Step 1: Understand the Problem
Keep the Old Car or Buy a Used Car
Manny is an online student who currently owns an older car that is fully paid for. He drives, on average, 190 miles per week to commute to work. With gas prices currently at $
2.9 per gallon, he is considering buying a more fuel-efficient car, and wants to know if it would be a good financial decision.



The old car Manny owns currently gets 18 miles per gallon for average fuel efficiency. It has been a great vehicle, but with its age, it needs repairs and maintenance that average $
770 per year (as long as nothing serious goes wrong).



The newer, more fuel-efficient car that he is looking at to purchase will cost a total of $
6,500 over a three-year loan process. This car gets 32 miles per gallon and would only require an average of $
10 per month for general maintenance. To help make a decision, Manny wants to calculate the total cost for each scenario over three years. He decides to use the quantitative reasoning process to do this.


Step 3: Apply Quantitative Tools
Use computational and algebraic tools to quantify the total costs (gas, maintenance/repairs, purchase price) for each scenario over the three years.

Round your answers to the nearest dollar.

Scenario

Total Cost for Three Years

Keep the old car

$
Number


Buy the fuel-efficient used car

$
Number

Here is the truth table for the statement (~ p V q) → q:

```

p     q     ~p    ~p V q   (~p V q) → q

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

T     T     F      T           T

T     F     F      F           T

F     T     T      T           T

F     F     T      F           T

```

In the table, `p` and `q` represent the truth values of the propositions `p` and `q`, respectively. The symbol `~` represents negation (i.e., "not"). The symbol `V` represents the logical connective "or" (i.e., "inclusive or"). The symbol `→` represents the conditional connective "implies" (i.e., "if...then").

To fill in the truth table, we first evaluate `~p` and `~p V q` for each combination of truth values for `p` and `q`. Then, we evaluate `(~p V q) → q` for each combination of truth values.

We can see that the statement is always true, regardless of the truth values of `p` and `q`, except for the case where `p` is true and `q` is false.

The height of the pyramid formed by the patio heater is 89.07 inches

An equation is an expression that shows how numbers and variables using mathematical operators.

Pythagoras theorem shows the relationship between the sides of a right angled triangle.

To find the height (h) of the pyramid. A right angled is form with base (b) = side of square / 2 = 22.8 / 2 = 11.4 in

Slant height (S) = 89.8 in

Using Pythagoras:

S² = h² + b²

89.8² = h² + 11.4²

h = 89.07 inches

The height of the pyramid is 89.07 inches

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Answer:

w≤9

Step-by-step explanation:

8w ≤ 72

Divide by 8 on both sides

w ≤ 9

For graph, draw a number line and label a point as 9. Draw a solid line facing to the left and having a closed point at 9. The line should not extend past 9.

40(2.25)ˣ function equation is represented by the graph transformation.

What do graph transformations mean?

Changes to the graph of a parent function are referred to as transformations. Transformations may be divided into three categories: translations, reflections, and dilations. There are two types of transformations that may be done to a function: vertical (which affects the y-values) and horizontal.

                               (affects the x-values). The following guidelines apply to the function translation and transformation: The function is moved up b units by f (x) + b. The function is moved downward by the notation f (x) b. The function is moved left by the value of f (x + b).

Let the function be of the form

  f(X ) = a(b)ˣ

Then the point (0,40) lies on the graph of this function. This point must satisfy the equation of the function.

We substitute the point to get,

  40 = a(b)⁰

This implies that,

   40 = a(1)

    40 = a

We substitute a=40 into the function equation to get,

         f(x) = 40(b)ˣ

The point (1,90) also lies on the graph of the function. This gives us,

       90 = 40(b)¹

       90 = 40b

      b = 90/40

       b = 2.25

We substitute the value of b also into function to get,

    f(X) = 40(2.25)ˣ

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Answer:

[tex]\large\boxed{\tt No.}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to determine if the equation has a solution given a value.}[/tex]

[tex]\textsf{Since x is given to us, we can use the \underline{Substitution Property of Equality} to evaluate}[/tex]

[tex]\textsf{the equation.}[/tex]

[tex]\large\underline{\textsf{What is the Substitution Property of Equality?}}[/tex]

[tex]\textsf{The Substitution Property of Equality is a Property that allows us to substitute}[/tex]

[tex]\textsf{a known term, or expression for an unknown variable, or some kind of placeholder.}[/tex]

[tex]\textsf{After Substitution, we can prove that the expressions are still equal with this}[/tex]

[tex]\textsf{property.}[/tex]

[tex]\large\underline{\textsf{Solving;}}[/tex]

[tex]\tt \frac{1}{4} x + 3 = " 3 = " 4x+1[/tex]

[tex]\textsf{There's likely a typo in your equation given, hence I will fix it.}[/tex]

[tex]\tt \frac{1}{4} x + 3 = 4x+1[/tex]

[tex]\textsf{Substitute 8 for x in both sides of the equation using our property.}[/tex]

[tex]\underline{\textsf{Use the Substitution Property of Equality;}}[/tex]

[tex]\tt \frac{1}{4} (8) + 3 = 4(8)+1[/tex]

[tex]\underline{\textsf{Follow PEMDAS, multiply first;}}[/tex]

[tex]\tt 2 + 3 = 32+1[/tex]

[tex]\underline{\textsf{Evaluate;}}[/tex]

[tex]\tt 5 \neq 33[/tex]

[tex]\textsf{If the "typo" was the actual equation;}[/tex]

[tex]\tt 5 \neq 3 \neq 33[/tex]

[tex]\large\boxed{\tt No.}[/tex]

Answer:

10

Step-by-step explanation:

When you think of the mean of a data set, think of the word average. 'Mean' and 'average' are the same thing when you're talking about a set of data.

To find the mean, you have to divide the sum of all values in your data set, by the number of values.

7 + 9 + 10 + 11 + 11 + 12 = 60

60/6 = 10

Therefore, the mean of this data set would be 10.

The interest rate if Ted made $56 in interest for 1 year on a principle of $700 will be 8%.

Given, the interest amount is $56.

The principal amount is $700.

The time period of investment is 1 year.

Now, we have to find the interest rate.

The formula to find the simple interest is,

  S.I = PTR/100.

Now, we need to substitute the values in the above formula.

S.I = PTR/100

56 = 700×1×R / 100

56 = 700R/100

5600 = 700R

R = 5600/700

R = 8

Therefore, the interest rate is 8%.

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The percentage of females that do not satisfy the requirement is given as follows:

55.17%.

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation are given as follows:

[tex]\mu = 998, \sigma = 202[/tex]

The proportion that does not satisfy the requirement is the p-value of Z when X = 1025, hence:

Z = (1025 - 998)/202

Z = 0.13

Z = 0.13 has a p-value of 0.5517.

Hence the percentage is given as follows:

0.5517 x 100% = 55.17%.

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To determine Duane's taxable income, we need to know his tax rate. Let's assume that his tax rate is a flat 20%.

We can use the formula:

taxable income = tax bill / tax rate

Plugging in the values given, we get:

taxable income = $1000 / 0.20

taxable income = $5000

Therefore, Duane's taxable income was $5000.

Answer:

Without more information, it is not possible to determine Duane's taxable income.

The perimeter and area of the square are 24√2 units and 12 units² respectively

To determine the perimeter and area of the square, we have to find the lengths of the sides.

But since we have the midpoints of all the sides, we can assume they're equidistant from one another since the figure is a square.

SQ = √(x₂ - x₁)² + (y₂ - y₁)²

SQ = √(2 - (-4)² + (5 - 11)²

SQ = 6√2

Let's find for QU

QU = √(-4 - 2)² + (-1 - 5)²

QU = 6√2

Let's find for UA ;

UA = √(-10 - (-4))² - (5 - (-1)²

UA = 6√2

And the distance from AS = 6√2

The perimeter of the square = 4 * (6√2)

Perimeter = 24√2 units

The area of the square is l²

Area of the square = (6√2)²

Area of square = 12 units²

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If Alonso pays the monthly payment each month for the full term. Alonzo will repay a total of $59,389.2 over the 6-year term of the loan if he makes the monthly payment each month.

To calculate the total amount that Alonzo will repay over the 6-year term of the loan, we need to use the formula for the monthly payment on an amortized loan:

M = P * (r/12) * (1 + r/12)^n / ((1 + r/12)^n - 1)

where:

M is the monthly payment

P is the principal amount

r is the monthly interest rate (which is the annual interest rate divided by 12)

n is the total number of payments (which is the number of years multiplied by 12).

Using the values given in the problem, we have:

P = $49,000

r = 6.55% / 12 = 0.005458333 (monthly interest rate)

n = 6 years * 12 months/year = 72

Substituting these values into the formula, we get:

M = $49,000 * 0.005458333 * (1 + 0.005458333)^72 / ((1 + 0.005458333)^72 - 1)

= $824.85 (rounded to the nearest cent)

To find the total amount he will repay, we can simply multiply the monthly payment by the number of payments:

Total amount =$824.85 * 72

Total amount = $59,389.2

Therefore, Alonzo will repay a total of $59,389.2 over the 6-year term of the loan if he makes the monthly payment each month.

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The probability that a randomly selected point within the red-shaded square is 0.16

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

The areas of the above shapes are

Red square = 6 * 6 = 36

White square = 15^2 = 225

The probability is then calculated as

P = Red square /White square

So, we have

P = 36/225

Evaluate

P = 0.16

Hence, the calculated value of the probability that the point falls in the red square is 0.16

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A measure of variability which the charity should use to accurately represent the data include the following: D. The range of 45 is the most accurate to use to show that they have plenty of money.

In Mathematics and Statistics, a box plot is a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the data set, the five-number summary for the given data set include the following:

Minimum (Min) = 10.

First quartile (Q₁) = 13.75.

Median (Med) = 20.

Third quartile (Q₃) = 26.25.

Maximum (Max) = 55.

For the IQR, we have:

IQR = Third quartile (Q₃) - First quartile (Q₁)

IQR = 26.25 - 13.75

IQR = 12.50

For the range, we have;

Range = maximum - minimum

Range = 55 - 10

Range = 45.

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Answer:

  [tex]x=-\dfrac{\sqrt{51}}{10}[/tex]

Step-by-step explanation:

You want the value of x in the third quadrant of a unit circle where y = -7/10.

The radius of a unit circle is 1, so the Pythagorean theorem tells you the magnitude of the value of x is ...

  1² = x² +(7/10)²

  x = -√(1 -49/100) = -√(51/100) . . . . . . x < 0 in the third quadrant

  x = (-√51)/10

Gabriela walks 1 mile distance on her way home from school as she is running part of the way.

The distance between two things is measured in distance units like miles, kilometres, or metres. Given that it only possesses magnitude and not direction, distance is a scalar number.

We must ascertain Gabriela's total trip distance before we can respond to this query.

We can get the overall distance by adding the distance she runs to the distance she walks because she runs a portion of the route and walks the other.  

Gabriela commutes two miles to school, so

She needs to travel a total of 2 miles.

We need to figure out how far she runs because she is running a portion of the way.

Since running is typically faster than walking, we can infer that she covers the distance in one mile by running half of it.

She will therefore need to walk the additional mile.  

Gabriela walks a mile on her way home from school, which is the answer to the question.

This is because she walks the remaining mile and runs the first mile.

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The exact value of sin 2θ is 12/37

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

We have:

cot θ = 6

Thus, tan θ = 1/6

Using the given information, we can sketch the location of the angle θ in the quadrant (See the attached image).

Thus, we can calculate the value of the hypotenuse using the Pythagoras theorem. That is:

hypotenuse = √((-6)² + (-1)²) = √37

sin θ = -1/√37

cos θ = -6/√37

Using trig. identity:

sin 2θ = 2sinθ·cosθ

sin 2θ = 2 * (-1/√37) * (-6/√37)

sin 2θ = 12/37

Therefore, the exact value of sin 2θ is 12/37

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Complete Question

If cot θ = 6,and θ lies in quadrant 3, find the exact value of sin 2θ

The bar graph from the results of the following survey question titled "Hours spent using your cell phone each week" is found in the attachment.

A bar chart, also called a bar graph, is a visual representation of categorical data that uses rectangular bars with heights or lengths that correspond to the measure of the values they represent.

The bars in a bar chart can be plotted either vertically or horizontally.

Vertical bars, horizontal bars, grouped bars, or stacked bars can all be used to make bar graphs.

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Answer:

Step-by-step explanation:

The common difference between the number of jumping jacks is 50, as you correctly stated. This means that the number of jumping jacks increases by 50 for each additional minute.

The slope of the line that connects any two points on this table represents the rate of change of the number of jumping jacks with respect to time. Since the common difference is constant, the slope of the line that connects any two points on this table will be the same. In this case, the slope is 50.

The slope of the line that connects the first and fourth points is calculated as (200 - 50) / (4 - 1) = 150 / 3 = 50, not 150.

The common difference (aka slope aka rate of change) is constant because the number of jumping jacks increases by the same amount for each additional minute. This means that the relationship between time and the number of jumping jacks is linear.

The probability that selected team includes more women then men is [tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

The likelihood that the selected team contains more women than males matches the probability that the team contains 5, 6, 7, or 8 women (there would therefore be 4, 3, 2, or 1 man in the squad, respectively).

The likelihood of there being 5 women on the team is equal to

[tex]\frac{(^{10} _{5} )(^{9} _{4} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

Since, we can choose 5 women out of 10 in [tex](^{10} _{5} )[/tex] ways, 4 men out of 9 in [tex](^{9} _{4} )[/tex] ways, and 9 persons out of 19 in [tex](^{19} _{9})[/tex] - [tex](^{10} _{9})[/tex] - [tex](^{9} _{9})[/tex] ways (we must reject the possibility of having all men ( [tex](^{9} _{9})[/tex] ways) or all women ( [tex](^{10} _{9})[/tex] ways) because it is a must).

Similarly, we compute the odds of having 6, 7, or 8 women on the team, and because these are disjoint alternatives, the final result is the total of their probabilities, which is

[tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

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The option that is not a run-on sentence is this: According to data collected throughout film history, the average length for a feature film is between 80 and 100 minutes, although some far exceed the standard running time.

A run-on sentence is a sentence that continues without using appropriate punctuation to join independent clauses. In the sentence above, the comma is used in the right place to join the independent clause, so we may not refer to this as a run-on sentence.

The other three options continue on end without appropriate punctuation to break off the independent clauses.

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The amount of time it takes the horse to complete a full cycle of motion is known as the period of the motion and can be calculated as follows:

T = 2π/ω.

Let's assume that a straightforward harmonic motion can be used to simulate the horse's movements. The simple harmonic motion equation is:

y = A sin(ωt + φ)

where:

The deviation from equilibrium is represented by the number y.

The maximum displacement's amplitude is A.

The angular frequency is equal to two times the frequency ω.

It's time t

The phase angle is φ

The equation can be written as follows, assuming that the horse is moving vertically and that its highest point corresponds to y = 0.

y = A sin(ωt)

where is the angular frequency and A is the motion's amplitude.

On a graph, where the horizontal axis denotes time and the vertical axis denotes displacement, we can draw this function as a sine wave. The

amount of time it takes the horse to complete a full cycle of motion is known as the period of the motion and can be calculated as follows:

T = 2π/ω

We need to know the duration of the horse's motion in order to predict when Ivan will have another chance to make a flawless shot. It's challenging to determine the period precisely in the absence of more details.

You can name the graph of the carousel equation θ = A cos(ωt) as follows:

The time, t, is displayed on the x-axis as seconds.

The angular position from the midline, θ, expressed in degrees, is shown on the y-axis.

The horizontal axis at θ = 0 degrees is the midline.

The largest angle of departure from the midline, or the distance between the wave's crest (highest point) or trough (lowest point), is known as the amplitude, or A.

The interval between two consecutive wave crests or troughs is known as the period, or T, which is the length of time it takes for one full cycle of motion.

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The solution is,

there are children = 12, teenagers= 66, adults = 102,

so, adults = 102 is the largest group represented

Here, we have,

given that,

The ratio of children: teenagers: adults in a survey sample are 2: 11:17.

now, we have,

If there are 180 people surveyed

now, let , we have,

children = 2x

teenagers= 11x

adults = 17x

now, we have,

2x + 11x + 17x = 180

or, 30x = 180

or. x = 6

Hence, The solution is,

there are children = 12, teenagers= 66, adults = 102,

so, adults = 102 is the largest group represented.

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The surface area of the given composite solid is calculated as: 200.24 square units.

Surface area of the composite solid = surface area of rectangular prism + surface area of cylinder - 2(area of the base of the cylinder)

Surface area of rectangular prism = length * width * height = 10 * 3 * 5

= 150 square units.

Surface area of cylinder = 2πr(h + r)

height (h) = 4

Radius (r) = 2

Plug in the values:

Surface area of cylinder = 2 * 3.14 * 2(4 + 2) = 75.36 square units

Area of the base of the cylinder = πr² = 3.14 * 2²

= 12.56 square units

Thus, we have:

Surface area of the composite solid = 150 + 75.36 - 2(12.56) = 200.24 square units.

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To prove that the left cosets of H constitute a partition of G, we need to show that they satisfy two conditions:

1) Each element of G belongs to some left coset of H.

2) Any two distinct left cosets of H are disjoint.

Let g be any element of G. Then, by definition of a left coset, there exists an element h of H such that g = h·x for some x in G. Therefore, g belongs to the left coset H·x. Since this holds for any element g of G, we have shown that the left cosets of H cover all of G, satisfying the first condition.

Now, let H·x and H·y be two distinct left cosets of H. Suppose there exists an element z in both H·x and H·y.

Then, z = [tex]h_{1}[/tex]·x = [tex]h_{2}[/tex]·y for some [tex]h_{1}[/tex], [tex]h_{2}[/tex] in H.

Solving for x,

x = [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] · y.

Since H is a subgroup, [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] is also in H, and therefore x belongs to H·y.

Thus, H·x is contained entirely in H·y, and the two cosets cannot intersect.

This satisfies the second condition, and hence the left cosets of H constitute a partition of G.

A similar argument can be used to show that the right cosets of H also constitute a partition of G. This completes the proof.

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Answer:

C

Step-by-step explanation:

Positive numbers represent a positive situation. And negative numbers coincide with a negative situation.

 Ex: An increase in temperature corresponds to a positive number, and a decrease in temperature with a negative number.

It is a point and point is zero dimensional. Therefore the right options are A and C respectively.

Point is a basic geometric object that represents a location or a position in space. It is often represented as a dot or a small circle.

Points have no size, shape, or dimension. They are considered to be zero-dimensional objects, and they are often used as a building block for more complex geometric structures.

In addition to representing locations in space, points are also used in other areas of mathematics, such as in coordinate systems and in topology. In geometry, points are used to define lines, planes, and other geometric shapes.

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a) Yes, the function is in standard form.

  Degree = 2

  Exponent = 2

  Constant term = -1

b) No, the function is not in standard form.

   And it doesn't have the degree, exponent and constant term.

Firstly, let's consider the function f(x) = x(x-1). To determine if it is a polynomial function, we need to check if all of its terms have non-negative integer exponents. In this case, we see that f(x) can be written as x² - x, which has only two terms, and each term has a non-negative integer exponent. Therefore, f(x) is a polynomial function.

In this case, the degree of f(x) is 2 because the highest exponent of x is 2.

In this case, the leading term of f(x) is x², and its leading coefficient is 1.

Finally, the constant term is the term that does not have the variable in it. Here, the constant term of f(x) is -1.

Next, let's consider the function f(x) = 3 - 1/2 x. We can see that f(x) is not a polynomial function because it has a term with a negative exponent.

Polynomial functions must have non-negative integer exponents on all terms.

Therefore, the answer to the first question is 'no.'

Since f(x) is not a polynomial function, it does not have a degree, leading term, or constant term.

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(c) Total interest accrued until payments begin = $1023.33 (loan 1) + $2362.33 (loan 2) ≈ $3385.66.

(a) For loan 1, $3200 at 6.7% for 4 years and 7 months (4 years in college + 3 months deferment), the simple interest = 3200 * 0.067 * (55/12) ≈ $1023.33.

(b) For loan 2, $12,000 at 7.3% for 2 years and 7 months (2 years in college + 3 months deferment), simple interest = 12000 * 0.073 * (31/12) ≈ $2362.33.

(c) Total interest accrued until payments begin = $1023.33 (loan 1) + $2362.33 (loan 2) ≈ $3385.66.

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Approximately 19142 people were in the arena at the exact moment the event ended.

To decipher an exponential equation synchronous with the provided data, we will employ the equation of [tex]F(t) = a * b^t.[/tex]

Herein, 'F(t)' stands for the total number of attendees in the stadium at 't' minutes after the stoppage of the event, with 'a' symbolizing the initial value and 'b' being the base.

Exploiting the regression aptitudes of a graphing calculator, the exponential equation that most adequately applies to the listed information is: [tex]F(t) = a * b^t[/tex] with three decimal points specified.

F(t) = 19141.846 * 0.718^t

Using the equation:

[tex]F(23) = 19141.846 * 0.718^2^3 = 2726[/tex]

Answer: Approximately 2726 people were in the arena 23 minutes after the event ended.

How many people were in the arena 60 minutes after the event ended?

[tex]F(60) = 19141.846 * 0.718^6^0 = 123[/tex]

Answer: Around 123 people were in the arena a full hour after the occurrence concluded.

How many people were present precisely when the event wrapped up?

F(0) = 19141.846 * 0.718^0 = 19141.846 * 1 = 19141.846

Answer: Approximately 19142 people were in the arena at the exact moment the event ended.

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