An exercise machine with an original price of $890 is on sale at 24% off.
a. What is the discount amount?
b. What is the exercise machines sale price?

The domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is

[ -4 , ∞ ).

Given, a graph of the function,

{ x | x ≥ -4 }

{ y l y ≥ 2 }

Now, we have to find the domain of the function

As, we know that the domain of a function is the set of real values on which the given function is defined.

so, it is clear that the function is defined for all the real values greater than or equal to -4.

Hence, the domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is

[ -4 , ∞ ).

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The exact interval notation of given equation is  x > - 8.

What is interval notation for compound inequalities?

The numbers that bound a continuous set of real numbers can be used to express the continuous set in interval notation. Intervals have many characteristics in common with ordered pairs when written.

We have,

            3x+1 > 2x-3 > x-11

Apply the rule that if m > v > n, then m > v and v > n to the given compound inequality to find the solution.

∴ m = 3x + 1,

   v = 2x - 3,

   n = x - 11.

Hence, by applying the rule of compound inequality we get,

          3x + 1 > 2x - 3 and 2x - 3 > x - 11

3x + 1 > 2x - 3

3x + 1 - 1 > 2x - 3 -1

3x > 2x - 4

3x - 2x > 2x - 4 - 2x

x > - 4

2x - 3 > x - 11

2x - 3 + 3 > x - 11 + 3

2x  > x - 8

2x  - x > x - 8 - x

x > - 8

The two computations above indicate that the acquired intervals, x > - 4 and x > - 8, will be combined to yield the answer to the given inequality.

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Yes, Dan productivity is constant and slope is given as 3/2.

What is slope?

The slope of a line is also called its gradient or rate of change. The slope formula is the vertical change in y divided by the horizontal change in x, sometimes called rise over run. The slope formula uses two points, (x1, y1) and (x2, y2), to calculate the change in y over the change in x.

Let we take one pair of point (4,6) and (2,3) and other pair of point is (6,9) and (0,0)

The slope between  (4,6) and (2,3)  is

m = y2-y1/x2-x1

m = 3-6/2-4

=3/2

The slope between (6,9) and (0,0)

m = 0-9/0-6

=3/2

Hence, both the slopes are equal, therefore his work is constant.

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Answer: 28 and 45

Step-by-step explanation: you add 11 and 17 to get 28 and then add 28 and 17 to get 45

The graph of the system of equations shows that there is no solution to the system.

To solve a system of equations by graphing, plot each of the lines on a graph. Then, find the point where both lines intersect each other.

The x-coordinate and the y-coordinate of the point where both lines of the equations intersect is the solution to the system.

Given the system of equations, x = -y and x + y = 7, both lines have been plotted on a graph shown below. It shows that there are is no point of intersection which means the system of equations have no solution.

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The number of cars that Tim has in all is 215.

The boxcars that Tim has is written in standard form. While the boxcars that Tim's grandfather gave him is written in scientific form. The first step is to convert the scientific form to standard form.

Number of boxcars that Tim's grandfather gave him: 1.62 x 10²

= 1.62 x 100

= 162

The next step is to add the boxcars that Tim's grandfather gave him to the number of boxcars that Tim has.

Total boxcars that Tim has = 162 + 53 = 215 box cars

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The mean of the given scores is 15.43

It is the average value of the set given.

It is calculated as:

Mean = Sum of all the values of the set given / Number of values in the set

We have,

Score       Mid-values       Frequency      Total score

1 - 7                  4                  19                 19 x 4 = 76

8 - 10               9                  16                  16 x 9 = 144

11 - 15              13                  6                   13 x 6 = 78

16 - 20            18                  14                  18 x 14 = 252

21 - 35            28                 15                  28 x 15 = 420  

36 - 50           43                  4                   43 x 4 = 172

Total                                     74                  1142

Mean = 1142 / 74 = 15.43

Thus,

The mean of the given scores is 15.43

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The equation for the situation is written as y = 35x + 10

Information gotten from the question include

a tank is filing with petrol at rate of 35 liter per minute

the tank already had 10 liters in it before the filling began

A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.

y = mx + c

Definition of variables to suit the problem to be solved

y = total petrol in  the tank in  liters

m = rate of filling liter per minute = 35

x = number of liters

c = quantity prior to filling = 10

The equation is written as

y = 35x + 10

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The equation has a unit rate of 3/4

From the question, we have the following parameters

16y = 12x

The above equation is a linear equation

In fact, it is a linear equation that shows a proportional relationship

Note that all linear equations have a constant rate of change

So, the proportional relationship is given as

16y = 12x

Make y the subject

y = 12/16x

This gives

y = 3/4x

As a general rule

A proportional relationship is represented as y = kx

Where k represents the unit rate

So, we have the following representation

y = 3/4x and y = kx

By comparing the equations

k = 3/4

Hence, the unit rate of the equation is 3/4

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Possible question

Find the unit rate.

16y = 12x

The geometric sequence in which the 3rd term is of 18 and the 7th term is of 29/8 is defined as follows:

[tex]a_n = 40(0.67)^{n - 1}[/tex]

A geometric sequence is a sequence in which the result of the division of consecutive terms yields always the same result, called common ratio q.

The nth term of a geometric sequence is given by the rule presented as follows:

[tex]a_n = a_1q^{n-1}[/tex]

In which [tex]a_1[/tex] is the first term.

The rule for the nth term is also called the general rule.

The general takes the first term as the reference term, however, the 7th term can be written as a function of the 3rd term, as follows:

[tex]a_7 = a_3q^{4}[/tex]

The fraction that represents the mixed number 3 and 5/8 is:

(3 x 8 + 5)/8 = 29/8.

Then the common ratio of the sequence is calculated as follows:

[tex]\frac{29}{8} = 18q^{4}[/tex]

[tex]q^4 = \frac{29}{144}[/tex]

[tex]q = \sqrt[4]{\frac{29}{144}}[/tex]

[tex]q = \left(\frac{29}{144}\right)^{\frac{1}{4}}[/tex]

q = 0.67.

Then the first term of the sequence is obtained as follows:

[tex]a_3 = a_1q^2[/tex]

[tex]a_1 = \frac{a_3}{q^2}[/tex]

[tex]a_1 = \frac{18}{(0.67)^2}[/tex]

[tex]a_1 = 40[/tex]

Then the definition of the geometric sequence is:

[tex]a_n = 40(0.67)^{n - 1}[/tex]

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The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10

Given that,

The function is f(x)= -2x² + 3x + 10

We have to find f(1) and f(3).

The core concept of mathematics' calculus is functions. The functions are special types of relations. In mathematics, a function is represented as a rule that produces a distinct result for each input x.

Take the function

f(x)= -2x² + 3x + 10

Take x as 1

f(1)= -2(1)² + 3(1) +10

f(1)= -2+3+10

f(1) = 11

Now, Take x as 3

f(3)= -2(3)² + 3(3) +10

f(3)= -18+9+10

f(3) = 1

Therefore, The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10

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Answer:

===================================

Continuous compounding equation:

Now find the time t for the balance to grow to $51806.67:

The time required is about 9 years.

Answer:

9 years

Step-by-step explanation:

Continuous Compounding Formula

[tex]\large \text{$ \sf A=Pe^{rt} $}[/tex]

where:

Given values:

Substitute the given values into the formula and solve for r to find the annual interest rate:

[tex]\implies \sf 39786.15=36000 \cdot e^{2.5r}[/tex]

[tex]\implies \sf \dfrac{39786.15}{36000}= e^{2.5r}[/tex]

[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=\ln e^{2.5r}[/tex]

[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=2.5r[/tex]

[tex]\implies \sf \dfrac{1}{2.5}\ln \left(\dfrac{39786.15}{36000}\right)=r[/tex]

[tex]\implies \sf r=\dfrac{2}{5}\ln \left(\dfrac{39786.15}{36000}\right)[/tex]

[tex]\implies \sf r=0.03999996933...[/tex]

[tex]\implies \sf r=0.04\;(2\;d.p.)[/tex]

Therefore, the annual interest rate is 0.04 = 4%.

To calculate how many years it will take for $36,000 to grow to $51,806.67, substitute the values into the formula and solve for t:

[tex]\implies \sf 51806.67=36000 \cdot e^{0.04t}[/tex]

[tex]\implies \sf \dfrac{51806.67}{36000}= e^{0.04t}[/tex]

[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= \ln e^{0.04t}[/tex]

[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= 0.04t[/tex]

[tex]\implies \sf \dfrac{\ln\left(\dfrac{51806.67}{36000}\right)}{0.04}=t[/tex]

[tex]\implies \sf t=9.10 \; (2\;d.p.)[/tex]

Therefore, it will take about 9 years for $36,000 to grow to $51,806,67.

The probability of more than 5 people is 0.4729

Since 49% of the people in a community use the emergency room at a hospital in one year, a sample of 11 people use the emergency room. To find the probability of more than 5 people use the emergency room, we use binomial probability.

The binomial probability of x is given by  P(X = x) = ⁿCₓpˣ(1 - p)ⁿ⁻ˣ

where

Since

We determine the probability of x ≤ 5.

So, P(X ≤ 5) = ¹¹C₀p⁰(1 - p)¹¹⁻⁰ + ¹¹C₁p¹(1 - p)¹¹⁻¹ + ¹¹C₂p²(1 - p)¹¹⁻² + ¹¹C₃p³(1 - p)¹¹⁻³

+ + ¹¹C₄p²(1 - p)¹¹⁻⁴ + ¹¹C₅p⁵(1 - p)¹¹⁻⁵

= ¹¹C₀(0.49)⁰(1 - 0.49)¹¹ + ¹¹C₁(0.49)¹(1 - 0.49)¹⁰ + ¹¹C₂(0.49)²(1 - 0.49)⁹ + ¹¹C₃(0.49)³(1 - 0.49)⁸

+ ¹¹C₄(0.49)⁴(1 - p)⁷ + ¹¹C₅(0.49)⁵(1 - 0.49)⁶

= (1)(1)(0.51)¹¹ + (11)(0.49)(0.51)¹⁰ + (55)(0.49)²(0.51)⁹ + (165)(0.49)³(0.51)⁸

+ (330)(0.49)⁴(0.51)⁷ + (462)(0.49)⁵(0.51)⁶

= 0.0006071 + (11)(0.49)(0.0011904) + (55)(0.2401)(0.0023341) + (165)(0.117649)(0.0045768)

+ (330)(0.0576480)(0.0089741) + (462)(0.0282475)(0.0175963)

= 0.0006071 + 0.0064163 + 0.0308230 + 0.0888452

+ 0.1707218 + 0.2296378

= 0.5270512

≅ 0.5271

So, P(X ≤ 5) = 0.5271, the probability of more than 5 people is P(X ≥ 5) = 1 - P(X ≤ 5)

= 1 - 0.5271

= 0.4729

The probability is 0.4729

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The projected cash receipts for April and the projected cash receipts for May can be found to be:

The projected cash receipts in April would be:

= ( 25% x Projected sales for April ) + ( 75% x Projected sales for March)

= (25% x 210, 000) + ( 75% x 180, 000)

= 52, 500 + 135, 000

= $187, 500

The projected cash receipts for May is then:

= ( 25% x Projected sales for May ) + ( 75% x Projected sales for April )

= ( 25% x 195, 000) + ( 75% x 210, 000)

= $ 48, 750 + 157, 500

= $206, 250

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Answer:1 hour and 15 minutes

Step-by-step explanation:

so it takes her 34 minutes for 4 problems so for 2 problems its 17 minutes

and then you do the 34 times 2 because its 4 problems so you add that and that = 68 minutes and then you add 17 for the 2 problems and thats 85 minutes or 1 hour and 15 minutes

Answer:

1 hour & 25 minutes (85 minutes)

Step-by-step explanation:

Time taken for 1 problem,

→ 34 ÷ 4

→ 8.5 minutes

Now time taken for 10 problems,

→ 8.5 × 10

→ 85 minutes

Hence, answer is 85 minutes.

Answer: f(x) = x³ + 7x² -8x

Step-by-step explanation:

Zeros means that at that x-value, the graph crosses the x-axis and y = 0. Hence, to meet those criteria, the function would be:

[tex]f(x)=x(x+8)(x-1)[/tex]

At those three x-values, the corresponding y-values would equal 0.

Multiply them out to find the polynomial function:

[tex]f(x)=x(x+8)(x-1)=x(x^{2} -x+8x-8)=x^{3} +7x^{2} -8x[/tex]

Answer:

3

Step-by-step explanation:

[tex]27^{\frac{1}{3}}[/tex]

1 ) Factor the number [tex]27=3^{3}[/tex]

[tex]=\left(3^3\right)^{\frac{1}{3}}[/tex]

2 ) Simplify...

3

Hope this helps! :)

A is the Point of representation of the colder midnight temperature in degrees Celsius.

The temperature in Quebec City was 5°C at 4 pm. By midnight, the temperature dropped 15 degrees.

To find out which point is a representation of the colder midnight temperature in degrees Celsius:

We know that the temperature dropped 15 degrees, and that it was initially of 5°C. So, by midnight, the temperature will be of -10°C (5-15=-10).

We also know that the distance between points is of 5°C. So, the point that we have to mark is point A, which represents -10°C.

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The number of packages that need to be delivered daily by UpEx to make a profit of $32 is 115 packages

First, find the contribution margin of each package:

= Charge to deliver package - Cost to deliver package

= 3.50 - 1.50

= $2.00

To find the number of packages needed to make a profit of $32, the formula is:

= (Fixed cost + projected profit) / Contribution margin

= ( 198 + 32) / 2

= 230 / 2

= 115 packages

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Using the distance formula, the distance between A to point Z to the nearest tenth is 4.1 .

The 2D distance formula gives the shortest distance between two points in a two-dimensional plane.

Using the distance formula,

d = √(y₂ - y₁) + (x₂ - x₁)²

let's use the distance formula, to find the distance between point A and XZ.

Using the coordinates,

A(3, 3) and XZ(4, -1)

Therefore,

x₁ = 3

x₂ = 4

y₁ = 3

y₂ = -1

d = √(4 - 3)² + (-1 - 3)²

d = √1² + (-4)²

d = √1 + 16

d = √17

Therefore,

d = 4.12310562562

d = 4.1

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The addition using Least multiple, [tex]\frac{2}{8}+\frac{1}{2}[/tex] is equal to [tex]\frac{3}{4}[/tex].

When two or more fractions are there, the least common denominator is the lowest number among all the common multiples of the denominators.

To find the LCD of numbers, factorization method can be used and if the numbers are co-prime then the LCD is just the multiplication of those numbers.

The denominator of given fractions is 8 and 2.

Now, finding the lowest common multiple of 8 and 2,

LCD is 8,

Now, renaming the fraction by multiplying 4 on numerator and denominator of [tex]\frac{1}{2}[/tex],

[tex]\frac{1 \times4}{2 \times4} =\frac{4}{8}[/tex]

Now, denominator is same, so adding the fractions,

[tex]\frac{2}{8}+\frac{4}{8} =\frac{2+4}{8} \\\\=\frac{6}{8}[/tex]

Dividing the numerator and denominator by 2,

[tex]\frac{6}{8} =\frac{6/2}{8/2}\\ =\frac{3}{4}[/tex]

Therefore, by adding the given fractions we get [tex]\frac{3}{4}[/tex].

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Answer: 12,000pd=6 tons

Step-by-step explanation:

Answer:

a)  Line r

b)  Line q

c)  Line s

d)  Line q

e)  Line p

Step-by-step explanation:

A transversal is a line that crosses two other lines in the same plane at two distinct points.

a) The transversal connecting ∠1 and ∠5 is line r.

b) The transversal connecting ∠7 and ∠14 is line q.

c) The transversal connecting ∠8 and ∠11 is line s.

d) The transversal connecting ∠6 and ∠15 is line q.

e) The transversal connecting ∠3 and ∠9 is line p.

The probability is 0.5881.

From the question, we have

n = 20

P(X[tex]\geq[/tex]10) = 1-P(X<10)

P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)

P(X[tex]\leq[/tex]9) = P(X=0)+P(X=1)+..................+P(X=9)

=0.4119

P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)

=1-0.4119

=0.5881

Probability:

Probability refers to possibility. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is.

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Answer:

105 Children and 3 cooks

Step-by-step explanation:

I took the test :)

a) The total number of children watching the movie is 105 children

b) The total number of cooks that worked at the event is 3 cooks

What is an Equation?

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.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

a)

Let the number of children be = a

Let the number of adults be = b

Let the total number of people in the theater be = 140

Now , the equation will be

a + b = 140   equation (1)

And , the number of children present is 3 times the number of adults

So ,

a = 3b

Divide by 3 on both sides of the equation , we get

b = a/3

Substituting the value of b in equation (1) , we get

a + b = 140

a + a/3 = 140

( 4a ) / 3 = 140

Multiply by 3 on both sides of the equation , we get

4a = 420

Divide by 4 on both sides of the equation , we get

a = 105

Therefore , the value of a is 105 and the number of children watching the movie is 105 children

b)

Let the number of cooks be = x

Let the number of waiters be = y

Let the amount for each cook be = $ 85

Let the amount for each waiter be = $ 45

The total number of workers = 10

Now , the equation will be

x + y = 10   be equation (1)

The total amount paid to the employees be = $ 570

Now , the equation will be

85x + 45y = 570   be equation (2)

On simplifying equation (1) and equation (2) , we get

Multiply equation (1) by 45 , we get

45x + 45y = 450   be equation (3)

Subtract equation (3) from equation (2) , we get

40x = 120

Divide by 3 on both sides of the equation , we get

x = 3

Therefore , the value of x is 3 and the number of cooks that worked in the event is 3 cooks

Hence ,

a) The total number of children watching the movie is 105 children

b) The total number of cooks that worked at the event is 3 cooks

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The number of tapes Melanie must sell to break even is 360 tapes.

To find the number of tapes Melanie must sell to break even, we must follow these steps.

We must identify how much she earns for each tape that she sells, for this we must subtract the sale price of each tape, by the value that she pays to make each tape.

Now we must divide the total cost of the mother tape by the profit she makes on each tape she sells.

Based on the above, we can infer that she must sell 360 tapes to cover the value of the mother tape and each of the tapes that she needs to sell. Next it is checked:

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In order to hit the UFO, the Anti Aircraft gun should be set an an angle of elevation of approximately 63.435° above the horizontal in the direction of the UFO

An angle of elevation is an angle that the line of sight makes with the horizontal.

The location of the UFO are;

The angle, θ, to which the anti aircraft gun needs to be set is found from the trigonometric ratio for tangent as follows.

[tex]tan(\theta) = \dfrac{Length \, of \, the \, side \, opposite \, to \, angle \, \theta}{Length \, of \, the \, side \, adjacent \, to \, \theta}[/tex]

The length of the side opposite to the angle θ to which the gun should be directed is the length of the height of the UFO from the observer, therefore;

Length of the side opposite to θ = 100 feet

Length of the side adjacent to θ is the length of the horizontal distance from the observer;

Length of the adjacent side to θ = 50 feet

Therefore;

[tex]tan(\theta) = \dfrac{100\, feet}{50\, feet} = 2[/tex]

tan(θ) = 2

θ is found using the inverse tangent function (arctangent) as follows;

θ = arctan(2) ≈ 63.435°

The direction the anti aircraft un should be set is at angle of elevation of approximately 63.435° above the horizontal

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Answer:

  (-12, 10)

Step-by-step explanation:

You want to solve by elimination the system of equations ...

The point of elimination is to make one of the variables disappear from the equation.

Here, we observe that the x-coefficients are opposites in the two equations. When we add them, the result is 0: the x-term disappears. Adding the two equations, we have ...

  (2x +4y) +(-2x -3y) = (16) +(-6)

  y = 10 . . . . . . . . simplified

Using this value for y, we can find x. Substituting into the first equation gives ...

  2x +4(10) = 16

  2x = 16 -40 = -24 . . . . . subtract 40 and simplify

  x = -12 . . . . divide by 2

The ordered pair that is the solution is (x, y) = (-12, 10).