object's capability on occasion.
How is volume measured?Volume, which is measured in cubic units, is the three dimensional volume occupied by matter or encircled by a surface. The cubic centimeter (m3), a derived unit, is the Standard unit of volume.
Substituting in the given information, we have:
V(m, θ) = sqrt(θ) * ln(m)
m(t) = t^2 + 3
θ(t) = e^t
Therefore:
dV/dm = (1/m) * sqrt(θ)
dm/dt = 2t
dV/dθ = (1/2sqrt(θ)) * ln(m)
dθ/dt = e^t
plugging in the values of m(t) = t^2 + 3 and θ(t) = e^t
dV/dt = (1/(t^2+3)) * sqrt(e^t) * 2t + (1/2sqrt(e^t)) * ln(t^2+3) * e^t
So the rate at which the volume is changing at time t is given by dV/dt
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At the store, there are 11 shelves with 27 pictures books and 12 shelves with 21 chapter books. How many fewer chapter books thank picture books does the store have
Answer:
45 fewer chapter books
Step-by-step explanation:
Picture books: 27 x 11= 297
Chapter books: 21 x 12= 252
297-252= 45
the linear system l has augmented matrix which is row equivalent to the following. how many solutions does l have?
The linear system l has one solution. This can be determined by examining the augmented matrix to find the row echelon form. The row echelon form is a matrix with zeroes below the leading entries and all the leading entries are in different columns.
By converting the augmented matrix into the row echelon form, we can identify the number of variables and the number of equations. Since there are three equations and three variables in the row echelon form, the linear system l has one solution. The row echelon form of the given augmented matrix is
[tex]$\begin{bmatrix}1&-3&2&3\\0&-5&-9&-10\\0&0&4&5\end{bmatrix}$[/tex]
By counting the number of equations and variables in the row echelon form, it can be seen that there is only one solution for the linear system l. This is because the number of equations is equal to the number of variables, indicating that there is a single solution.
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the linear system l has an augmented matrix which is a row equivalent to the following. how many solutions does l have?
[tex]$\begin{bmatrix}1&-3&2&3\\-3&8&-11&-14\\2&-7&5&7\end{bmatrix}$[/tex]
• The floor in a computer lab has a width of 6x – 9 and a length of 6x + 9. Find the area of the floor. Each tile that will cover the floor has an area of 24x2- 12x – 36. Divide the two areas and tell how many tiles will be needed to cover the floor?
Answer: The area of the floor can be found by multiplying the width and the length.
Area of the floor = (6x - 9)(6x + 9) = 36x^2 - 63x - 81
The area of each tile can be found by evaluating the expression 24x^2 - 12x - 36.
Area of each tile = 24x^2 - 12x - 36
To find out how many tiles are needed to cover the floor we need to divide the area of the floor by the area of each tile.
Number of tiles needed = (36x^2 - 63x - 81) / (24x^2 - 12x - 36)
The division is simplified by canceling out common factors between the numerator and denominator, the number of tiles is a fraction that can't be simplified any further.
So the answer is (36x^2 - 63x - 81) / (24x^2 - 12x - 36) tiles are needed to cover the floor.
Step-by-step explanation:
given triangle xyz and triangle pqr with measurements as marked below. determine whether or not triangle xyz is similar to triangle pqr
The Pythagorean Theorem holds true for both triangles, then the triangles are not only similar, but also congruent.
Triangle xyz has sides of length x, y, and z and triangle pqr has sides of length p, q, and r.
To determine if triangle xyz is similar to triangle pqr, we must calculate the ratios of corresponding sides. This is done using the formula:
Ratio of sides = (Side length x) / (Side length y)
We can calculate the ratios for each side of the triangles and compare them. If the ratios are equal for all sides, then the triangles are similar. For example, if the ratio of side x to side y in triangle xyz is equal to the ratio of side p to side q in triangle pqr, then the triangles are similar. If the ratios are not equal for all sides, then the triangles are not similar.
We can use the Pythagorean Theorem to determine if the triangles are not only similar, but also congruent. The Pythagorean Theorem states that a2 + b2 = c2, where c is the longest side and a and b are the other two sides. If the Pythagorean Theorem holds true for both triangles, then the triangles are not only similar, but also congruent.
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a researcher wants to examine the effect of fertilizer and the effect of sunlight on the yield of tomatoes. she bought 60 tomato plants at a local garden store. she randomly assigned 30 tomato plants to be planted on the sunny side of the hill and 30 to be planted on the shady side. the 30 plants which are planted on the shady side are randomly assigned to one of three groups. the first group are grown with no fertilizer, the second group with a small amount of fertilizer, and the third group with a large amount of fertilizer. the 30 plants which are planted on the sunny side are likewise randomly assigned to one of three groups. the first group are grown with no fertilizer, the second group with a small amount of fertilizer, and the third group with a large amount of fertilizer. all tomato plants are planted at the same time and are all treated alike (in terms of how much they are watered, weeded, etc.), and the person who tends the plants does not know which group each is in. each plant is grown to maturity. the total weight of tomatoes obtained from each plant is recorded. Describe the design of the experiment (completely randomized or blocked).
Completely randomized over one factor (location), blocked by fertilizer
Blocked by fertilizer, blocked by location
Completely randomized over one factor (fertilizer)
Completely randomized over one factor (fertilizer), blocked by location
Completely randomized over two factors (fertilizer and location)
The design of the experiment is Completely randomized over two factors (fertilizer and location).
Randomization is a statistical technique used in experimental design to ensure that all possible combinations of the factors being studied are represented in the study groups. It is used to eliminate bias and control for confounding variables that may affect the outcome of an experiment.
The researcher randomly assigned the tomato plants to be planted on either the sunny or the shady side of the hill. This creates two groups or two levels of the factor "location". She also randomly assigned the tomato plants to one of three groups based on the amount of fertilizer given, creating three groups or three levels of the factor "fertilizer".
The random assignment of plants to the different groups for both factors helps to ensure that any differences in the yield of tomatoes is due to the independent variables (fertilizer and location) being tested, and not due to other extraneous factors.
Therefore, The design of the experiment is Completely randomized over two factors (fertilizer and location).
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the quality control manager at a factory records the number of equipment breakdowns each day. let the random variable y represent the number of breakdowns in one day. the standard deviation of y is 0.28. which of the following is the best interpretation of the standard deviation? responses the number of breakdowns on a randomly selected day is expected to be 0.28.
The best interpretation for the given standard deviation as per given information is given by :
Option D. On an average, the number of breakdowns per day varies from the mean by about 0.28.
As given in the question,
Standard deviation represents the measure of the data in a given set which varies or deviates from the mean. In the given information, the standard deviation of y is 0.28 for the number of equipment breakdowns each day means that on an average, the number of breakdowns each day deviates by 0.28 about the mean.This represents that the most of the data exists in the range of 0.28 above or below the mean. Standard deviation is a method to present the data which spread , it also help in learning the distribution of the given data set.Therefore, the best interpretation of standard deviation is :
Option D. On an average, the number of breakdowns per day varies from the mean by about 0.28.
The above question is incomplete , the complete question is:
The quality control manager at a factory records the number of equipment breakdowns each day. Let the random variable Y represent the number of breakdowns in one day. The standard deviation of Y is 0.28. Which of the following is the best interpretation of the standard deviation?
A. The number of breakdowns on a randomly selected day is expected to be 0.28.
B. The number of breakdowns on a randomly selected day will be 0.28 away from the mean.
C. The average number of breakdowns per day for a random sample of days is expected to be 0.28
D. On average, the number of breakdowns per day varies from the mean by about 0.28.
E. The number of breakdowns per day for a random sample of days is expected to be 0.28 away from the mean.
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Reuben is putting money into a checking account. Let y represent the total amount of money in the account (in dollars). Let x represent the number of weeks Reuben has been adding money. Suppose that x and y are related by the equation 750=40x.
Based on equation 750=40x, the value of x is 18.75 weeks.
What is an equation?An equation is a mathematical statement that two or more algebraic expressions are equal or equivalent.
Equations must have the equation symbol (=) to show the equality of the mathematical expressions.
Equations are started with let statements, defining the variables.
Let the total amount of money in the account = y (in dollars)
Let the number of weeks Reuben has been adding money = x
Equation:750 = 40x
40x = 750
x = 18.75 (750/40)
Thus, based on the solution to the equation, we know that Reuben has added money into the account for 18.75 weeks.
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Question Completion:Compute the number of weeks Reuben has been putting money into the account.
5. If the bakery sold 144 cupcakes, write an inequality to represent the possible number of cakes sold
The required inequality to represent the possible number of cakes sold is Cupcakes ≥ 144.
What is inequality?The idea of inequality, which is the state of not being equal, especially in terms of status, rights, and opportunities1, is at the core of social justice theories. However, because it frequently has diverse meanings to different people, it is prone to misunderstanding in public discourse.
According to question:
Write an inequality to illustrate the potential amount of cakes sold if the business sold 144 cupcakes.
So,
Bakery can sold 144 cupcakes and more
Cupcakes ≥ 144
Thus, required inequality is Cupcakes ≥ 144.
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Examine quadrilateral FGHI below. In order to determine the length of each * 5 points
side of the parallelogram, which of the following equations would need to
be used? Select two that apply.
The equations for the sides of the parallelogram would be
8x - 6 = 9x
y + 10 = 11y
Option (A) and (C) are correct.
What is a parallelogram?
A parallelogram is a four-sided shape with opposite sides that are parallel and congruent. In other words, opposite sides of a parallelogram are of equal length and parallel to each other. Additionally, opposite angles in a parallelogram are also congruent. It has two pairs of parallel sides. It can be thought of as a rectangle with opposite sides that are not necessarily at right angles to each other.
As the property of the parallelogram says
"Opposite sides of the parallelogram are equal"
So we can write
8x - 6 = 9x
y + 10 = 11y
Hence, The equations for the sides of the parallelogram would be
8x - 6 = 9x
y + 10 = 11y
Option (A) and (C) are correct.
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nscribed and Circumscribed Circles: Tutorial
?
Answer the question below. Type your response in the
space provided.
In quadrilateral QRST, m¿Q > is 68°, m¿R is (3x + 40)º, and m¿T is (5x
52)°. What are the measures of ZR, ZS, and
values in that order with the measures separated by commas.
Done
Clear
The angle ∠R, ∠S, and ∠T for the quadrilateral are ∠R, ∠S, ∠T = 112°, 112°, 68°.
what are quadrilaterals?
A quadrilateral is a polygon with four sides, four vertices, and four angles.
It is formed by joining four non-collinear points. The sum of the interior angles of quadrilaterals is always equal to 360 degrees. The word quadrilateral is derived from the Latin words ‘Quadra’ which means four and ‘Latus’ means ‘sides’. Not all the four sides of a quadrilateral need to be equal in length. Hence, we can have different types of quadrilaterals based on sides and angles.
Five special quadrilaterals are shown below in the image, along with their definitions and pictures, and points for them to get proven.
Now,
TQRS is an inscribed quadrilateral.
5 x - 52° + 3 x + 40° = 180°
8 x - 12° = 180°
8 x = 180° + 12°
8 x = 192°
x = 192° : 8 = 24°
m∠ R = 3 · 24° + 40° = 112°
m∠ T = 5 · 24° - 52° = 68°
m∠ S = 360° - ( 68° + 68° + 112° ) = 112°
Answer:
∠R, ∠S, ∠T = 112°, 112°, 68°.
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Right question:-
"In quadrilateral QRST, m∠Q is 68°, m∠R is (3x + 40)°, and m∠T is (5x − 52)°. What are the measures of ∠R , ∠S , and ∠T ? Write the numerical values in that order with the measures separated by commas.
Find the slope of a line perpendicular to the line whose equation is 4x - 3y = -3.
Fully simplify your answer.
The slope of the line having the equation 4x - 3y = -3 is calculated to be
4/3
How to find the slope of the lineThe slope of the line is gotten by rewriting the equation of the line in the standard form Ax + By = C. to equation of a line in slope intercept form which is y = mx + c
The equation of line in the problem is 4x - 3y = -3
The equation is rewritten as follows
4x - 3y = -3
-3y = -3 - 4x
Isolating y by dividing trough by -3
y = 1 + 4/3 x
y = 1.33x + 1
comparing with y = mx + c where m is the slope we can see that m = 4/3
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The linear equation y=2x represents the cost y for x pounds of apples. Which ordered pair lies on the graph of the equation
The ordered pair that lies on the graph of the equation is (1, 2)
How to determine the ordered pair lies on the graph of the equationFrom the question, we have the following parameters that can be used in our computation:
y = 2x
The point in the ordered pair lies on the graph of the equation is the solution set
Take for instance, we set x = 2
So, we have
y = 2 * 1
Evaluate the product
This gives
y = 2
When repesented as an ordered pair, we have
(1, 2)
Hence, the point is (1, 2)
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In the preivous activity you used a table of coordinates to determine that the baseball Billy hit followed a nonlinear path.I’m this activity you will use a graphical means to make the same determination
This equation represents the path of the ball that billy
Y=-1/300 x2 +x+3
Answer:
Part a: Answer is 1/2
78-3/1500 = 75/150
= 1/2
Part b: Answer is -1/2
3-78/300-150
=-75/150
= -1/2
Part c: No, the two rates aren't the same,
because one is positive and the other is
negative.
Part d: The rate of change of the function
determined by using the two pairs of
coordinates is not the same. So, the
function modeling the movement of the
ball is a nonlinear function
Find the equation of the regression line for the given data and use it to predict the value of y for x = −2.5. Round to the nearest thousandth.
x: -5, -3, 4, 1, -1, -2, 0, 2, 3, -4
y: -10, -8, 9, 1, -2, -6, -1, 3, 6, -8
For your data, the regression equation for Y is:
Y = 2.09697X - 0.55152
What is Linear function?
A linear function in mathematics is one that has either one or two variables and no exponents. It is a function with a straight line as its graph.
Given, a data set
x: -5, -3, 4, 1, -1, -2, 0, 2, 3, -4
y: -10, -8, 9, 1, -2, -6, -1, 3, 6, -8
Sum of X = -5
Sum of Y = -16
Mean X = -0.5
Mean Y = -1.6
Sum of squares (SSₓ) = 82.5
Sum of products (SP) = 173
Regression Equation = Y = bX + a
b = SP/SSₓ = 173/82.5 = 2.09697
a = Mₓ - bMₓ = -1.6 - (2.1*-0.5) = -0.55152
Y = 2.09697X - 0.55152
For x = 2.5 the value of y:
y = 2.09697* 2.5 - 0.55152
y = 4.69
Therefore, For your data, the regression equation for Y is:
Y = 2.09697X - 0.55152
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The temperature, t, of water and the number of minutes, m, the water is in the freezer
Which is the dependant variable
Answer:
Temperature.
Step-by-step explanation:
The temperature of the water depends on the minutes the water is in the freezer. If the water is frozen for longer, the water will be colder and if the water is frozen for less time, the water will be less cold.
if sum a n and sum b n are series with positive terms, and lim a n/b n does not exist, and sum b n diverges, then sum a n diverges
Since lim a n/b n does not exist, it means that the ratio between a n and b n can be larger or smaller depending on the terms.
This implies that the rate of growth of a n and b n are not necessarily the same. If the series b n diverges, it means that the terms of b n increase without bound. Therefore, the terms of a n will also increase without bound, and sum a n will also diverge.
Formally, if lim a n/b n = ∞, then ∑ a n is divergent. We can prove this by contradiction. Suppose ∑ a n converges. Then by the comparison test, lim a n/b n must converge and be finite, which is a contradiction. Therefore, ∑ a n must diverge.
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what is the sum of -3 and -12 ?
Answer: The sum is -15
Step-by-step explanation:
Doug smalley, a meteorologist, is married, earns $1,352.75 a week, and claims 1 allowance
The federal income tax withheld for Doug, given the fact Doug is married and claims 1 allowance is $ 149.
How to find the federal income tax withheld ?Using the Married Persons Weekly Payroll table, you can find the bracket that Doug Smalley belongs in which is $1, 350 - $ 1, 360.
When looking at this bracket, we see that the federal income tax withheld for a person in this bracket, who claims 1 withholding allowance, is $ 149.
The federal income tax for Doug Smalley who earns $ 1, 352.75 a week in 2016, is therefore $ 149.
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The full question is:
Use the Married Persons Weekly Payroll table to find the federal income tax withheld. Doug smalley, a meteorologist, is married, earns $1,352.75 a week, and claims 1 allowance.
Select all of the conditions that prove that a quadrilateral is a rectangle.
A) Both pairs of opposite sides are congruent.
B) One pair of opposite sides is longer than the other pair of sides.
C) The diagonals bisect each other.
D) The diagonals are congruent.
E) The diagonals are perpendicular.
only C) The diagonals bisect each other, and E) The diagonals are perpendicular are the two conditions that prove that a quadrilateral is a rectangle.
What is the quadrilateral?
A quadrilateral is a four-sided polygon, which means it has four straight edges and four angles. Quadrilaterals can come in many different shapes and sizes and can be classified based on their properties. Some common examples of quadrilaterals include squares, rectangles, parallelograms, rhombi, and trapezoids.
A quadrilateral is a rectangle if it satisfies the following conditions:
C) The diagonals bisect each other.
E) The diagonals are perpendicular.
A) Both pairs of opposite sides are congruent, is not a necessary condition for a quadrilateral to be a rectangle, as a rectangle can have different side lengths.
B) One pair of opposite sides is longer than the other pair of sides, is not a necessary condition for a quadrilateral to be a rectangle, as a rectangle can have equal side lengths.
D) The diagonals are congruent, is not a necessary condition for a quadrilateral to be a rectangle, as a rectangle can have diagonals of different lengths.
So, only C) The diagonals bisect each other, and E) The diagonals are perpendicular are the two conditions that prove that a quadrilateral is a rectangle.
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Quadrilateral GFHE and quadrilateral TRQS are similar. Which pair of angles are corresponding congruent angles of these quadrilaterals?
The following pair of angles are corresponding congruent angles of these quadrilaterals:
∠G ≅ ∠T
∠F ≅ ∠R
∠H ≅ ∠Q
∠E ≅ ∠S
What is quadrilateral?A quadrilateral in geometry is a four-sided polygon with four edges and four corners.
Given:
Quadrilateral GFHE and quadrilateral TRQS are similar.
The figures are given in the attached image.
The following pair of angles are corresponding congruent angles of these quadrilaterals:
∠G ≅ ∠T
∠F ≅ ∠R
∠H ≅ ∠Q
∠E ≅ ∠S
Therefore, all the pairs are given above.
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Indicate the equation of the given line in standard form. The line containing the diagonal, BD, of a square whose vertices are A(-3, 3), B(3, 3), C(3, -3), and D(-3, -3). Find two equations, one for each diagonal. Show calculations.
The linear functions for each diagonal are given as follows:
Diagonal BD: y = x.Diagonal AC: y = -x.How to define the linear functions?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.For diagonal BD, the points are given as follows:
D(-3,-3) and B(3,3).
The slope is given by the change in y divided by the change in x, hence:
m = (3 - (-3))/(3 - (-3)) = 1.
Hence:
y = x + b.
When x = 3, y = 3, hence the intercept b is given as follows:
y = x.
The diagonals are perpendicular, meaning that the multiplication of their slopes is of -1, hence the slope of AC is of:
m = -1.
Hence:
y = -x + b
When x = -3, y = 3, hence the intercept b is given as follows:
3 = 3 + b
b = 0.
Hence the equation is:
y = -x.
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