The table below displays the distances driven during each 1-hour interval of a 4 hour trip.


Which is closest to the total distance driven?

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 3\%\to \frac{3}{100}\dotfill &0.03\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.03}{1}\right)^{1\cdot 4}\implies A=3000(1.03)^4 \implies A \approx 3376.53[/tex]

Using the given information, we calculated the following prices:

a) The retail price is $420

b) The wholesale price is $55

What are retail and wholesale prices?

The processes of wholesale and retail are essentially distinct from one another because they entail the transfer of commodities from manufacturing to distribution. Purchasing products and selling them to clients constitute retail.

Retailers pay wholesale pricing to producers or distributors. Then, the shop charges customers a higher price—the retail price—for the identical product. Wholesale pricing is what you charge retailers who buy things in large amounts. With wholesale pricing, products are sold for more money than it costs to produce them in order to generate a profit.

The final selling price that retailers decide to charge customers is known as the retail price. In retail pricing, the customer is everything.

Let,

The retail price = x

The wholesale price = y

Then,

x = y + 0.4y = 1.4y

a) When y = $300

x = 1.4 * 300 = $420

b) When x = $77

77 = 1.4y

y = 77/1.4 = $55

Therefore using the given information, we calculated the following prices:

a) The retail price is $420

b) The wholesale price is $55

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thy be c matey truth me ARG!!??

Answer:7$

Step-by-step explanation:$4+3$=7$

The shaded area to the right of 4 indicates all values larger than 4 being inequality included in the solution set, while the closed circle at 4 represents the number 4 being included in the solution set.

A relationship between two terms or values which is not equal is known as an inequality in mathematics. Inequality follows imbalances, therefore. In mathematics, an inequality makes a link among two quantities that are not equal. Egality and inequality are different. Use the not similar symbol most frequently when two components are not equal (). Various inequalities are employed to contrast values of all sizes. By changing the two parts until only the variables are left, one can solve many straightforward inequalities. However a variety of factors support inequality: Negative values are split or added on either side. Switch between the left & right.

The graph below shows the set of solutions to the inequality where x + 2 is larger than or equal to 6:

From negative 9 to positive 9 is a number line. At positive 4, a closed circle appears. Positive numbers four through nine are shaded on the number line.

This means that the inequality will be satisfied by any value of x larger than or equal to 4. The graph includes the value 4 in the solution set since it has a closed circle there, and because it is shaded to the right of 4, it also includes all values higher than 4.

Here is an illustration of the graph:

  |----|----|----|----|----|----|----|----|----|

     -9   -8   -7   -6   -5   -4   -3   -2   -1    0

                                           o        

                                         ---->      

                                           4        

                                         ----]      

                                           >        

      |----|----|----|----|----|----|----|----|----|

      -9   -8   -7   -6   -5   -4   -3   -2   -1    0

The shaded area to the right of 4 indicates all values larger than 4 being included in the solution set, while the closed circle at 4 represents the number 4 being included in the solution set.

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The smaller acute angle of the right triangle is x-3 = 12.6 degrees.

What is Right Triangle?

A triangle in which one angle is 90 degree is called a Right Angled Triangle.

In a right triangle, if one acute angle is x-3 degrees, then the other acute angle is 90-(x-3) = 93-x degrees.

Using the given equation, we have:

sin(x-3) = cos(6x+9)

Taking the sine of both sides:

sin(x-3) = sin(90 - (6x+9)) = sin(81-6x)

Now we have:

sin(x-3) = sin(81-6x)

Since the two angles have the same sine, they must differ by a multiple of 360 degrees. Thus:

x-3 = 81-6x + 360n or x-3 = 6x-81 + 360n

where n is an integer.

Simplifying each equation:

7x = 84 + 360n or 5x = 78 + 360n

Dividing both sides by 7 or 5, respectively:

x = 12 + 51.43n or x = 15.6 + 72n

The first equation gives us values of x that are too large for an acute angle, so we use the second equation:

x = 15.6 + 72n

The smallest such x that satisfies 0 < x < 90 is when n=0, which gives:

x = 15.6 degrees

Therefore, the smaller acute angle of the right triangle is x-3 = 12.6 degrees.

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The population parameter of interest in this study is the proportion or percentage of all female adults in the entire population who support a ban on public smoking. The Gallup poll was conducted to estimate this proportion or percentage by collecting data from a sample of female adults. By using statistical inference methods, the researchers can use the sample data to make inferences about the population parameter with a certain level of confidence.

Answer:

Nonexamples of base and exponent in math would be any mathematical expressions that do not involve raising a base to an exponent. For example, 2 + 3 or √4 do not involve base and exponent.

An example of coefficient and exponential form is 5x², where 5 is the coefficient and x² is the exponential form. A nonexample could be x²/5, which does not have a coefficient in front of the exponential form.

An example of expanded form is 5x + 2, where the expression is fully written out. The standard form of the same expression would be 2 + 5x. A nonexample of expanded form is 2(x + 3), where the expression is not fully written out.

Examples of exponent laws include the product rule (a^m * a^n = a^(m+n)), quotient rule (a^m / a^n = a^(m-n)), and power rule ((a^m)^n = a^(mn)). A nonexample of an exponent law could be the sum of powers rule, which does not exist in traditional exponent laws.

The solutions for all the parts are given below.

Exponentiation is one of the mathematics operations.

Let mᵃ, where m and a are the real numbers.

And m is multiplied by 'a' times to itself.

So, a is the exponent of m.

1) Non-examples of a base and exponent in math would be: "2 + 3" or "5 - 7" as these are expressions and not a base raised to an exponent.

2) An example of a coefficient and exponential form would be "2x^3" where 2 is the coefficient and x^3 is the exponential form. A nonexample would be "3x + 2" as it is an expression and not in exponential form.

3) An example of an expanded form would be 235 = 2 x 100 + 3 x 10 + 5 x 1, where each digit is multiplied by its corresponding place value. An example of standard form would be 235 written as is, without being broken down into its place values. A nonexample of expanded form would be 235 written as 2 + 3 + 5, as this is not a multiplication of the digits with their place values.

4). Examples of exponent laws include:

Product law: aᵇ x aⁿ = a⁽ᵇ ⁺ⁿ⁾

Quotient law: [tex]a^m / a^n = a^{(m-n)}\\[/tex]

Power law: [tex](a^m)^n = a^{(m \times n)}[/tex]

Negative exponent law: [tex]a^{(-n)} = 1/a^n[/tex]

Zero exponent law: [tex]a^0 = 1[/tex].

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The three accοmpanying questiοns, assuming that the firm is prοfit-maximizing and dοes nοt shut dοwn in the shοrt run

1)Firm's Tοtal Revenue=  $78000

2)Firm's Tοtal Cοst =$128700

3)Firm's Lοss = - $50700

Wοrd prοblems are οften described verbally as instances where a prοblem exists and οne οr mοre questiοns are pοsed, the sοlutiοns tο which can be fοund by applying mathematical οperatiοns tο the numerical infοrmatiοn prοvided in the prοblem statement. Determining whether twο prοvided statements are equal with respect tο a cοllectiοn οf rewritings is knοwn as a wοrd prοblem in cοmputatiοnal mathematics.

Here in the given graph,

Cοst per unit = $300

Equilibrium quantity = $260 Then,

Firm's Tοtal Revenue = P * Q

=> $300*260 = $78000 (Equilibrium where, MR = MC)

Firm's Tοtal Cοst = Cοst per Unit * equilibrium quantity

=> $495*260 = $128,700

Firm's Lοss = TR - TC

=> $78000 - $128,700 = - $50700

Hence the answers are,

1)Firm's Tοtal Revenue=  $78000

2)Firm's Tοtal Cοst =$128700

3)Firm's Lοss = - $50700

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Answer:

95.522

Step-by-step explanation:

First, we have to find the area of the triangle...

Next, we have to find the area of the sector...

Now we have to add our results together...

Hope this helps, have a lovely day! :)

Answer:

Step-by-step explanation:

To fill in the table, we can use the following steps:

Calculate the interest for each month, which is the prior balance multiplied by the monthly interest rate (1.75%).

Subtract the monthly interest from the prior balance to get the new balance before the monthly payment.

Subtract the monthly payment from the new balance to get the ending balance.

Month         Prior     1.75% Interest on   Monthly                Ending

                Balance     Prior Balance      Payment              Balance

1                $5000           $87.50              $325                $4762.50

2               $4762.50 $83.39              $325                $4520.84

3               $4520.84 $79.06              $325                $4276.90

4               $4276.90 $74.50              $325                $4029.23

5               $4029.23 $69.69             $325                $3777.87

Therefore, after 5 months of making $325 payments, the remaining balance on the loan is $3777.87.

After solving the given problem, we found that the difference between the area of the dilated square and the original square is 32 square centimeters.

The area of the original square with a side length of 2 centimeters can be calculated as:

A = s²

A = 2²

A = 4 square centimeters

When this square is dilated by a scale factor of 3, the new side length will be:

s' = 3s

s' = 3(2)

s' = 6 centimeters

The area of the dilated square can be calculated as:

A' = s'²

A' = 6²

A' = 36 square centimeters

The difference between the area of the dilated square and the original square is:

A' - A = 36 - 4

A' - A = 32 square centimeters

Therefore, the difference between the area of the dilated square and the original square is 32 square centimeters.

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Answer: Let's assume that the speed of the plane in still air is represented by p and the speed of the wind is represented by w.

When the plane is flying with the tailwind, its speed relative to the ground is the sum of its speed in still air and the speed of the wind, or (p + w). Similarly, when the plane is flying against the wind, its speed relative to the ground is the difference between its speed in still air and the speed of the wind, or (p - w).

We can set up two equations based on the given information:

(p + w) = 158 (1) (when flying with the tailwind)

(p - w) = 112 (2) (when flying against the wind)

To solve for p and w, we can add equations (1) and (2):

2p = 270

p = 135

So the speed of the plane in still air is 135 km/h.

We can then substitute this value of p into equation (1) to solve for w:

(p + w) = 158

(135 + w) = 158

w = 23

So the speed of the wind is 23 km/h.

Therefore, the plane flies at 135 km/h in still air and the wind blows at 23 km/h.

Step-by-step explanation:

Given statement: Here's the table with the values organized:
X | Y
-----
1 | 5
-3 | 2
7 | 2
6 | 3
7 | -4
5 | 9
8 | 4
1 | 7
1 | 0

This statement is False.

Because, The table provided is not a function because it does not have a clear and unique output value for each input value.

Since the X values 1 and 7 have multiple Y values, the given table is not a function.

In a function, each input value (X) must have a unique output value (y). However, in the given table, there are some input values, such as X = 7 and X = 1, that have multiple output values. For example, when X = 7, the table provides two output values, 2 and -4. Similarly, when X = 1, the table provides two output values, 5 and 0.

A function is a mathematical relationship between the input and output values, where each input value produces only one unique output value. Functions are used to represent many real-world scenarios, including calculating distances, temperatures, and profits. Therefore, it is crucial to ensure that the provided table represents a function by ensuring that each input value has a unique output value.

In conclusion, the table provided is not a function as it violates the one-to-one mapping between input and output values.

A function must have a clear and unique output value for each input value.

To determine if the given table represents a function, we need to make sure that each input (X value) corresponds to only one output (Y value).
Now let's check for any repeating X values with different Y values:
1 corresponds to both 5 and 7
7 corresponds to both 2 and -4.

False.

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1) The nth term if the sequences are (a) 9a-7  (b)  13-6a  

A sequence is a list of numbers or objects in a special order.1 It is an enumerated collection of objects in which repetitions are allowed and order matters.

The given sequences are

1) (a)  2,11,20,...

the first term a= 2, the common difference d = 11-2 =9

The nth term is given as

Tn = a + (n-1)d

Tn = 2 + (n-1)*9

Opening the brackets to get

2 + 9n -9

Rearrange to have

9n -7

(b)  the sequence is 7,1,-5,...

a = 7 ,

d= 1-7 = -6

Tn = a + (n-1)d

Tn = 7 + (n - 1)-6

Tn = 7 + -6n +6

Tn = 7+6 -6n

Tn = 13 -6n

2) Remember for every Fibonacci sequence

1st+2nd=3rd

2nd+3rd=4th

3rd+4th=5th

4th+5th=6th

Remember for every arithmetic Progression

Tn = a + (n-1)d

For every Geometric Progression,

Tn = arⁿ⁻¹

First five terms        Next three terms      Name of sequence(A,G,F,Q)

  10,6,2,-2,-6                 -10,-14,-18                    arithmetic

  2,8,32,126512        2048, 8192,32768          Geometric

  20,13,33,46,79           125,204,329                 Fibonacci

  200,100,50,25,12.5   6.25, 3.125, 1.5625      Geometric

46,39,32,25,18,              11,4 , -3                         Arithmetic

-2,,2,0,2,2,4                  6,10,16                            Fibonacci

3,6,10,15, 21                 28,35,42                        Arithmetic

25,15,0,-20,-45,           -80,-125,-175                Arithmetic

2,5,7,12,19                    31,50,81                       Fibonacci

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The probability that exactly 2 of the workers interviewed are union members is approximately 0.044.

To solve this problem, we can use the binomial probability formula, which is:

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

where:

n is the sample size (number of workers being interviewed)

k is the number of successes we are interested in (in this case, 2 of the workers being union members)

p is the probability of success (in this case, the probability that a worker is a union member)

(n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items (this can be calculated using the formula n! / (k! * (n - k)!))

Plugging in the values, we get:

[tex]P(X = 2) = (4 choose 2) * (0.96)^2 * (1 - 0.96)^(4 - 2)[/tex]

[tex]= (6) * (0.96)^2 * (0.04)^2[/tex]

= 0.04403136

Therefore, the probability that exactly 2 of the workers interviewed are union members is approximately 0.044.

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The directiοnal derivative οf [tex]f(x,y) = 2ln(x^2 + y^2)[/tex] at the pοint (2,1) in the directiοn οf the vectοr v = (-1,2) is 0.

A directiοnal derivative shοws hοw quickly a functiοn changes in any directiοn. A fοrmula that incοrpοrates the gradient can be used tο determine the directiοnal derivative. The gradient οf a functiοn οf mοre than οne variable shοws the directiοn οf greatest change.

We must cοmpute the dοt prοduct οf the gradient οf f at (2,1) with the unit vectοr in the directiοn οf v in οrder tο determine the directiοnal derivative οf the functiοn  [tex]f(x,y) = 2ln(x2, + y2)[/tex]  at the pοsitiοn (2,1) in the directiοn οf the vectοr v = (-1,2).

Then, we determine f's gradient:

Grad(f) is equal tο [tex](f/x, f/y) = (4x/(x2+y2), 4y/(x2+y2)[/tex]

Hence, the gradient at lοcatiοn (2,1) is as fοllοws:

grad [tex](f)(2,1) = (4(2)/(2^2+1^2), 4(1)/(2^2+1^2)) = (8/5, 4/5)[/tex]

The unit vectοr in the directiοn οf v is then lοcated:

||v|| = √((-1)²+ 2²) = √5\su = v/||v|| = (-1/√5, 2/√5)

We calculate the directiοnal derivative last.

Def = grad(f) (2,1) · u\s= (8/5)(-1/√5) + (4/5)(2/√5)\s= -8/5√5 + 8/5√5\s= 0

Hence, at pοint (2,1) in the directiοn οf the vectοr v = (-1,2), the directiοnal derivative οf f(x,y) = 2ln(x², + y²) is equal tο 0.

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Step-by-step explanation:

Assuming that the grain required to produce 100 liters of ethanol could feed one person for a year, we can calculate the number of people that could have been fed by the additional 49 billion liters of ethanol produced in the US from corn between 2001 and 2018.

First, we need to determine the amount of grain required to produce 1 liter of ethanol. This can vary depending on a number of factors, including the type of grain and the production method, but a commonly cited estimate is that it takes around 1.4 kilograms of corn to produce 1 liter of ethanol.

Therefore, to produce 49 billion liters of ethanol, we would need:

49,000,000,000 liters x 1.4 kg of corn per liter = 68,600,000,000 kg of corn

To convert this to the amount of grain required to feed people, we need to divide by the amount of grain needed to feed one person for a year. Again, this can vary depending on the person's age, sex, and level of activity, but a commonly cited estimate is that an adult needs around 700 kilograms of grain per year.

Therefore, the amount of grain required to feed the number of people who could have been fed by the grain used to produce the additional ethanol is:

68,600,000,000 kg of corn / 700 kg of corn per person per year = 98,000,000 people

So the additional 49 billion liters of ethanol produced in the US from corn between 2001 and 2018 could have fed around 98 million people for a year.

The additional ethanol production from corn in 2018 could have potentially fed 49 billion people

To determine how many people could have been fed with the additional 49 billion liters of ethanol produced in the US from corn in 2018 compared to 2001, we need to calculate the amount of grain saved by producing ethanol instead of using it for direct consumption.

According to the information given, the grain required to produce 100 liters of ethanol can feed a person for a year. Therefore, for each liter of ethanol produced, the grain equivalent can sustain one person for a year.

To find the number of people fed, we can multiply the additional 49 billion liters of ethanol produced by the grain equivalent for each liter:

49 billion liters * 1 person/year per liter = 49 billion people

Hence, the additional ethanol production from corn in 2018 could have potentially fed 49 billion people based on the assumption that the grain used to produce ethanol would have been used for direct consumption instead.

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The first derivative of y is: y' = [4tan(πt)sec(πt) - πsec²(πt)]/(π³) and

The second derivative of y is: y'' = [4sec(πt)(π²sec²(πt) - 3πtan(πt))]/(π⁴)

The first derivative of a function represents the rate of change or slope of the function. The second derivative represents the rate of change of the first derivative or the curvature of the function.

To find the first derivative of y, we use the quotient rule:

y = [tan(πt)/π²] + [4 sec(πt)/π²]

y' = [π²(sec²(πt)) - 2tan(πt)sec(πt)]/(π⁴) + [4πsec(πt)tan(πt)]/(π⁴)

Simplifying this expression, we get:

y' = [(πsec(πt))(4tan(πt) - πsec(πt))]/(π⁴)

y' = [4πtan(πt)sec(πt) - π²(sec²(πt))]/(π⁴)

To find the second derivative of y, we again use the quotient rule:

y'' = [(π³(sec³(πt)) - 12πtan(πt)sec²(πt)) - 2π(4tan(πt)sec(πt) - πsec²(πt))(πsec(πt))]/(π⁶)

Simplifying this expression, we get:

y'' = [(4πsec(πt))(π²(sec²(πt) - 3tan(πt)))]/(π⁶)

y'' = [4sec(πt)(π²sec²(πt) - 3πtan(πt))]/(π⁴)

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Answer:

9000000 Milligrams

Step-by-step explanation:

We know that,

1 kilogram = 10^6 Miligram

So, 9  Kilpgrams = 9×10^6 = 9000000 Milligrams

The equations of the other two sides are AD: y = -3x + 3 and DC: y = 1/2x - 1/2.

What is point-slope form?

The point-slope form is a way to write the equation of a straight line in algebra. It is used when you know the coordinates of a point on the line and the slope of the line. The point-slope form of a linear equation is:

y - y₁ = m(x - x₁)

where m is the slope of the line, and (x1, y1) are the coordinates of a point on the line.

Since AB and BC are two sides of a parallelogram, they are parallel to each other. Thus, we can use the slope formula to find the slopes of these sides.

Slope of AB:

m = (y₂ - y₁)/(x₂ - x₁)

= (6 - 3)/(6 - 0)

= 3/6

= 1/2

Slope of BC:

m = (y₂ - y₁)/(x₂ - x₁)

= (3 - 6)/(7 - 6)

= -3/1

= -3

Now that we know the slopes of AB and BC, we can use the point-slope form to find the equations of the other two sides.

Equation of AD:

We know that AD is parallel to BC and passes through point A. So, we can use the point-slope form with the slope of BC and point A.

y - y₁ = m(x - x₁)

y - 3 = -3(x - 0)

y - 3 = -3x

y = -3x + 3

Equation of DC:

We know that DC is parallel to AB and passes through point C. So, we can use the point-slope form with the slope of AB and point C.

y - y₁ = m(x - x₁)

y - 3 = 1/2(x - 7)

y - 3 = 1/2x - 7/2

y = 1/2x - 1/2

Therefore, the equations of the other two sides are:

AD: y = -3x + 3

DC: y = 1/2x - 1/2

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bc=√{ac²-ab²}

bc=√{16²-11²}

bc=√{256-121}

bc=√{135}

bc=11.62

Answer:

x+3+3

Step-by-step explanation:

So bassicly you  have to add everything up ad then divide by 1

The value of x is 15 inches, we find this by Pythagoras theorem and property of isosceles triangle.

Pythagoras theorem states that sum of square of base and square of perpendicular is equal to square of hypotenuses.

So , it can be formula is given below ,

  [tex]perpendicular^{2} + base^{2} = hypotenuse^{2}[/tex]

And In ab isosceles triangle two equal side vertex divide base in two equal parts.

So, here base = 40.

Therefore it is divided in 20-20 inches,

Now putting the formula of Pythagoras theorem, we get

[tex]20^{2} + x^{2} = 25^{2}[/tex]

[tex]x^{2} = 25^{2} - 20^{2}[/tex]

[tex]x^{2} = 225[/tex]

So we get, x = 15 inches.

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The sequence of transformations is:

Let's only follow the coordinates of one of the vertices to identify the transformations, we clearly have a reflection over a vertical line and a reflection over a horizontal line, so first let's apply thes two.

Vertex A starts at (1, -4)

First a reflection over the x-axis will change the sign of the y-component, then we get:

A₁ = (1, 4)

Then a reflection over the y-axis changes the sign of the x-component to:

A₂ = (-1, 4)

Finally we have a translation, we can see that:

A' = (2, 4).

Then we have a translation (a, b) such that:

(-1 + a, 4 + b) = (2, 4)

So we can see that a = 3 and b = 0, then we have a translation of 3 units to the right.

That is the sequence of transformations.

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The two column proof that shows that ΔADC ~ ΔDBC is as shown below.

A two-column proof is defined as a geometric proof that is said to consist of a list of statements, and the reasons for which  those statements are true. The statements are thus listed in a column on the left, with the reasons for which the specific statements can be made are now listed in the right column.

The complete two column proof is expressed as:

Statement 1: ∠ADC is a right angle

Reason 1: Given

Statement 2: DB is perpendicular to AC

Reason 2: Given

Statement 3: ∠DBC is a right angle

Reason 3: Definition of Perpendicular

Statement 4: ∠ADC ≅ ∠DBC

Reason 4: All right angles are congruent

Statement 5: ∠C ≅ ∠C

Reason 5: Reflexive Property of Congruence

Statement 6: ΔADC ~ ΔDBC

Reason 6: AA Similarity

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Answer:

158.4 ft/sec

Step-by-step explanation:

multiply the speed value by approximately 1.467

Answer:

Common ratio r = [tex]\dfrac{2}{3}[/tex]

Sequence  is described by
[tex]a_n = \dfrac{3}{2}\cdot \left(\dfrac{2}{3}\right)^{n-1}\\[/tex]

Step-by-step explanation:

[tex]S_n = \dfrac{a_1(1-r^n)}{1-r}[/tex]

where
r = common ratio
a₁ = first term

Sum of first two terms

[tex]S_2 = \dfrac{a_1(1-r^2)}{1-r}[/tex]

Sum of first four terms:
[tex]S_4 = \dfrac{a_1(1-r^4)}{1-r}[/tex]

[tex]\dfrac{S_4}{S_2} = a_1 \cdot \dfrac{1-r^4}{1-r} \div a_1 \cdot \dfrac{1-r^2}{1-r}\\[/tex]

To divide, flip the divisor and multiply

[tex]\dfrac{S_4}{S_2} =\dfrac{a_1(1-r^4)}{1-r} \times \dfrac{1-r}{a_1(1-r^2)}[/tex]

The a₁ and (1-r) terms cancel out from numerator and denominator leaving

[tex]\dfrac{S_4}{S_2} =\dfrac{1-r^4}{1-r^2} \cdots [1][/tex]  

Using the identity
[tex]a^2 - b^2 = (a- b)(a+b)[/tex]

[tex]1- r^4 = 1^4 - r^4 = (1^2 - r^2)(1^2+r^2) = (1-r^2)(1+r^2)[/tex]

Plugging this into equation 1 we get
[tex]\dfrac{S_4}{S_2} =\dfrac{(1-r^{2})(1+r^{2})}{1-r^{2}}[/tex]

The [tex]1- r^2[/tex] terms cancel out leaving:
[tex]\dfrac{S_4}{S_2} =1 + r^2[/tex]

We are given

[tex]S_4 =\dfrac{65}{18}\\\\S_2 = \dfrac{5}{2}[/tex]

[tex]\dfrac{S_4}{S_2} = \dfrac{65}{18} \div \dfrac{5}{2}[/tex]

To divide, flip the denominator [tex]\dfrac{5}{2}[/tex] and multiply
[tex]\dfrac{S_4}{S_2} = \dfrac{65}{18} \times \dfrac{2}{5}\\\\= \dfrac{13}{9}[/tex]

Therefore
[tex]1 + r^2 = \dfrac{13}{9}\\\\r^2 = \dfrac{13}{9} - 1\\\\= \dfrac{13}{9} - \dfrac{9}{9}\\\\= \dfrac{4}{9}[/tex]

[tex]r = \sqrt{\dfrac{4}{9}}\\\\= \dfrac{\sqrt{4}}{\sqrt{9}}\\\\= \dfrac{2}{3}[/tex]

So the common ratio
[tex]r = \dfrac{2}{3}[/tex]

To find the first term  we have sum of first two terms = 5/2

[tex]S_2 = \dfrac{a_1(1-r^2)}{(1-r)} = a_1 (1+ r)[/tex]

Plugging in knowns
[tex]\dfrac{5}{2} = a_1(1+\dfrac{2}{3})\\\\= a_1 \cdot \dfrac{5}{3}[/tex]

Multiply both sides by 3/5 to get
[tex]a_1 = \dfrac{5}{2} \times \dfrac{3}{5}\\\\a_1 = \dfrac{3}{2}[/tex]

The nth term of a GP is
[tex]a_n = a_1 \cdot r^{n-1}\\[/tex]

Plugging in the values obtained
[tex]a_n = \dfrac{3}{2}\cdot \left(\dfrac{2}{3}\right)^{n-1}\\[/tex]

Answer:

x = 36/7 (or approximately 5.143); y = 19/70 (or approximately 0.271)

Step-by-step explanation:

First, we can distribute the 2 to have both equations look similar:

[tex]2(0.5x-y)=4.6\\x-2y=4.6[/tex]

Now, we can eliminate the ys by multiplying the entire first equation by 2.  This will allow us to solve for x.

[tex]2(0.2x+y=1.3)\\\\0.4x+2y=2.6\\x-2y=4.6\\\\1.4x=7.2\\x=\frac{36}{7}\\ x=5.142857143[/tex]

Now, we can plug in 36/7 for x in any of the two equations:

[tex]0.2(\frac{36}{7})+y=1.3\\ \frac{36}{35}+y=1.3\\ y=\frac{19}{70}\\ y=0.2714285714[/tex]

If you use the decimal answers, you can round as much as you need.