What is the independent variable in a linear function?; How do the independent and dependent values change in linear functions?; Which function has a constant additive rate of change of?; Which must be true of the data set if a linear function can be used to represent the data?

The formula for the slope of parallel lines is m1 = m2, and the slopes of parallel lines are equal. The slope of parallel lines is identical because all of the parallel lines are similarly inclined with regard to the positive x-axis.

A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance.

Equations come in a variety of forms, including linear, quadratic, cubic, etc.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation.

To determine the value of an unknown variable that represents an unknown quantity, equations can be solved. The statement is not an equation if there is no "equal to" symbol in it.

Hence, The formula for the slope of parallel lines is m1 = m2, and the slopes of parallel lines are equal. The slope of parallel lines is identical because all of the parallel lines are similarly inclined with regard to the positive x-axis.

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The shorter piece is 56 inches and the longer piece is 180 inches when cut into two halves is a 224-inch pipe.

Given that,

Cut into two halves is a 224-inch pipe. The length of one component is three times that of the other.

We have to determine the shorter piece's length.

We know that,

Let's use the variable L to represent the longer pipe and the variable S to represent the shorter pipe.

Since the pipe will be the sum of L and S, we can create the following equation:

L+S=224

The longer piece is three times the shorter piece, which is another thing we know.

L=3S

Substituting the second equation for the first equation

L+S=224

3S+S=224

4S=224

S=224/4

S=56

The shorter piece is 56 inches.

The lengthier component should be used to replace the following:

L=3S

L=3(56)

L=168

The longer piece is 180 inches.

Therefore, The shorter piece is 56 inches and the longer piece is 180 inches when cut into two halves is a 224-inch pipe.

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Check the picture below.

The sum can be written in logarithms as 6z log 6

What is logarithm ?

The word "logarithm" comes from the combination of the Greek words "logos," which means "principle or concept," and "arithmos," which means "number." A logarithm is the power that must be increased to obtain a given number. It is shown by a number's log.

Use of property

log (ab) = log a + log b

log a^b = b log a

Given that,

log 6^8 + log 6^z

=log 6^8 + log 6^z

=log (6^8*6^z)

=log (6^6z)

=6z log 6

Therefore, the sum can be written in logarithms as 6z log 6

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From the given information, Therefore, the function [tex]= > y = 4(\frac{5}{6} )^x[/tex] is an exponential decay.

The process of decreasing a sum at a constant percentage rate over time is referred to in mathematics as exponential decay. It may be written as the formula y=a(1-b)x, where y represents the final amount and a, b, and x, respectively, represent the initial amount and the decay factor.

The formula for an exponential function is where b is a positive integer that is not equal to 1.

Whether a function is an exponential growth or a decay depends on the value of b. If b > 1 or if b<  1, an exponential function exhibits exponential growth and exponential decline, respectively.

given,

[tex]= > y = 4(\frac{5}{6} )^x[/tex]

[tex]= > b = \frac{5}{6} < 1[/tex]

Therefore, the function [tex]= > y = 4(\frac{5}{6} )^x[/tex] is an exponential decay.

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Answer: -12

Step-by-step explanation:

The weight of clothing is the ratio of weight to dye is different from the other ratios is B. 42 pounds.

Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).

It should be noted that from the information given, others have a ratio of 1:7. This is different for 42 pounds since the ratio is:

= 42/9

= 14:3

In conclusion, the correct option is B.

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The dimensions of the rectangular package with the condition of maximum volume is equal to  x = 19inches and y = 38inches.

As given in the question,

Let us consider length of the cross section which is square be x

Perimeter of the cross section is 4x

And y be the length of the rectangular package

Maximum combined ( length + perimeter of the cross section ) = 114

y + 4x = 114

⇒ y = 114 -4x

Volume of the package 'V' = x²y

⇒V = x²(114 -4x)

⇒V = 114x² -4x³

Differentiate both the side of the equation with respect to x,

dV/dx = 228x - 12x²

For maximum Volume

dV/dx =0

228x -12x² = 0

⇒12x( 19 -x ) =0

⇒12x = 0 or 19 - x =0

⇒x =0 or x =19

x =0 is not possible

⇒x =19inches

y = 114 - 4(19)

  = 38inches

Therefore, the dimensions of the rectangular package with the given condition of volume is equal to x = 19inches and y = 38inches.

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Answer: 45

Step-by-step explanation:

Mrs Kim will pay 103 dollars for her son's party.

What is a substitution method?

The algebraic approach to solving simultaneous linear equations is known as the substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.

Given, the total price for the party with c children = 8.5c + 35 dollars

For 8 children, c = 8 then,

Total price = 8.5*8 + 35 = 103 dollars

Hence, Mrs Kim will pay 103 dollars for her son's party.

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The proportionality constant of the given graph is k = 16.

The proportionality constant of a graph is the ratio of the y-coordinates to the x-coordinates.

Thus, it is written as k = y1/x1 = x2/y2 = ... = yn/xn

The given graph has the coordinates as

x = 0; y = 0 i.e., (0, 0)

x = 0.5; y = 8 i.e., (0.5, 8)

x = 1; y = 16 i.e., (1, 16)

x = 1.5; y = 24 i.e., (1.5, 24)

and so on.

Then, the ratio of y - coordinate to the x - coordinate is

k = 8/0.5 = 16

or

k = 16/1 = 16

or

k = 24/1.5 = 16

and so on.

So, the proportionality constant for the given graph is k = 16.

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The equation of the regression line is defined as follows:

y = 0.71x + 8.27.

The estimates are given as follows:

a) x = 140 cm: 107.67 cm.

b) x = 172 cm: 130.39 cm.

c) x = 164 cm: 124.71 cm.

d) x = 158 cm: 120.45 cm.

The linear regression equation is obtained inserting the points of the data-set into a linear regression calculator.

From the table given in this problem, the points are given as follows:

(138, 106), (168, 130), (151, 116), (144, 105), (160, 124), (159, 119), (125, 103), (137, 103), (154, 120), (147, 110), (146, 108), (147, 110).

Inserting these points into a calculator, the equation is given as follows:

y = 0.71x + 8.27.

Then the estimates are obtained replacing x into the equation and obtaining the value of y, for example, for x = 140 cm:

y = 0.71(140) + 8.27 = 107.67 cm.

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1) Initial value of the linear equation is 0. and 2) Rate of change is equal to -3.

The initial value of a linear equation is the value of the y-intercept, which is the point where the equation intersects the y-axis. It is the constant term in the equation and is represented by the letter "b" in the form y=mx+b. Therefore, the initial value of a linear equation is the value of the constant "b".

The initial value of a linear equation is the y-intercept of the line.

The given linear equation is y=-3x
1) Initial value of the linear equation is the y-intercept.  
   the y-intercept of y=-3x is 0.
   Therefore the initial value of the given equation is 0.

2) Rate of change of y=-3x is the derivative of y.
    [tex]\frac{dy}{dx}=-3\frac{dx}{dx}[/tex]
    ⇒[tex]\frac{dy}{dx}=-3[/tex]
   Therefore the rate of change is -3.

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Burton Bush has to invest $93,199 to get $400,000 at the end. Hence, option D is correct.

Interest that is added to a loan and deposit sum is known as compound interest. In our everyday lives, it is the notion that is employed the most frequently. Compound interest is calculated as a sum using the interest and principal accrued over time. Compound interest versus simple interest differ primarily in this way.

As per the given information in the question,

Rate, r = 6%

Number of years, n = 25

Present value, PV = ?

Final value, FV = $400,000

Use the equation of compound interest:

[tex]PV = FV (1 + r)^-^n[/tex]

Put values from above in the given equation,

PV = 400,000 (1 + 0.06)⁻²⁵

PV = 400,000 × 0.23

PV = $93,199.

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Cumulative markup percent for department at the end  is 23.1%.

The men's furnishing department had an opening inventory of $168,000 at cost  with a markup of 48.5%

Cumulative markup is  the difference between the total wholesale cost and total retail price of all merchandise handled during a given period of time.

cumulative markup dollars are the total markup dollars on all merchandise for a selected selling period. To calculate cumulative markup dollars and cumulative markup percent, the following formulas are utilized by the retailer:

Cumulative Markup $ = Total Retail $ - Total Cost $

Cumulative Markup % = Cumulative Markup $ ÷ Cumulative Retail $

Steps for calculating cumulative markup percentage for the specified period of time:

Step 1. Calculate the unknown retail value for either opening inventory or purchases.

Step 2. Calculate total retail of opening inventory and purchases.

Step 3. Calculate the unknown cost value for either opening inventory or purchases.

Step 4. Calculate total cost of opening inventory and purchases.

Step 5. Calculate total markup dollars.

Step 6. Calculate cumulative markup percent for the specified period of time.

For opening inventory

retail = $168,000

cost = 168,000*48.5/100 = $81480

For purchases

retail = 80,000*52.5/100 = $42000

cost  = $80,000

total retail of opening inventory and purchase =

168,000+42000 = $210000

total cost of opening inventory and purchase =

81480+80,000 = $161480

total markup = total retail - total cost

total markup  = 210000-161480 = 48520

cumulative markup percent = (total markup/ total retail)

= (48520/210000)*100

= 0.23104761904*100

=23.1%

So cumulative markup percent for department at the end  is 23.1%.

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Linear, polynomial, and quadratic functions can be both increasing or decreasing functions. Exponential function is monotonically increasing function. The derivative of a function may be used to determine whether the function is increasing or decreasing. Exponential functions eventually exceed all other increasing functions.

Function, in mathematics, refers to an expression or rule that describes a relationship between two or more variables. A function is said to be an increasing function when the y-value increases as the x-value increases and vice versa. In order to determine whether the function is increasing or decreasing, the derivative of a function may be used on any intervals in its domain.  If f′(x) > 0 at each point in an interval I, then the function is said to be increasing on I. f′(x) < 0 at each point in an interval I, then the function is said to be decreasing on I. From the given function options, Exponential functions eventually exceed all other increasing functions as the one of variable is in the exponential leading an exponential increase and not a constant increase.

Note: The question is incomplete. The complete question probably is: From the following, Which functions are increasing? How to know if a function is increasing or decreasing?; Which section of the function is increasing?; Which type of increasing function eventually exceeds all other increasing functions? A) linear B) exponential C) polynomial D) quadratic.

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The minimum sample size needed in order for us to be 90% confident that our estimate is within 1.2 hours per week of is 55.

Statistics is the science concerned with developing and studying methods for collecting, analyzing, interpreting and presenting empirical data.

Assuming that the standard deviation of the number of hours worked by college graduates is 6.30 hours per week

We need to find the minimum sample size needed in order for us to be 90% confident that our estimate is within 1.4 hours per week of.

The formula to determine sample size is n = [zs/E]²

z is standardization value indicating confidence level.

s is sample standard deviation.

E is acceptable acceptance error.

n = [invNorm(0.95)×6.3/1.4]²

n = [1.645×6.3/1.4]²

n=54.79=55

Hence, 55 is the minimum sample size needed in order for us to be 90% confident that our estimate is within 1.2 hours per week of.

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The sinusoidal function is y = -4.5cos(8(x + π/2)) + 1/2

From the question, we have the following parameters that can be used in our computation:

Maximum = (-π/2, 5)

Minimum = (π/4, -4)

A cosine function is represented as

y = A cos(B(x+C)) + D

Where

The cosine function has a max and min of 1 and -1.

So, we have

Difference = |-4 - 5|

Difference = 9

This means that

Amplitude, A = 9/2

A = 4.5

Also, we have

Vertically shift = (5 - 4)/2

Vertically shift = 1/2 up

So, the function becomes

y = 4.5cos(θ) + 1/2

Next, we calculate the value of θ

The period is calculated as

Period = π/4 + π/2

Period = -π/4

This means that

2π/B = -π/4

So, we have

B = -8

This gives

y = 4.5cos(-8(x + c)) + 1/2

cos(0) = 1,

So we want -8(-π/2) - 8C = 0

Solving for c, we have

8C = 8(π/2)

Divide

C = π/2

Substitute C = π/2 in y = 4.5cos(-8(x + c)) + 1/2

y = 4.5cos(-8(x + π/2)) + 1/2

y = -4.5cos(8(x + π/2)) + 1/2

Hence, the function is y = -4.5cos(8(x + π/2)) + 1/2

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Complete question

The function has a maximum point at (-pi/2, 5) and a minimum point at (pi/4, -4)

Determine a cosine equation for this function.

Coverage Error results from the exclusion of certain groups of subjects from a population frame. The answer for the given statement is Coverage Error.

Coverage errors come from failure to cover sufficiently all components of the population being studied. Incomplete sampling frames usually result in coverage errors. Coverage errors happen because of the divergences between the target population and the frame.

Coverage error also may happen when not all members of the population of interest have an equal chance of being selected for the survey. For example, if we use e-mail lists or phone lists to build the sample and not all potential respondents are on those e-mail or phone lists.

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The investment options in order from the least risk to the greatest is:

From the options, the least risky investment option would be a b - rated bond as this bond is relatively safe. Following this is property as property is relatively safe because it has tangible value.

Starting a business is quite risky as many new businesses fail. However, investing in speculative stock is the riskiest as the value can fall without warning.

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Answer:

=5x2-(20-5)x-20

=5x2-20x+5x-20

=5x(x-4)+5(x-4)

=(x-4)(5x+5)

Answer:-5(2+3x)

Step-by-step explanation:

Based on the information , the store that gives the best deal is Store A .

Inn the question ,

it is given that ,

the original price for the shoes that Samuel is buying is = $80 .

the discount offered by Store A is = 25% ,

So , the effective price is = 80 - 0.25 of original price  

Substituting the values  ,

we get ,

= 80 - 20

= $60

the discount offered by Store B is = $10

So , the effective price is = 80 - 10 = $70 .

So , the price of shoe is less in store A in comparison to store B .

Therefore , the best deal is given by Store A .

The given question is incomplete , the complete question is

Samuel is buying a pair of boots with an original price of $80. Store A is offering a 25% discount off the original price of all boots. Store B is offering a $10 discount off the original price of all boots .

Based on this information, which store gave the best deal ?

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The answer will be [tex]a^{-20-12a}[/tex] that can be find out using exponential function.

What are exponential function ?

Exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change.

Like, f(x) = [tex]x^{a}[/tex]

x is base and a is power.

Basic rules to solve this function :

1. If we take the product of two exponentials with the same base, we simply add the exponents:

x^ a * x^ b = x^ a + b.

2. If we take the quotient of two exponentials with the same base, we simply subtract the exponents:

x^ a / x^ b =  x^ a-b

3. We can raise exponential to another power, or take a power of a power. The result is a single exponential where the power is the product of the original exponents:

x^ (a)^ b = x^ (ab).

In given equation, [tex]{a^{2^{-2^5}}/{a^{4a^3}}[/tex]

So the answer is a^(-20-12a).

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The time period the projectile at least 160 feet above the ground is 2 sec to 5sec.

What is time period?

The duration of time that a behavior lasts or a condition persists. Depending on the type of activity or situation under consideration, it may be measured in seconds or in millions of years.

Given function that represents height of the projectile above the ground is s feet is s = - 16t² + 112t.

To find the time period is the projectile at least 160 feet above the ground, make an inequality:

- 16t² + 112t  ≥ 160

Subtract 160 from both sides:

- 16t² + 112t - 160 ≥ 0

Divide both sides by 16:

- t² + 7t - 10 ≥ 0

- t² + 2t + 5t - 10 ≥ 0

-t(t -2) + 5(t - 2) ≥ 0

(t - 2)( -t+ 5) ≥ 0

Either, t - 2 ≥ 0 → t ≥ 2

or, -t + 5  ≥ 0 → t ≤ 5

2 ≤ t ≤ 5

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The expression that Rick choose is 99 + 53.

What is parentheses?

The well-known () symbols, often known as "round brackets," are used in pairs to group things together or designate the order of operations in an equation. You will frequently need to employ brackets in math when forming or resolving equations. They support number grouping and operation definition.

Given expression is 64 + 88

The simplest form of the expression 64 + 88 is 152.

152 = 99 + 53.

152 = 151 + 1

152 = 149 + 3

152 = 139 + 13

152 = 137 + 15

152 = 131 + 21

152 = 127 + 25

152 = 113 + 39

152 = 109 + 43

152 = 107 + 45

152 = 103 + 49

152 = 101 + 51

etc.

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The number of buttons A will be equal to 11. And the number of buttons B will be  equal to 11.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

Given that container of buttons contains a similar number of each kind of button it contains 220g of buttons.

The weight of the button is given as follows;

Button A=8 grams

Button B = 12 Grams

Consider x be the number of each button. Then the equation is;

8x + 12x = 220

20x = 220

x = 11

The number of buttons A is 11. And the number of buttons B is 11.

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Answer:

8

Step-by-step explanation:

28/3.5=8

(a) The required sample size is 1.96.

(b) Sample size = n = 89

(c) Sample size = n = 62

What is standard deviation?

The standard deviation in statistics is a measurement of how much a set of values can vary or be dispersed. A low standard deviation suggests that values are typically close to the set's mean, whereas a high standard deviation suggests that values are dispersed over a wider range.

(a) Population standard deviation = σ = 3

Margin of error = E = 0.253

At 95% confidence level the z is,

α = 1 - 95%

α = 1 - 0.95 = 0.05

α/2 = 0.025

Zα/2 = 1.96

sample size = n = [Zα/2* σ / E]^2

n = [1.96 * 3 / 0.253]2

n = 540.14

Sample size = n = 541

b) Population standard deviation = σ = 24

Margin of error = E = 5

sample size = n = [Zα/2* σ/ E] 2

n = [1.96 * 24 / 5]2

n = 88.51

Sample size = n = 89

c) Population standard deviation = σ = 8

Margin of error = E = 2

sample size = n = [Zα/2* σ / E] 2

n = [1.96 * 8 / 2]2

n = 61.46

Sample size = n = 62

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Therefore , the best choice is you to join jmu and after that you will able to earn the same amount you spend on the graduation in 2 year.

Equation: A statement that two variable or integer expressions are equal. In essence, equations are questions, and the motivation for the development of mathematics has been the systematic search for the answers to these questions.

Here,

Given:

You would earn $37,870 per year at walmart

Tuition at jmu is $20,000 per year, textbook expenses are $2,699 per year, and room and board expenses are $15,000 per year.

and you will get a $75,000 per year after graduation.

So the total expense is = 20,000 + 15000 + 2699 =37,699

Therefore , the total expense at jmu = $37,699

and total 4 year graduation course fees = 37,699 * 4=1,50,796

Thus, total fees you have to pay = $1,50,796

and after this you will get a job  of $75,000 per year.

you will able to earn the same amount you spend on the graduation in 2 year.

Therefore , the best choice is you to join jmu and after that you will able to earn the same amount you spend on the graduation in 2 year.

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