The beginning steps for determining the center and radius of a circle using the completing the square method are shown in the table.

Step 1
[original equation]: x2 + 8x + y2 − 6y = 11
Step 2
[group like terms]: (x2 + 8x) + (y2 − 6y) = 11
Step 3
[complete the square]:

Which of the following is the correct equation for Step 3?

Group of answer choices

(x2 + 8x + 4) + (y2 − 6y + 3) = 11 + 4 + 3

(x2 + 8x + 16) + (y2 − 6y + 9) = 11 + 16 + 9

(x2 + 8x + 16) + (y2 − 6y − 9) = 11 + 16 − 9

(x2 + 8x + 4) + (y2 − 6y − 3) = 11 + 4 − 3

The equation in slope - intercept form would be y = 0.6x + 1.1.

Given is that Mr. Phan fills a portion of his pool with water on Monday. Then on Tuesday, he fills the pool with water coming from a hose at a constant rate. The table shows the number of feet of water in the pool as a function of time on Tuesday -

Time on Tuesday after 9 AM (hr) →   0    1      2

Amount of Water in Pool (ft) →          1.1   1.7   2.3​

Let the equation be -

y = mx + c

We can write the slope as -

{m} = (1.7 - 1.1)/(1 - 0)

{m} = 0.6

and

{c} = 1.1        

So, the equation in slope - intercept form would be -

y = 0.6x + 1.1

Therefore, the equation in slope - intercept form would be y = 0.6x + 1.1.

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The relative change in the annual salaries when comparing Fort Wayne to New York City is 66. 7 %

The relative change in the annual salaries when Fort Wayne and New York City are compared can be found by the formula :

= Difference between the annual salaries of both cities / Salary in Fort Wayne

Difference between the annual salaries of both cities:

= 92, 190 - 55, 300

= $ 36, 890

The relative change is:

= 36, 890 / 55, 300

= 66. 7 %

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Using concept of linear equation, Pedro hit runs=1483 and Ricky hit runs=1198.

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

e.g. x-10=0. Here, x is a variable, A is a coefficient and B is constant.

The standard form of a linear equation in two variables is of the form

Ax + By = C

e.g. 2x-4y=10. Here, x and y are variables, A and B are coefficients and C is a constant.

Now,

Given,  

Together, teammates Pedro and Ricky got 2681 base hits last season. i.e. P+R=2681 -->1 where P=Pedro and R=Ricky

Pedro had 285 more hits than Ricky i.e. P-R=285 --->2

Solving equation 1 and 2 we get

P=1483 and R=1198

hence,

          Pedro hit runs=1483 and Ricky hit runs=1198.

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2x+6 is the required weight.

The weight of an object is defined as the force acting on the object due to gravity. It is a scalar quantity. Weight is measured in newtons.

In Physics, the gravitational force with which the earth attracts all the masses to its center is called weight.

In the given question,

Let the weight of an object is x.

6 pounds more than twice the weight means the required weight of an object will be the 2×weight of the object(x) + 6 pounds.

Therefore,

2x+6 is the required weight of the object.

The complete question is,

The weight of 1st object is x. The weight of the second object is 6 pounds more than twice the weight. What is the weight of the second object?

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The needed weight is 2x+6.

The force exerted on an object by gravity is known as the weight of the object. It has a scalar value. Newtons are a unit of weight.

Weight is the term used in physics to describe the gravitational force that pulls all masses toward the center of the earth.

In the posed query,

Let's say that an object weighs x pounds.

The needed weight of an object is the weight of the object(x) plus 6 pounds, which is 6 pounds more than twice the weight.

Therefore,

The object must weigh at least 2x+6.

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The complete question is:-

The first item weighs x pounds. The second object weighs 6 pounds more than it does, or more than twice as much. What does the second object weigh?

The class intervals would be:

Class 1: 2-10

Class 2: 11-19

Class 3: 20-28

Class 4: 29-37

Class 5: 38-46

Class interval can be determined by dividing the range of the data by the desired number of intervals (also known as classes).

First, we need to find the range of the data:

Maximum value: 47

Minimum value: 2

Range: 47 - 2 = 45

For this example, let's choose 5 classes.

Class interval = Range / Number of classes = 45 / 5 = 9

The class intervals would be:

Class 1: 2-10

Class 2: 11-19

Class 3: 20-28

Class 4: 29-37

Class 5: 38-46

Note: The upper limit of each class is always rounded up.

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The method of moment estimators of a and b are [tex]\hat{a}=\bar{X}-\sqrt{3}S_x[/tex] and[tex]\hat{b}=\bar{X}+\sqrt{3}S_x[/tex] . And, the maximum likelihood estimators of a and b are  [tex]\hat{b}=x(n)[/tex] and [tex]\hat{a}=x(1)[/tex].

The random sample given is X₁, X₂, X₃,..........., Xₙ from the uniform distribution on [a, b]. The probability density function (pdf) of this is written as,

[tex]f(x)=\begin{cases} \frac{1}{b-a},&a < x < b\\0,&\text{otherwise} \end{cases}[/tex]

a) Now, let's calculate the expected value of X E(X). We get,

[tex]\begin{aligned}\mu _1=E(X)&=\int_a^b x\cdot\frac{1}{b-a}\;dx\\&=\frac{1}{b-a}\left[\frac{x^2}{2}\right]_a^b\\&=\frac{b+a}{2}\end{aligned}[/tex]

Let's calculate the second moment of X. We get,

[tex]\begin{aligned}\mu_2=E(X^2)&=\int^b_ax^2\cdot\frac{1}{b-a}\;dx\\&=\frac{1}{b-a}\left[\frac{x^3}{3}\right]_a^b\\&=\frac{b^3-a^3}{3(b-a)}\\&=\frac{b^2+a^2+ab}{3}\end{aligned}[/tex]

From the above values, variance is calculated as follows,

[tex]\begin{aligned}\mu_2-\mu_1^2=E(X)-E(X^2)&=\frac{b^2+a^2+ab}{3}-\frac{a^2+b^2+2ab}{4}\\&=\frac{4(b^2+a^2+ab)-3(a^2+b^2+2ab)}{12}\\&=\frac{b^2-2ab+a^2}{12}\\&=\frac{(b-a)^2}{12}\end{aligned}[/tex]

Now, using the method of moment, we denote [tex]E(X^2)=\mu_r[/tex] and [tex]M_n=\frac{1}{n}\sum_{i=1}^{n}X_i^2[/tex]. Then, [tex]\hat{\mu_r}=\frac{1}{n}\sum_{i=1}^{n}{X_i} ^r[/tex].

When [tex]\hat{a}[/tex] and [tex]\hat{b}[/tex] are the method of moment estimator of a and b, then, [tex]\hat{\mu_1}=\frac{\hat{a}+\hat{b}}{2}=\bar{X}[/tex] and [tex]\hat{\mu_2}=\frac{1}{n}\sum_{i=1}^{n}{X_i}^{2}[/tex].

Then,

[tex]\begin{aligned}\hat{\mu_2}-\hat{\mu_1^2}&=\left(\frac{1}{n}\sum_{i=1}^{n}({X_i}^{2}\right)-\bar{X}^2\\&=\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar{X})^2\\&={S_x}^{2}\\\frac{(\hat{b}-\hat{a})^2}{12}&={S_x}^{2}\end{aligned}[/tex]

From [tex]\hat{b}+\hat{a}=2\bar{X}[/tex] and [tex]\hat{b}-\hat{a}=\sqrt{12{S_x}^{2}}=2\sqrt{3}S_x[/tex].

Then, the moment estimators of a and b are [tex]\hat{a}=\bar{X}-\sqrt{3}S_x[/tex] and[tex]\hat{b}=\bar{X}+\sqrt{3}S_x[/tex] .

b) Using the joint distribution of [tex]X=(X_1,........, X_n)[/tex] be

[tex]\begin{aligned}L(x_i\; a,b)&=\frac{1}{(b-a)^n}, a < x_i < b\\&=\frac{1}{(b-a)^n}, a < x(1) < x(n) < b\\&=\frac{1}{(b-a)^n}\cdot I\{x(n) < b\}\cdot I\{x(1) > a\}\end{aligned}[/tex]

where [tex]I_A=\begin{cases}1\;\text{if A is true}\\0\;\text{if not}\end{cases}[/tex]

The joint distribution of likelihood is maximum when b-a is minimum under the constraints b>x(n) and a<x(1). Then, [tex](b-a) > x(n)-x(1)[/tex]. So the minimum value is attained when [tex]b-a=x(n)-x(1)[/tex] that is [tex]\hat{b}=x(n)[/tex] and [tex]\hat{a}=x(1)[/tex]. Hence, the maximum likelihood estimate of a and b.

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The total weight of the wire is 24.32 ounces.

Multiplication of two numbers is defined as the addition of one of the number repeatedly until the times of the other number.

a × b means that a is added to itself b times or b is added to itself a times.

Given that,

Total length of the wire = 7.6 meters

Weight of each meter of wire = 3.2 ounces

Total weight of the wire = Total meter of the wire × Weight of each meter

                                        = 7.6 meters × 3.2 ounces

                                        = 24.32 ounces

Hence the weight of the total length of the wire is 24.32 ounces.

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y=-2x+5 is the slope-intercept equation for a line that goes through the point (4,-3) and 5 is the value of b.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

We have to find the slope-intercept equation for a line that goes through the point (4,-3)--with a slope of -2

m=-2

Now let us find the y intercept

-3=-2(4)+b

-3=-8+b

-3+8=b

5=b

Hence, y=-2x+5 is the slope-intercept equation for a line that goes through the point (4,-3) and 5 is the value of b.

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Answer: t= 12.9

Step-by-step explanation:

14.2=-11.6+2t

25.8=2t

t=12.9

The percentage multiplier to find 26% of an amount is, 0.26.

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

We know that;

Percentage multiplier calculates the percentage of an amount. It is also used to increase or decrease an amount by a percentage.

Now, In order to find the percentage multiplier of 26% of an amount we can multiply the number by a multiplier.

⇒ 26%

⇒ 26/100

⇒ 0.26

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The piecewise function that defines the graph is given as follows:

A piecewise function is a function that has different definitions, according to the input values assumed by the function.

The different definitions for the function in this problem are given as follows:

To the left of x = -2, the function is increasing with a slope 1, hence it would cross the y-axis at y = 1, and the definition is given as follows:

y = x + 1, x < -2.

At x = 2 and between x = -2 and x = 1, the function is constant.

The piecewise function is given by the image presented at the end of the answer.

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For a, the percentage of given angle with respect to whole rotation is 0.36% and 1.3° is equivalent to 0.0072π radian.

What is the percentage?

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word percent means per 100. It is represented by the symbol “%”.

Examples of percentages are:

10% is equal to 1/10 fraction.

20% is equivalent to ⅕ fraction.

25% is equivalent to ¼ fraction.

Now,

Given angle=1.3°

Total rotation = 360°

So, percentage of given angle=1.3/360*100=0.36%

and 1.3° in radian=1.3*π/180=0.0072π

Hence,

            the percentage of given angle with respect to whole rotation is 0.36% and 1.3° is equivalent to 0.0072π radian.

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The correct answer is A, Canada. This is because the select() function allows you to specify the exact variables to be included in the data frame, and Company.Location is one of the variables that was specified in this case.

The first row of the table created using the select() function contains the data for the three selected variables from the original flavors_df data frame. The select() function was used to create a new data frame with only three variables: Rating, and Cocoa.Percent, and Company.Location. This function was used to filter out all the other variables from the original flavors_df data frame, leaving only the three variables of interest.

When looking at the first row of the table, the value in the Company.Location column is Canada, which is the country in the company producing the chocolate is located in.

Complete question:

After previewing and cleaning your data, you determine what variables are most relevant to your analysis. Your main focus is on Rating, Cocoa.Percent, and Company.Location. You decide to use the select() function to create a new data frame with only these three variables.

Assume the first part of your code is:

trimmed_flavors_df <- flavors_df %>%

Add the code chunk that lets you select the three variables.

select(Rating, Cocoa.Percent, Company.Location)

What company location appears in row 1 of your table?

Single Choice Question. Please Choose The Correct Option &#x2714;

A. Canada

B. Scotland

C. France

D. Columbia

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Answer:

at least 160 students have to buy a ticket to cover the show's cost.

Step-by-step explanation:

1200 divided by 7.50 = 160.

A two-way table is a statistical tool used to summarize and display the relationship between two categorical variables.

Below is a two-way table that shows the data from the question:

                Popcorn            No Popcorn    Total

Drink            84 - 10 = 74            15              89

No Drink      10                            6              16

Total            84                           21              105

It is a table that displays frequencies (counts or percents) for all possible combinations of two categorical variables which are popcorn and no popcorn and also drink and no drink.

The above table shows that 74 people had both popcorn and a drink, 10 had popcorn but no drink, 6 had neither drink nor popcorn, and 15 had a drink but no popcorn.

The total number of people who had popcorn is 84, and the total number who had a drink is 89.

The total number  of people who's not had popcorn is 21, and the total of  people who's not had dring is 21

The total number of people in the theater is 105.

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The total receipts after the end of 1,5,7 years are 124 , 604, 844 million respectively.

Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given here: The function T(x)= 120x²+4

Thus after 1 year the receipts is T(1)=120×1+4

                                                          =124 million

After 5 years T(5)=120×5+4

                            =600+4

                            =604 million

Similarly for t=7 years T(7)=840+4

                                          =844 million

Hence, The total receipts after the end of 1,5,7 years are 124 , 604, 844 million respectively.

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1. The function is continuous everywhere because it is defined for all values of x, and there are no points where f(x) is undefined or where the limit as x approaches a specific value does not exist.

2. Interval of continuity: (-∞, ∞)

Discontinuities: None

Conditions of continuity not satisfied: None

Hence, option (d) is correct.

If and only if a function is continuous on every point in the open interval (a,b), it is continuous over that interval (a,b).

If and only if it is continuous on (a,b), then is continuous over the closed interval [a,b].

The right-sided limit of at x=a is (a), while the left-sided limit of at x=b is (b).

If a function is continuous across the whole open interval, it is said to be continuous over the interval. If it is continuous at each point in its interior and at both of its ends, it is continuous over a closed interval.

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For a car worth $38,890.00 a person should invest $23,403.660 for a rate of 10.2% for 5 years if compounded monthly.

What is Compound Interest?

The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest. Compound interest is commonly abbreviated C.I. in mathematics.

The amount is A = $38,890.00

The time period in years is t = 5 years

The rate is given as r = 10.2%

The amount is compounded monthly, so n = 12.

First, convert R as a percent to r as a decimal -

r = R/100

r = 10.2/100

r = 0.102 rate per year,

The formula for compound interest is -

A = P(1 + r/n)^nt

Where,

A = amount

P = principal

r = rate of interest

n = number of times interest is compounded per year

t = time (in years)

Substitute the values into the equation -

A = P(1 + 0.102/12)^(12)(5)

38,890 = P(1 + 0.0085)^(60)

38,890 = P(1 + 0.0085)^(60)

38,890 = P(1.661705)

P = 38,890 / 1.661705

P = 23,403.660

Therefore, the principal value is $23,403.660.

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Answer: L=3, W=

Step-by-step explanation: Considering it is a rectangle

P = 2 (l+b) where L equals length and B breadth width)

10 = 2( L + (2/3×L)

Divide by 2

5= x + 2/3×x

5= x/1 + 2/3×x

5= x/1 + 2x/

expand the fraction to get the MCD

3x/3×1 + 2x/3

3x/3 + 2x/3

3x + 2x / 3

5x/3

5= 5x/3

multiply them by 3/5

3/5×5 = 3/5×5/3x.

cancel first 5s

3=3/5 × 5/3x.

cancel 3s

3= 1/5 ×5x

cancel 5s

3=x.

3 is length, so let's add everything up.

p = 2.(l+b)

10= 2. (3+b)

Divide this by

5=3+x

highlight the x by changing order of the numbers.

-x=3-5

-x = -2

then multiply by (-1) to make them positive:

x=

Therefore, length is 3 and width is 2.

Her total pay for last week is given as follows:

£506.25.

The total pay is obtained applying the proportions in the context of this problem.

The rates are given as follows:

The number of hours is given as follows:

Hence the total earnings are given as follows:

5 x 7.5 x 10.8 + 7.5 x 13.50 = £506.25.

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The function represents an exponential decay, as it is a decreasing function.

A function is classified as exponential growth if it is an increasing function, that is, the function is increasing over it's entire domain.

A function is classified as exponential decay if it is an decreasing function, that is, the function is decreasing over it's entire domain.

As the function in this problem is decreasing, it is an exponential decay function.

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The area of the octagon is 11+ 12√2 square units.

The length of the sides of an equiangular octagon is 1, 2, 3, 4, 1, 2, 3 and 4 in that order clockwise.

Suppose it is in a rectangle where the 4 unit sides and the 2 unit sides lie on the sides of the rectangle.

The area of the outer rectangle is shown in the upper-right corner of the sketch.

As per the question, the octagon is equiangular, so all its internal angles are 135 degrees. This means it fits neatly in a rectangle of dimension 3+3√2 by 1+3√2. The area of the octagon is the area of the rectangle minus the area of the four right isosceles triangular corners = 11+12√2.

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Answer:

[tex]C = 16.8954 ^\circ[/tex]

Step-by-step explanation:

Law of Sines

The rule of sines states: If a, b, and c are, respectively, the lengths of a triangle's legs on either side of angles A, B, and C:

[tex]\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}[/tex]

Given:

[tex]C = \sin^{-1} \left[ \dfrac {c \sin B}{b}\right][/tex]

[tex]= \sin^{-1}\left[\frac {{13}{\sin}(28 ^\circ)} {21}\right][/tex]

[tex]= 16.8954 ^\circ[/tex]

Thanks.

From the options given in the problem statement the values of f(x) and g(x) that satisfy equation (2) i.e. y = f(g(x)) = (10)/(√(2x + 9))are:f(x) = 10/√(x) and g(x) = 1/(2x + 9)

What is function ?

Function can be defined in which it relates an input to the output.

In this problem y is called a composite function of f(x) and g(x). So in the options given in the problem statement which option will satisfy y = f(g(x)) is to be found out.

Let f(x) be defined as some function then if ‘x’ is replaced by g(x) in f(x) i.e. f(g(x)) then its value will be y. Hence we can write:

y = f(g(x)) --- (1)

We are also given that

y = (10)/(√(2x + 9))

Let us consider each of the cases given under options, one by one. We have to find out the nature of the function f(x) as well as g(x) so that we substitute g(x) for x in f(x) we have:

y = (10)/(√(2x + 9))

(i) Consider f(x) = 10 and g(x) = √(2x + 9)

Here f(x) cannot take g(x) as its inputs.

(ii) Consider f(x) = 10/√(x) , g(x) = 2x + 9

Here we determine that f(g(x)) = (10)/(√(2x + 9))

(iii) Consider f(x) =10/x , g(x) = 2x + 9

Here f(g(x) = 10/(2x+9)

(iv) Consider f(x)= √(2x + 9) , g(x) = 10

f(g(x) = f(10) = √(2(10) + 9) = √29

Hence, From the options given in the problem statement the values of f(x) and g(x) that satisfy equation (2) i.e. y = f(g(x)) = (10)/(√(2x + 9))are:

f(x) = 10/√(x) and g(x) = 1/(2x + 9)

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Using algebraic equation, the number of hot dogs sold is 40 and the number of  sodas sold is 90.

algebraic equation, statement of the equality of two expressions formulated by applying to a set of variables the algebraic operations, namely, addition, subtraction, multiplication, division, raising to a power, and extraction of a root.

Given that they sold one hot dogs at $3 and one sodas at $2.

The number of sodas sold was 10 more than two times the number of hot dogs sold.

Let the number of hot dogs sold is x

The number of sodas sold is 2x + 10

The school made total $300. SO that,

x*$3 + (2x+10)*$2 = $300

⇒ 3x + 4x + 20 = 300

⇒ 7x + 20 = 300

subtracting 20 from both side of equation

⇒ 7x = 300 - 20

⇒ 7x = 280

divided both side by 7

⇒ x = 280/7

⇒ x = 40

so the number of hot dogs sold is 40.

and the number of sodas sold is 2x + 10

= 2*40 + 10

= 80 + 10

= 90

The number of sodas sold is 90.

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Miss Saver needs to withdraw $391 each year from her bank.

To solve this problem, we need to

First, we need to determine the total amount of money Miss Saver will have saved using:

Total = $1000 * (1 + 0.05)^30 = 4321.94

Next, we need to determine the future value of this amount after 30 years at an effective annual interest rate of 2%.

This is done using

$4321.94 * (1 + 0.02)^30 = $7828.60

Now, we can determine the annual withdrawal amount, X as follows

X = $7828.59/20 = $391.4295

Rounding up to the nearest integer, X = $391

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The solution for the equation 2cos4x - 1 = 0 is x = 3π/35.

For the values:

x = π/5 + 2kπ and also;

x = π/7 + kπ

then we can solve for k as follows:

π/5 + 2kπ = π/7 + kπ

collect like terms

2kπ - kπ = π/7 - π/5

kπ = (5π - 7π)/35 {simplifying the right hand side to a single fraction}

kπ = -2π/35

k = -2/35 {divide through by π}

put -2/35 for k in the equation x = π/7 + kπ to get the value of x;

x = π/7 - 2π/35 {simplifying the right hand side to a single fraction}

x = (5π - 2π)/35

x = 3π/35

Therefore, the solution for the equation 2cos4x - 1 = 0 is x = 3π/35.

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Answer:

The most the couple can spend for the meal is $82.82 to the nearest cent.

Step-by-step explanation:

Let x be the greatest amount (in dollars) that the couple can spend for the meal.

If a sales tax of 5% is added to the bill, then the bill will be:

[tex]x+0.05x=1.05x[/tex]

If a tip of 15% is then added to the bill, the bill will be:

[tex]1.05x+0.15 \cdot 1.05x=1.2075x[/tex]

If the maximum the couple can spend is $100 then:

[tex]1.2075x\geq100[/tex]

Solve the equation for x:

[tex]\begin{aligned}1.2075x &\geq 100\\\\x&\geq \dfrac{100}{1.2075}\\\\x&\geq 82.8157349...\end{aligned}[/tex]

Therefore, the most the couple can spend for the meal is $82.82 to the nearest cent.

Step-by-step explanation:

The equation being

D = -65m

They provides us with m

We can plug it in to find out the value of D

The first one is 0

D = -65(0) = 0m

The second is 1

D = -65(1) = -65m

And the last one is 2

D = -65(2) = -130m