Find the average rate of change for the following functions over the interval.

Answer:

The sides of the triangle are 15 inches and the sides of the square are 15 - 6 = 9 inches

Step-by-step explanation:

It is given that the perimeter of an equilateral triangle is 9 inches more then the perimeter of a square, and side of the triangle is 6 inches longer then the side of the square.

Now,

An equilateral triangle has all 3 sides the same length

Let one side of the triangle be x so the perimeter will be 3x.

Further, if one side of the square is 6 longer than a side of the triangle

then the square has side length x - 6 and perimeter is;

=> 4(x - 6)

=> 4x - 24

Since, the perimeter of the triangle is 9 more than the perimeter of the square we get the equation as;

=> 3x = 4x - 24 + 9

=> x =  15

Therefore,

The sides of the triangle are 15 inches

The sides of the square are 15 - 6 = 9 inches

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Answer: The monthly payment assuming that each month he paid the same amount is = 362.5 $

Step-by-step explanation:

Given data,

Johnny bought a used car for  $7,500

He had $4600 and borrowed the rest from his parents.

Johnny 8 months to pay back the money he borrowed

So, we can write ,

Let us assume,   x1 = $7,500  and  x2 = $4600

Based on the given conditions,

we use formula is :

                          x = ( x1 - x2 ) / pay back months

                          x = ( 7,500 - 4,600 ) / 8

we can subtract the values,

we can write,

                    x = 2900/8

                      x = 725/2

                      x = 362.5 $

Therefore,

The monthly payment assuming that each month he paid the same amount is = 362.5 $.

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the extraneous solution for the equation √ (x + 3) - 1 = x is x = 1 or x = - 2.

We are given an equation:

√ (x + 3) - 1 = x

√ (x + 3) = x + 1

Raise both sides to the power 2, we get that:

x + 3 = x² + 1 + 2 x

x² + 2 x - x + 1 - 3 = 0

x² + x - 2 = 0

x² + 2 x - x - 2 = 0

x ( x + 2) - 1 ( x + 2) = 0

(x - 1)(x + 2) = 0

x = 1 or x = - 2.

Therefore, we get that, the extraneous solution for the equation √ (x + 3) - 1 = x is x = 1 or x = - 2.

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Brianna would receive 28 votes from students.

So, a recurring decimal at the number "2" is "0.6222.

Let x = 0.62222....

Subtract 10x from 100x as follows:

Therefore, Brianna would receive 28 votes from students.

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The correct question is given below:
Brianna asked 45 students if they would vote for her to be student council president. She used her calculator to compare the number of students who said yes with the total number of students. Her calculator showed the result as 0.6222...

1. Write this number as a fraction (Steps)

2. How many students said they would vote for Brianna?

The amount in a continuously compounded account for the given conditions is $2330.64 .

It is given that principal is 2000 , annual interest rate is 5.1 % , time is 3 years

Putting [tex]P=\$2000 , \ r= 5.1\% , \ t=3 \ years[/tex] in equation (1) , we get

  [tex]A=2000\times 2.71^{0.051 \times 3}\\A=200\times 2.71^{0.153}\\ \ A=2000\times1.1653\\A =\$2330.64[/tex]

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A number line plot is a graph that displays the frequency a number occurs in a set of data. Dot plot is another name for a number line plot.

The number line is a one-dimensional graph that can display all of the rational numbers (whole numbers, integers, and rational numbers).

The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1,2,3,...} where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero , 0,1,2,3,...

The set of integers adds the opposites of the natural numbers to the set of whole numbers: −3,−2,−1,0,1,2,3,

It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.,−3,−2,−1,negative integers

0,zero1,2,3, positive integers.

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The mentioned number in word form is three hundred ten, seven hundred sixty three, one hundred thirty six.

The arithmetic value of a number is one that is used to denote quantity. As a result, a number is a mathematical concept used for counting, measuring, and labeling. As a result, mathematics is based on numbers.

Early humans used a variety of symbols to represent numbers, as evidenced by the inscriptions discovered at archaeological sites. For instance, farmers, traders, and merchants in antiquity used tally marks to indicate quantities. A standing line is drawn for each count in tally marks, and the fifth count is indicated by crossing out the first four lines. However, this was a laborious method and it was impractical to display quantities.

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Answer:

4(2)³+A×2²+7(2)-1=7

4×8+4A+14-1=7.

32+14-1-7=-4A..

38=-4A.

A=38/-4.

The original value of the machine is $ 30 , 720 , 000 .

Given : Decay factor = 0.75 per year

           Present value of machine ( after 6 yrs . ) = $ 7500 .

According to ques .

The original value of machine be worth $ x .

After nth yr. the present value of machine will be

( x - 0.75 x ) [ [tex]( 1 - 0.75 ) ^{n-1}[/tex]  ]  .

Therefore , after 6 yrs. the present value of machine will be

( x - 0.75 x )  [ [tex]( 1 - 0.75 ) ^{5}[/tex]  ]  = 7500

0.25 x * ( [tex]0.25 ^{5}[/tex] ) =7500

Hence , x = 30 , 720,000

Therefore , the original value of the machine is $ 30 , 720 , 000 .

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Answer:

Alternate Exterior Angles

Answer:

alternate interior angles

The required value of z using given conditions and functions is 7.92.

What is a function?

A function is defined as an expression or rule that specifies the relationship between two variables or objects, out of which one is dependent and the other is an independent variable. The image of each and every element is unique.

Calculation for the value of z

Given value of z = 1.5, x = 9, y = 4

Consider z as a function varying directly with x and inversely with y i.e.

z = kx / y        —-- 1

Putting the values of x, y, z, we get,

1.5 = k × 9 / 4

6 = 9 × k

k  = 6 / 9

k = 2 / 3 = 0.66

Now, calculating z for x = 6, k = 0.66 and y = 0.5, using function,

z = (0.66 × 6) / 0.5

= 3.96 / 0.5

= 7.92

Hence, the required value of z = 7.92.

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After solving the equation the value of x in (4x - 1) = 36 is  37/4

We have been asked to find x in (4x - 1) = 36

⇒ (4x - 1) = 36

⇒ 4x = 36 + 1

⇒ 4x = 37

⇒ x = 37/4

So, x is equal to 37/4

To solve linear equations, utilize the steps that are provided below.

Lets take an example of solve z for 7z – (3z – 4) = 12

Here is no multiplication or division, so this will be easy to solve

⇒ 7z – (3z – 4) = 12

⇒ 7z – 3z + 4 = 12

⇒ 4z = 12 – 4

⇒ 4z = 8

⇒ z = 8/4

⇒ z = 2

See, that was so simple, using the mentioned methods, every linear equation can be solved easily.

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Answer:x=21

Step-by-step explanation:

10/14=15/x

5/7=15x^-1

(5/7)/15=ans

ans=x^-1

ans^-1=21

Answer: x=21

Step-by-step explanation:

[tex]\frac{10}{14} =\frac{15}{x}[/tex]

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 14x, the least common multiple of 14,x.

x×10=14×15

Multiply 14 and 15 to get 210.

x×10=210

Divide both sides by 10.

x=

10

210

​

Divide 210 by 10 to get 21.

x=21

Answer:

= 17

Step-by-step explanation:

If f(x)=3x+2, then replacing x=5 we get f(5)=3x5+2=15+2=17

Answer:

the answer to the equations is 17

Step-by-step explanation:

Answer:

Triangular inequality:  Length of third side must be less than the sum of the lengths of the other 2 sides.

x < 18 + 29

x < 47

so Maximum length of third side is 46 units.

Also 29 < 18 + x

x > 11    so minimum length = 12

Difference  = 46-11 = 35

The exponential function for the value of car is V = 25,000 × e^( r × T) and the value of the car after 5 years is $ 1109.26.

The original value of a car is $25,000.

The value of the car after one year is $21,250.

Let V be the value of the car after T years.

Let the rate at which the value of the car increase be r.

From, [tex]\[ A = Pe^{r \times T} \][/tex]

Then,

[tex]\[ V = 25,000 \times e^{ r \times T } \][/tex]

The value of the car after one year is $21,250.

[tex]\[ 21,250 = 25000 \times e^{ r \times T } \][/tex]

Divide both sides by 25000

[tex]\frac{21250}{25000} = e^{1 \times r }[/tex]

Now, taking the log of both sides

[tex]\[ \ln (21250/25000) =\ln e^{1 \times r} \][/tex]

By the property of logarithm,

[tex]\ln e^{(1 \times r)} = r[/tex]

Therefore,

㏑(21250/25000) = r

Now, the value of the car after 5 years will be:

[tex]\[ V = 25,000 \times e^{ r \times T} \]\\\\[/tex]

[tex]\[ V = 25000 \times e^{ \ln (\frac{21250}{25000} ) \times 5 } \][/tex]

V = 25000 × 0.4437053125

V = $ 1109.26

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Answer: Solution is t = 905/8

Step-by-step explanation:

Given data,

         ttt:-\dfrac{7}{4}=\dfrac{2}{5}t− 47 = 52​ t

We can write,

              - 7/4 = (2/5) t - 47

Rearrange variables to the left side of the equation

               - (2/5) t = - 47 + 7/4

Find common denominator and write the numerators above common denominator

                 - (2/5) t = ( -188 + 7 ) / 4

calculate the sum or difference

                   - (2/5) t = -181/4

Divide both sides of the equation by the coefficient of variable

                      t = (-181/4) x (-5/2)

Determining the sign for multiplication or division

                       t = 181/4 x 5/2

Write as a single fraction

                          t = ( 181 x 5 ) / ( 4 x 2 )

Calculate the product or quotient

                          t =  (905/8)

Therefore,   t = 905/8

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The answers will be :

a. The length of the line segment will be equal to 8.24 units.

b. The midpoint of the line segment will be ( -2 , -2 ).

c. The equation of line passing through points Y and Z will be 4y + x - 10 = 0

What is slope of a line segment ?

The ratio of the difference in y-coordinates over the equivalent x-coordinates between two different locations on a line.

It is given that a line segment has endpoints of Y(2,-3) and Z(-6,-1).

Let's solve the given parts based on the above data.

a.

The length of the line segment will be given by :

YZ = [tex]\sqrt({x_{2} - x_{1})^2 + (y_{2}- y_{1})^2}[/tex]

YZ = √ [( -6 -(2)]^2 + [ -1 - (-3)]^2}

YZ = √ (-8)² + 2²)

YZ = √68

YZ = 8.24 units

b.

The midpoint (say M) of the line segment will be given by :

M = ( [tex]\frac{x_{1} + x_{2} }{2} , \frac{y_{1} + y_{2} }{2}[/tex])

M = [(2-6)/2 , (-3-1)/2]

M = ( -2 , -2 )

c.

And the equation of line passing through points Y and Z will be :

[tex]y_{2} - y_{1}[/tex] = m ( [tex]x_{2} - x_{1}[/tex] )

Let's calculate value of slope (m) firstly which will be :

m =([tex]y_{2} - y_{1}[/tex]) / ( [tex]x_{2} - x_{1}[/tex] )

m = ( -1 + 3 ) / (-6 - 2)

m = 2 / -8

m = -1 / 4

or

m = -0.25

Using the value of slope (m) : we get the equation of line as :

[tex]y - y_{1}[/tex] = - 0.25 ( [tex]x - x_{1}[/tex] )

or

(y + 3) = - 0.25 ( x - 2)

y + 3 = -0.25 x + 0.50

or

y + 0.25 x - 2.5 = 0

If we multiply by 4 throughout the equation ; then the equation of line can also be written as :

4y + x - 10 = 0

Therefore the answers will be :

a. The length of the line segment will be equal to 8.24 units.

b. The midpoint of the line segment will be ( -2 , -2 ).

c. The equation of line passing through points Y and Z will be 4y + x - 10 = 0

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Expression as a single natural logarithm of 4 ln 3  is 4.39444915467

4 ln 3  

=[tex]ln 3^{4}[/tex]

= ln 81

=4.39444915467

The inverse of an exponential function, the natural log is the logarithm to the base of the number e. Natural logarithms are unique varieties of logarithms that are employed in the treatment of time and growth-related issues.

The distinction between log and ln is that log is expressed in terms of base 10, while ln is expressed in terms of base e. As an illustration, log of base 2 is denoted by log2 and log of base e by loge = ln (natural log).

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Answer:

B. x = (4/3)

Step-by-step explanation:

-x + (3/7) = 2x - (25/7)

(-x + (3/7) = 2x - (25/7)) × 7

-7x + 3 = 14x - 25

-14x        -14x

-21x + 3 = -25

         -3     -3

-----------------------

-21x = -28

÷-21    ÷-21

-----------------

x = (4/3)

I hope this helps!

The required expression that describes the shaded region is (A'∩B) ∪ (A ∩ B'). Option A is correct.

GIven that,
For the given Venn diagram which expression represents the shaded area

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
The total area of A and B
= AUB
Area of unshaded area,
= (A'∩B) ∪ (A ∩ B')

Thus, the required expression that describes the shaded region is (A'∩B) ∪ (A ∩ B'). Option A is correct.

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Answer:

2.5

Step-by-step explanation:

2 2/4

Simplify 2/4

2/4 = 1/2 = 0.5

2 2/4 = 2 + 0.5 = 2.5

2 2/4 = 2 1/2

2 1/2 = 2.50 (or just) 2.5

Answer:

a) 6 hours

b) 12 hours

c) 36 hours

Step-by-step explanation:

let hours taken to mow lawns = x

baby sit hours = 2x

walk dogs hours = 3(2x) = 6x

x+2x+6x = 54

thus, x = 6

Answer:

10 pencils

Step-by-step explanation:

16-x = 36-3x

16-36 = -3x+

thus x = 10

An explicit formula for Mallory's run will be 15 + 5x.

Mathematical expression is the combination of numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also be used to denote the logical syntax's operation order and other properties.

Given that on June 1st, she starts by running for 15 minutes, and every day after that she adds 5 minutes to her run.

Equation = 15 + 5x, where x is the number of days.

Therefore, the explicit formula for Mallory's run will be 15 + 5x.

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Answer:

0,896

Step-by-step explanation:

the slope is 3 for the equation

Answer:

the slope is 3 for the equation

Step-by-step explanation:

If the expression is log₂9-log₂3, then it will look like [tex]log_{2}3[/tex] in single logarithm.

Given that the expression is log₂9-log₂3.

We have to give the single logarithm for the given expression.

The quotient rule for logarithms tells us that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule we also can use the inverse property to derive the quotient rule. The quotient property basically says that log (m/n)=log m-log n.

The expression is log₂9-log₂3.

log₂9-log₂3=[tex]log_{2}[/tex](9/3)

We can write it as [tex]log_{2}3[/tex].

Hence if the expression is log₂9-log₂3, then it will look like [tex]log_{2}3[/tex] in single logarithm.

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