The water usage at a car wash is modeled by the equation W(x) = 3x^3 + 4x^2 − 18x + 4, where W is the amount of water in cubic feet and x is the number of hours the car wash is open. The owners of the car wash want to cut back their water usage during a drought and decide to close the car wash early two days a week. The amount of decrease in water used is modeled by D(x) = x^3 + 2x^2 + 15, where D is the amount of water in cubic feet and x is time in hours.

Write a function, C(x), to model the water used by the car wash on a shorter day.

By rewriting the 3 given systems, we will see that:

1) Independent and inconsistent.

2) Independent and consistent.

3) Independent and inconsistent..

Let's analyze each system.

Let's look at the first one, we can rewrite it as:

y = 3x - 1

And, 2y - 6x = 4

2y = 6x + 4

y = 3x + 2

Here both lines have the same slope, so the lines never do intersect, meaning that the system is inconsistent and independent.

Now let's look at the second system, we can rewrite it as:

2y = - x + 6

And, 4y + 2x = 12

y = - 1/2x + 3

So, the slopes are different, meaning that we have one solution, so this system is consistent and independent.

For the final system we have:

y = 2x + 3

x + y = - 9

So, the slopes are different, meaning that we have one solution, so this system is consistent and independent.

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[1st Picture] Answer: x = 3.98

Step-by-step explanation:

    We are given the opposite side and adjacent side from our reference angle. This will use the tangent function. tanθ = [tex]\frac{opposite}{adjacent}[/tex].

tan(53°) = [tex]\frac{x}{3}[/tex]

3tan(53°) = x

x = 3.98

[2nd Picture] Answer: x = 3.13

Step-by-step explanation:

    Here, we are given the opposite side from our reference angle and the hypotenuse. This will use the sine function. sinθ = [tex]\frac{opposite}{hypotenuse}[/tex].

sin(23°) = [tex]\frac{x}{8}[/tex]

8sin(23°) = x

x = 3.13

[3rd Picture - #6] Answer: θ = 63.4°

Step-by-step explanation:

    Again, we are given the opposite side from our reference angle and the hypotenuse. This will use the sine function. sinθ = [tex]\frac{opposite}{hypotenuse}[/tex]. To find the angle measurement of theta at the end, we will use the inverse of sine.

sin(θ) = [tex]\frac{6}{6.71}[/tex]

[tex]sin^{-1}[/tex](sin(θ)) = [tex]sin^{-1}[/tex]([tex]\frac{6}{6.71}[/tex])

θ = 63.4°

Answer:

x = 65 degrees

Step-by-step explanation:

The sum of all 4 interior (inside) angles in a quadrilateral = 360.

So 82+118+95+x = 360.

295+x = 360

x = 360-295

x = 65

Check that all the angles add up to 360:

82+118+95+65 = 360

The surface area of the box which is a rectangular prism would be = 24 square inches. That is option D.

To calculate the area of the given box the formula that should be used is the formula of the area of a rectangular prism. That is;

Area of rectangular prism = length×width×height

Where:

length = 4 inches

width = 3 inches

height = 2 inches

Area = 4×3×2 = 24 square inches

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The height of the beaker is 6.35 cm.

To find the height of the beaker, we can use the formula for the volume of a cylinder, which is given by:

Volume = π ×[tex]radius^2[/tex] × height

In this case, we have the volume of the beaker as 1,004.8 cubic centimeters and the radius as 4 cm.

1,004.8 = π × [tex]4^2[/tex] × height

Simplifying the equation:

1,004.8 = 16π × height

To solve for the height, divide both sides of the equation by 16π:

height = 1,004.8 / (16π)

Using a calculator for the approximation of π (pi) as 3.14159, we can calculate the height:

height ≈ 1,004.8 / (16 × 3.14159)

height ≈ 6.35 cm

The height of the beaker is approximately 6.35 cm.

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The length and width of another rectangular park is 110 meters by 65 meters

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

A rectangular park is 70 meters wide and 105 meters long

This means that

Width = 70

Length = 105

The area is calculated as

Area = 70 * 105

Area = 7350

For smaller area, we decrease the width and increse the length

So, we have

Width = 70 - 5

Length = 105 + 5

Evaluate

Width = 65

Length = 110

Hence, the length and width of another rectangular park is 110 by 65

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The all four data sets can be represented on a dot plot, as dot plots are both decimal and Integer values.

A dot plot is a graphical representation that displays data points as dots along a number line. It is commonly used to show the distribution of a small set of quantitative data.

The data set should be shown on a dot plot, we need to consider the characteristics of the data.

1. Data Set 1: {0.012, 0.078, 0.093, 0.147, 0.187}

This data set consists of decimal values. A dot plot can effectively represent decimal values, where each dot represents a data point. Therefore, Data Set 1 should be shown on a dot plot.

2. Data Set 2: {1.3, 1.9, 2.5, 2.7, 3.5, 4.8, 5.3, 7.9, 9.0}

Similar to Data Set 1, this data set consists of decimal values. Hence, it can also be effectively represented on a dot plot. Each value would be represented as a dot along the number line.

3. Data Set 3: {712, 722, 722, 723, 724, 724, 724, 725, 727, 728, 730}

This data set consists of integer values. Dot plots are commonly used for discrete data, including integers. Therefore, Data Set 3 can be appropriately represented on a dot plot.

4. Data Set 4: {16, 29, 31, 37, 44, 49, 58, 63, 69, 70, 83, 97}

This data set also consists of integer values dot plots can effectively display discrete data. Hence, Data Set 4 can be shown on a dot plot.

all four data sets can be appropriately represented on a dot plot, as dot plots are suitable for displaying both decimal and integer values.

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The first time that Account X will have more money than Account W, excluding the initial deposits, is in the fifth month.

To determine when Account X will contain more money than Account W, we need to compare the balances of the two accounts each month until Account X surpasses Account W.

Let's analyze the balances of the two accounts month by month:

Month 1:

Account W: $0 + $100 = $100

Account X: $1 + 80% × $1 = $1 + $0.80 = $1.80

Month 2:

Account W: $100 + $100 = $200

Account X: $1.80 + 80% × $1.80 = $1.80 + $1.44 = $3.24

Month 3:

Account W: $200 + $100 = $300

Account X: $3.24 + 80% × $3.24 = $3.24 + $2.592 = $5.832

Month 4:

Account W: $300 + $100 = $400

Account X: $5.832 + 80% × $5.832 = $5.832 + $4.6656 = $10.4976

Month 5:

Account W: $400 + $100 = $500

Account X: $10.4976 + 80% × $10.4976 = $10.4976 + $8.39808 = $18.89568

Please note that the values provided in this calculation assume no withdrawals and no interest earned on the balances.

As we can see, by the fifth month, Account X will contain more money than Account W. The first time that Account X will have more money than Account W, excluding the initial deposits, is in the fifth month.

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Answer:

C) (-2 , 4)

x = -2

y = 4

Step-by-step explanation:

    We can use substitution method.

     3x + 2y = 2 -------------(I)

    -2x + y  = 8             --------------(II)

Make y as the subject in equation (II)

              y = 8 + 2x ----------------(III)

Substitute y = 8 + 2x in equation (I) and we can find the value of 'x',

          3x + 2*(8 + 2x) = 2

     3x + 2*8 + 2*2x    = 2 {Distributive property}

           3x + 16 + 4x    = 2

           3x + 4x + 16    = 2

Combine like terms,

                      7x + 16 = 2

Subtract 16 from both sides,

                           7x   = 2 - 16

                           7x   = -14

Divide both sides by 7,

                             x  = - 14 ÷ 7

                             [tex]\boxed{\bf x = -2}[/tex]

Substitute x = -2 in equation (III)

           y = 2*(-2) + 8

              = -4 + 8

           [tex]\boxed{\bf y = 4}[/tex]

Answer:

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

The range of the data is the difference of maximum and minimum values.

According to description of the box plot, the minimum value is 6 and maximum value is 18.

Hence the range is:

Answer:

24 years old.

Step-by-step explanation:

Let's say that Elga's age is x years old. Then Alvin's age would be 11 years older than Elga, or x + 11 years old.

We know that the sum of their ages is 59 years old, so we can write an equation:

x + (x + 11) = 59

Simplifying the left side of the equation:

2x + 11 = 59

Subtracting 11 from both sides:

2x = 48

Dividing both sides by 2:

x = 24

Therefore, Elga's age is 24 years old.

Answer: 24

Step-by-step explanation:

Set Elga's age to x and Alvin's age to x+11...

(x+11)+x=59

2x=48

x=24

Answer:

-9

Step-by-step explanation:

Based οn bοth the median and mean values, it appears that Bus 47 typically has the faster travel time cοmpared tο Bus 18.

We know that

Statistics is the discipline of mathematics concerned with data collection, analysis, interpretation, presentation, and organisation. It includes approaches for summarising and describing data, as well as inferring and forecasting based on the data.

Tο determine which bus typically has the faster travel time, we need tο cοmpare the measures οf center (mean and median) οf the twο buses.

The median represents the dataset's midway, where half of the values are above and half are below. The mean represents the dataset's average value, which is computed by adding all of the values and dividing by the total number of values.

When the medians are compared, we can see that Bus 47 has a higher median of 16, while Bus 18 has a lower median of 13. This shows that Bus 47 may have a faster average trip time.

However, we must also consider the methods. When the means are compared, we can see that Bus 47 has a higher mean of 16, whereas Bus 18 has a lower mean of 13. This lends credence to the notion that Bus 47 may have a faster average travel time.

Therefοre, based οn bοth the median and mean values, it appears that Bus 47 typically has the faster travel time cοmpared tο Bus 18.

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The complete question is:

Compare the data and use the correct measure of center to determine which bus typically has the faster travel time. Round your answer to the nearest whole number, if necessary, and explain your answer.

Bus 18, with a median of 13

Bus 47, with a median of 16

Bus 18, with a mean of 13

Bus 47, with a mean of 16

The point estimate is given as 0.4688

The standard error = 0.051

P = x / n

= 45 / 96

= 0.4688

The point estimate is given as 0.4688

How to solve for the standard error

SE = √[ p(1 - p) / n ]

Substituting the values we have:

SE = √[ 0.46875 * (1 - 0.46875) / 96 ]

= √[ 0.2482 / 96 ]

The standard error = 0.051

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Question

For the given confidence level and values of x and n, find the following.

=x45, =n96, confidence level 95% find the point estimate and the standard error

Symmetry means,  hmmm usually a mirror image.

So when something is symmetry to another, is a way to say, one is the mirror image of the other, or namely they're just identical twins.

Now, if know that B is the midpoint, by definition is cutting AC into two equal halves, so AB is the mirror image of BC, so they're identical twins, there are no flaws, Q.E.D.

Check the picture below.

Answer:

26 units

Step-by-step explanation:

AD = 9 -2 = 7 units long. BC is the same. so we have 2 X 7 = 14.

DC is 9 - 3 = 6 units long. AB is the same. so we have 2 X 6 = 12.

perimeter = 14 + 12 = 26 units

The set of ordered pairs above: is a function.

The mapping diagram above: is a function.

y = x²: is a function.

The table above: is not a function.

In Mathematics and Geometry, a function is a mathematical equation which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair.

Based on the set of ordered pairs, mapping diagram, and the equation above, we can reasonably infer and logically deduce that it represent a function because the input values (domain, x-value, or independent values) are uniquely mapped to the output values (range, y-value or dependent values).

In this context, we can reasonably infer and logically deduce that the table does not represent a function because the input value (-1) has more than output values (2 and 4 respectively).

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The solution is:

[tex]\int\limits^a_b {\frac{x^{3} }{x^{2} + x -2 } } \, dx[/tex] =[tex]ln|x| + \frac{2}{5} ln|x+2| + \frac{1}{3} ln|x-1| + C[/tex]

The partial fraction expansion of a rational fraction is  described as an operation that consists of expressing the fraction as a sum of a polynomial and one or several fractions with a simpler denominator.

To solve the above partial fraction decomposition,

[tex]x^3 + x^2 - 2x = x(x^2 + x - 2) = x(x+2)(x-1)[/tex]

we then write the fraction as a sum of partial fractions:

[tex]fx-2dx = \frac{A }{x } + \frac{B}{x + 2 } + \frac{C }{x - 1}[/tex]

We  find the values of A, B, and C by  multiplying  both sides by the common denominator (x)(x+2)(x-1) and  substituting

fx-2dx = A(x+2)(x-1) + B(x)(x-1) + C(x)(x+2)

We set  x = 0

0 = -2A

A = 0.

We set  x = -2

-2 = 20B/(-2)(-4)

-2 = -5B

B = 2/5.

we set  x = 1

1 = 3C

C = 1/3.

fx-2dx = 0/x +  2/5/(x+2) +  1/3/(x-1)

We integrate each term separately and have the solution as :

[tex]\int\limits^a_b {\frac{x^{3} }{x^{2} + x -2 } } \, dx[/tex] =[tex]ln|x| + \frac{2}{5} ln|x+2| + \frac{1}{3} ln|x-1| + C[/tex].

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The probability of obtaining a success given that X is less than 5, P(X < 5) is 0.8906.

The probability of obtaining a success is calculated as follows:

Given data:

X has a binomial distribution with n = 6 trials and p = 0.5, the probability of obtaining a success.

Using the binomial distribution function:

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

P(X = k) = [tex](_nC_k) * p^{k} * (1 - p)^{(n - k)}[/tex]

Therefore;

P(X < 5) = (₆C₀) * 0.5⁰ * 0.5⁶ + ( 1) * 0.5¹ * 0.5⁵ + (₆C₂) * 0.5² * 0.5⁴ + (₆C₃) * 0.5³ * 0.5³ + (₆C₄) * 0.5⁴ * 0.5²

P(X < 5) = 0.015625 + 0.09375 + 0.234375 + 0.3125 + 0.234375

P(X < 5) = 0.8906

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The profit did you make from your investment is $20

To determine how much profit you made from your investment, we need to find the value of P that makes your friend's profit equal to the initial investment of $20.

In other words, we need to solve the equation:

0.4P - 20 = 20

Adding 20 to both sides, we get:

0.4P = 40

Dividing both sides by 0.4, we get:

P = 100

if the football is sold for $100, your friend's profit will be $100 - $50 = $50, and your profit will be 40% of $50, or $20.

This means that you have made back your initial investment of $20, and have earned an additional profit of $20.

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The ratio of literature books to children books is: a) 4 literature books: 1 children book.

Ratio is simply the numerical quantity that relates two amounts of quantities, showing how much one contains of the other.

The ratio of literature books to children books can be found by dividing the number of literature books by the number of children books:

Ratio of literature books to children books = Number of literature books / Number of children books

Plugging in the given values, we get:

Ratio of literature books to children books = 400 / 100 = 4

So the ratio of literature books to children books is 4:1. Therefore, the answer is option a) 4 literature books: 1 children book.

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The value of sin (2π/3) is √3/2.

The value of sin (5π/4) is -√2/2.

The value of sin (180) is 0.

The value of cos (π/4) is √2/2.

The value of cos (300) is ¹/₂.

The value of cos (7π/6) is -√3/2.

The value of sin (2π/3) is calculated as follows;

sin (2π/3) = sin (2 x 180 / 3) = sin (120)

sin (120) = sin (60) = √3/2

The value of sin (5π/4) is calculated as follows;

sin (5π/4) = sin (5 x 180 / 4) = sin (225)

sin (225) = - sin (45) = -√2/2

The value of sin (180) is calculated as follows;

sin (180) = sin (0) = 0

The value of cos (π/4) is calculated as follows;

cos (π/4) = cos (180/4) = cos (45)

cos (45) = √2/2

The value of cos (300) is calculated as follows;

cos (300) = cos (60)

cos (60) = ¹/₂

The value of cos (7π/6) is calculated as follows;

cos (7π/6) = cos (7 x 180 / 6) = cos (210)

cos (210) = - cos(30) = -√3/2

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Answer:

486 square feet

Step-by-step explanation:

To calculate the area of the path, first find the dimensions of the garden, including the path. Then find the area of this larger outer rectangle and subtract the area of the inner rectangle (the garden without the path).

The garden measures 28ft by 47ft.

The path is 3ft wide, so it adds a total of 6ft to the length (3ft on each side) and 6ft to the width.

The dimensions of the garden, including the path:

Length: 47ft + 6ft = 53ft

Width: 28ft + 6ft = 34ft

Find the area of the outer rectangle (including the path):

Area_outer = Length * Width = 53ft * 34ft = 1802 sq ft

Find the area of the inner rectangle (the garden without the path):

Area_inner = 28ft * 47ft = 1316 sq ft

Now subtract the area of the inner rectangle from the area of the outer rectangle to find the area of just the path:

Area_path = Area_outer - Area_inner = 1802 sq ft - 1316 sq ft = 486 sq ft

So, the area of the path is 486 square feet.

Answer: 486 square feet;

Step-by-step explanation:

Area of the garden = 28 x 47 = 1316ft.²

-

Length of the garden and path = Length of the garden + 3 ft. on both sides

Length of the garden and path = 28 + 3 + 3 = 34 ft.

-

Width of the garden and path = Width of the garden + 3 ft. on both sides

Width of the garden and path = 47 + 3 + 3 = 53ft.

-

Area of the garden and path = Length x Width

Area of the garden and path = 34 x 53 = 1802 ft²  

-

Area of the path = Area of the garden and path - Area of the garden

Area of the path = 1802 - 1316 =  486ft²

-

Answer: 486ft²

The book has a lower cost per minute than the movie.

To find the cost per minute of the book, we divide the cost by the number of minutes:

cost per minute of book = $15 / 200 minutes = $0.075 per minute

To find the cost per minute of the movie, we need to convert the time to minutes.

2 hours and 39 minutes is equal to:

2 hours x 60 minutes/hour = 120 minutes

39 minutes = 39 minutes

Total minutes = 120 + 39 = 159 minutes

cost per minute of movie = $14 / 159 minutes = $0.088 per minute

Therefore, the book has a lower cost per minute than the movie.

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The circumference of a circle with a radius of 21 centimeters is 132 centimeters.

The circumference is the distance around the boundary of a closed curve or shape. In the case of a circle, the circumference is the total length of the circular boundary.

To find the circumference of a circle with a radius of 21 centimeters, we can use the formula C = 2πR, where R is the radius and π is the mathematical constant approximately equal to 22/7.

Substituting the given values:

C = 2 x (22/7) x 21

C = (44/7) x 21

C = 924/7

C ≈ 132 centimeters

Therefore, the circumference of a circle with a radius of 21 centimeters is 132 centimeters.

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The z-score for the raw score of 39 is 1.The z-score for the raw score of 30 is -0.8.Z-scores measure the number of standard deviations a raw score is from the mean. A positive z-score indicates a raw score above the mean, while a negative z-score indicates a raw score below the mean.

To find the z-scores, we need to use the formula:

z = (x - μ) / σ

where:

z is the z-score,x is the raw score,μ is the mean, andσ is the standard deviation.

Let's calculate the z-scores for the given mean of 34 and standard deviation of 5.For a raw score of 39:

z = (39 - 34) / 5

z = 5 / 5

z = 1

So, the z-score for the raw score of 39 is 1.For a raw score of 30:

z = (30 - 34) / 5

z = -4 / 5

z = -0.8

The z-score for the raw score of 30 is -0.8.

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The slope (m) and y-intercept (b) with the equation of the line in slope-intercept form are given as: m = 2 and b = 5 is y = 2x + 5; m = 3 and b = 5 is y = 3x + 5; m = 5 and b = 4 is y = 5x + 4.

If m represents the slope of a line and b represents its y-intercept, the equation of the line in slope-intercept form is written as: y = mx + b.

Where m = 2 and b = 5, the equation of the line would be expressed as: y = 2x + 5.

Where m = 3 and b = 5, the equation of the line would be expressed as: y = 3x + 5.

Where m = 5 and b = 4, the equation of the line would be expressed as: y = 5x + 4.

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