i’m confused and would love help.

The largest number of members such a subset can have is 60.

Let's consider the numbers in the subset that are multiples of 5. If we have one of these multiples in the subset, then we cannot have any other multiple of that number in the subset, otherwise, one of them would be 5 times another.

So we can choose at most one number from each set of multiples of 5: either 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, or 100.

This gives us a maximum of 20 numbers that we can choose from these sets.

Now we consider the remaining numbers from 1 to 100 that are not multiples of 5. Each of these numbers can be expressed as a product of a power of 2 and an odd number. For example, 33 = 2^0 x 33, 72 = 2^3 x 9.

If we choose any number that has a power of 2 greater than or equal to 3, then we cannot choose any other number that has the same odd factor, otherwise, one of them would be 5 times another. This is because any number with a power of 2 greater than or equal to 3 is a multiple of 8, and any multiple of 8 has a factor of 5.

So we can choose at most two numbers from each set of numbers that have the same odd factor: either 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, or 99.

This gives us a maximum of 2 x 20 = 40 numbers that we can choose from these sets.

Therefore, the largest number of members such a subset can have is 20 + 40 = 60.

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Answer: (1) The scenario represents an arithmetic sequence because each time an omelet is made, the number of remaining eggs decreases by the same amount.

(2) Let's denote the number of remaining eggs after making the n-th omelet by a_n. We can see from the problem statement that the difference between any two consecutive terms in the sequence is constant. Let d be a common difference. Then, we have:

a_{n+1} = a_n - d

Using the information from the problem statement, we can find the value of d:

a_2 - a_1 = 174 - 177 = -3

a_3 - a_2 = 171 - 174 = -3

Since the difference is the same in both cases, we have d = -3. Therefore, the recursive formula is:

a_{n+1} = a_n - 3, with a_1 = 177.

(3) To find an explicit formula, we can use the recursive formula to derive a general expression for a_n. Starting with the recursive formula, we have:

a_2 = a_1 - 3

a_3 = a_2 - 3 = a_1 - 23

a_4 = a_3 - 3 = a_1 - 33

a_5 = a_4 - 3 = a_1 - 4*3

We can see that the general expression for a_n is:

a_n = a_1 - (n-1)*3

Substituting a_1 = 177, we get:

a_n = 180 - 3n

Therefore, the explicit formula for the number of remaining eggs after making the n-th omelet is a_n = 180 - 3n.

(4) To find the number of eggs Joe will have left after making 42 omelets, we can simply substitute n = 42 into the explicit formula:

a_{42} = 180 - 3*42 = 54

Therefore, Joe will have 54 eggs left after making 42 omelets.

Step-by-step explanation:

As a result, the answer is k = -100 a

A linear equation is one in which the variable's highest power is always 1. An additional name for it is a one-degree equation. The most common form of a linear equation with just one variable is Ax + B = 0. In this case, B is constant, A is a coefficient, and x is a variable.

given

Increasing the right side of the equation results in:

Because -(-x) = x, -4 = -(-k-86) + 10 translates to k + 86 + 4.

-4 = k + 96

By taking 96 away from both sides, we get at:

-4 - 96 = k + 96 - 96

-100 = k

As a result, the answer is k = -100 .

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Keep in mind that the numbers used in the example are arbitrary, and you will need to use the actual premium amounts to calculate the deductions

Hi! To find the annual premium deductions and weekly deductions for each insurance plan, follow these steps:

1. Calculate the annual deduction for each plan by multiplying the total annual premium by 30% (your contribution).
2. Calculate the weekly deduction by dividing the annual deduction by 52 (number of weeks in a year).

For example, if AAA+ annual premium is $1,000, Health Today annual premium is $1,200, and Best Insurance annual premium is $800:

AAA+ Annual Deduction = 30% of $1,000 = $300
Weekly Deduction = $300/52 ≈ $5.77

Health Today Annual Deduction = 30% of $1,200 = $360
Weekly Deduction = $360/52 ≈ $6.92

Best Insurance Annual Deduction = 30% of $800 = $240
Weekly Deduction = $240/52 ≈ $4.62

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Rowan should get the ice cream in a cup as the cup holds 26.2 in3 more ice cream than the cone.

First, we need to find the volume of ice cream that the cone holds:

we can use the formula for volume, we get:

Volume = π × r² × h/3

when we put the values in the formula, we get:

Volume = 3.14 × 2.5² × 5/3

solving the problem, we get:

Volume =32.7 inches³

Now we need to find the volume of ice cream that the cup holds,

we use the formula of volume, we get:

Volume = π × r² × h

Diameter = 5 inches ⇒ Radius = 2.5 inches

(since we know that the radius is the exact half of the diameter)

when we put the values in the formula, we get:

Volume = 3.14 × 2.5² × 3

Volume =58.9 inches³

Now that we have both the values, we can easily find the difference:

Volume of the cup - volume of the cone = 58.9 - 32.7 = 26.2 inches³

The cup holds 26.2 in³ more ice cream than the cone.

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Answer:

Step-by-step explanation:

For vectors a = {4, 5, 0} and b = {1, 0, 3}, you want the cross product and verification that the cross product is orthogonal to both 'a' and 'b'.

The cross product of 4i+5j+0k and 1i+0j+3k is the determinant ...

  [tex]\left|\begin{array}{ccc}i&j&k\\4&5&0\\1&0&3\end{array}\right|=15i-12j-5k[/tex]

As a list of coefficients, the cross product is c = {15, -12, -5}.

Vectors are orthogonal if their dot product is 0.

  a·c = {4, 5, 0}·{15, -12, -5} = (4·15) -(5·12) +(0·(-5)) = 60 -60 = 0

  b·c = {1, 0, 3}·{15, -12, -5} = (1·15) +(0·(-12)) +(3·(-5)) = 15 -15 = 0

The dot products are both zero, so the cross product is orthogonal to both of the vectors that created it.

The cross product of vectors a and b is 15i - 12j - 5k. To verify if it is orthogonal to both a and b, we calculate the dot products and find that they both equal zero.

The cross product of vectors a and b is given by:

a ⨯ b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k

Calculating the cross product of a and b using the given values:a ⨯ b = (5)(3) - (0)(0)i + (0)(1) - (4)(3)j + (4)(0) - (5)(1)k = 15i - 12j - 5k

To verify if the cross product is orthogonal to both a and b, we calculate the dot products:

(a ⨯ b) · a = (15)(4) + (-12)(5) + (-5)(0) = 0

(a ⨯ b) · b = (15)(1) + (-12)(0) + (-5)(3) = 0

Since both dot products equal zero, we can conclude that the cross product a ⨯ b is orthogonal to both a and b.

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Answer = (b) 4

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We Know ::

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In a Rectangle,

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So ::

[tex]\begin{gathered} \\ \\ \begin{gathered} \; \; :\longmapsto \; \sf{10 + 2x = 18} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{2x = 18 - 10} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{2x = 8} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{x = \cancel{\dfrac{8}{2}}} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered}\begin{gathered} \; \; :\longmapsto \;\underline{\boxed{\sf{\pmb{x = 4}}}} \; \pmb{\red{\bigstar}} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \qquad{\rule{100pt}{10pt}} \end{gathered}[/tex]

Also ::

[tex]\begin{gathered} \\ \\ \begin{gathered} \; \; :\longmapsto \; \sf{2y + 4 = 10} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{2y = 10 - 4} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{2y = 6} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered} \\ \begin{gathered} \; \; :\longmapsto \; \sf{y = \cancel{\dfrac{6}{2}}} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered}\begin{gathered} \; \; :\longmapsto \;\underline{\boxed{\sf{\pmb{y = 3}}}} \; \pmb{\red{\bigstar}} \\ \\ \end{gathered}\end{gathered}[/tex]

[tex]\begin{gathered}\begin{gathered} \\ {\underline{\rule{170pt}{9pt}}} \end{gathered}\end{gathered}[/tex]

If the new number is four times the reciprocal of the original number, then  the original number is 0.02.

Let us assume the original positive decimal number is = "x".

When we move the decimal point of x four places to the right,

We get,  "10000x", which is four times the reciprocal of x.

So, equation can be written as;

⇒ 10000x = 4(1/x),

On simplifying the equation,

We get,

⇒ 10000x² = 4,

Dividing both sides by 10000,

We get,

⇒ x² = 0.0004,

Simplifying further,

We get,

⇒ x = 0.02

Therefore, the original positive decimal number is 0.02.

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The probability that the next telephone call will arrive within 7 minutes is 0.9999 rounded to four decimal places.

Telephone calls arrive at an information desk at a rate of $\lambda$ per minute. Round to four decimal places as needed.The number of events in a Poisson distribution, $X$ , is dependent on the time interval or the area under consideration. When the mean number of events in the area is $\mu$, the probability of a specific number of events, $x$, is given by the Poisson distribution as$P(X=x)=\frac{\mu^xe^{-\mu}}{x!}$where x is a non-negative integer.Let's consider the given scenario for this particular question.

$\lambda$ is the rate of telephone calls arriving at an information desk. This rate can be utilized to find the mean of a Poisson distribution by multiplying it by the time interval. Let $\mu$ denote the mean number of calls arriving in 7 minutes. Then$\mu = \lambda \times 7$ $\mu$ = $7\lambda$When this mean value is substituted in the Poisson probability formula, the probability that a telephone call arrives within 7 minutes is obtained.$P(X \leq 1)$ =$P(X=0)+P(X=1)$ =$\frac{(7\lambda)^0 e^{-7\lambda}}{0!}+\frac{(7\lambda)^1 e^{-7\lambda}}{1!}$ =$e^{-7\lambda}+\frac{7\lambda}{e^{7\lambda}}$ = $0.9999$ (rounded to four decimal places).

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The two-digit number with a ones digit that is 3 more than its tens digit is 58.

We know that a number is a two-digit number and the sum of its two digits is 13. Based on this, some possible numbers that fit this description are:

Moreover, we know that the ones digit is 3 more than the tens digit. Based on this principle, the number that fits the description is 58.

5 (3 more than 8 ) + 8 = 13

Note: This question is incomplete; here is the complete question:

I am 2 digit number. My ones digit is 3 more than my tens and the sum of my digit is 13. Who am I?

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To find the volume of a sphere with a circumference of 20π, we first need to find its radius.

The formula for the circumference of a sphere is C = 2πr, where C is the circumference and r is the radius.

Therefore, we can rearrange this formula to solve for the radius:

r = C/(2π) = 20π/(2π) = 10

Now that we know the radius is 10, we can find the volume of the sphere using the formula:

V = (4/3)πr^3

Substituting r = 10 into this formula, we get:

V = (4/3)π(10)^3 = (4/3)π(1000) = 4000π/3

So, the volume of the sphere is 4000π/3, which is approximately equal to 4188.79 cubic units.

When the p-value is less than alpha, reject H_0.

When you conduct a two-tailed hypothesis test, you check if the sample falls outside of the two critical regions. You make use of a two-tailed test if you are required to check both sides of the data.

For example, in a two-tailed hypothesis test, suppose the null hypothesis is that a coin is fair. This means that you expect the coin to be heads half of the time and tails half of the time.

The alternative hypothesis is that the coin is biased. This means that it will be heads more than half of the time or tails more than half of the time.If you get a p-value that is less than your alpha, you can reject the null hypothesis.

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Given:

Robert gets a loan from his bank he agrees to borrow £6000 at a fixed annual simple interest rate of 7% he also agrees to pay the money back over a 10 year period.

To Find:

How much money in total will he have paid back at the end of the 10 years?

Solution:

We know that,

[tex]\text{Simple interest}=\dfrac{\text{P}\times\text{R}\times\text{T}}{100}[/tex]

Here, we have

[tex]\text{Principle}=\£6000[/tex]

[tex]\text{Rate}=7\%[/tex]

[tex]\text{Time}=10\ \text{years}[/tex]

Substituting these values, we get

[tex]\text{SI} = \dfrac{6000\times7\times10}{100}[/tex]

  [tex]= \dfrac{420000}{100}[/tex]

  [tex]= 4200[/tex]

[tex]\text{Amount = Principal + Simple Interest}[/tex]

              [tex]= 6000 + 4200[/tex]

              [tex]= \£10200[/tex]

Hence, Robert would have paid back £10200 at the end of the 10 years.

By formula:-

SI= PTR/100

A=I+P

Clara is earning money at a faster rate than Tiffany, with a difference of $4 per hour.

A graph in x-y axis plot is a visual representation of mathematical functions or data points on a Cartesian coordinate system.

To determine who is earning money at a faster rate, we need to calculate the hourly earnings for each person.

For Tiffany, the hourly earnings can be calculated as follows:

Hourly earnings = Earnings / Time

Hourly earnings = 40 / 5 = 8

Hourly earnings = 80 / 10 = 8

Hourly earnings = 120 / 15 = 8

So, Tiffany's hourly earnings are a constant $8 per hour.

For Clara, we can estimate her hourly earnings from the graph. The graph shows that Clara earns $60 for 5 hours of work, $120 for 10 hours of work, and $180 for 15 hours of work. Therefore, her hourly earnings can be calculated as follows:

Hourly earnings = Earnings / Time

Hourly earnings = 60 / 5 = 12

Hourly earnings = 120 / 10 = 12

Hourly earnings = 180 / 15 = 12

So, Clara's hourly earnings are a constant $12 per hour.

Thus, Clara is earning money at a faster rate than Tiffany, with a difference of $4 per hour.

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Complete question-

The answer is: g(x) increases by a factor of 9.

What is change in interval?

In mathematics, the term "interval" refers to a range of values between two endpoints. A change in interval refers to a shift or movement of this range along the number line.

There are two types of changes that can occur in an interval: a shift or a stretch.

A shift occurs when the interval is moved left or right along the number line. For example, if we start with the interval [0, 5] and shift it to the right by 2 units, we get the new interval [2, 7]. Similarly, if we shift it to the left by 3 units, we get the new interval [-3, 2].

A stretch occurs when the interval is expanded or compressed. For example, if we start with the interval [0, 5] and stretch it by a factor of 2, we get the new interval [0, 10]. If we compress it by a factor of 1/3, we get the new interval [0, 5/3].

Changes in interval are important in many areas of mathematics, including calculus, where they are used to describe the domain and range of functions, and in geometry, where they are used to define the length and area of shapes.

The function g(x) = 3ˣ represents exponential growth.

To determine how g(x) changes over the interval from x=8 to x=10, we can evaluate g(10) and g(8) and compare the ratios:

g(10) = 3¹⁰ = 59,049

g(8) = 3⁸ = 6,561

The ratio of g(10) to g(8) is:

g(10)/g(8) = 59,049/6,561 = 9

Therefore, g(x) increases by a factor of 9 over the interval from x=8 to x=10.

The answer is: g(x) increases by a factor of 9.

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The possible function f(x) is C. [tex]f(x) = (x + 6)/(x - 6)[/tex] because C   satisfy the limit conditions as well.

To determine which function could be function f, we need to check if it satisfies the given domain and limit conditions.

First, let's check the domain.

Option A: The function is undefined at x = 6, but is defined for all other values of x. Therefore, the domain of f is (-∞, 6) U (6, ∞), as required.

Option B: The function is undefined at x = ±6, but is defined for all other values of x. Therefore, the domain of f is (-∞, -6) U (-6, 6) U (6, ∞), which is not the required domain.

Option C: The function is undefined at x = 6, but is defined for all other values of x. Therefore, the domain of f is (-∞, 6) U (6, ∞), as required.

Option D: The function is undefined at x = -6, but is defined for all other values of x. Therefore, the domain of f is (-∞, -6) U (-6, 6) U (6, ∞), which is not the required domain.

Therefore, options A and C are the only ones that satisfy the domain condition. Now let's check the limit conditions.

Option A:

[tex]\lim_{x \to -\infty} [(x^2 -36)/(x - 6)] = \lim_{x \to -\infty} [(x+6)(x-6)/(x-6)] \\= \lim_{x \to -\infty} [x+6] = -\infty[/tex]

[tex]\lim_{x \to \infty} [(x^2 -36)/(x - 6)] = \lim_{x \to \infty} [(x+6)(x-6)/(x-6)] \\= \lim_{x \to \infty} [x+6] = \infty[/tex]

Option C:

[tex]\lim_{x \to -\infty} [(x+6)/(x - 6)] = \lim_{x \to -\infty} [x/x] = -\infty[/tex]

[tex]\lim_{x \to \infty} [(x+6)/(x - 6)] = \lim_{x \to \infty} [x/x] = \infty[/tex]

Therefore, both options A and C satisfy the limit conditions as well.

Therefore, the possible function f is either A or C.

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The complete question is :

The domain of function f is (-∞, 6) U (6, ∞). The value of the function -∞ approaches as x approaches -∞ , and the value of the function approaches ∞ as x approaches ∞ . Which function could be function f?

A. [tex]f(x) = (x^2 -36)/(x - 6)[/tex]

B.[tex]f(x) = (x -6)/(x^2 - 36)[/tex]

C. [tex]f(x) = (x + 6)/(x - 6)[/tex]

D. [tex]f(x) = (x - 6)/(x + 6)[/tex]

Daniel

Step-by-step explanation:

student of account

Answer:  it's T

Step-by-step explanation:

There are 90 possible combinations of marble color and die number that are possible from a bag of 15 different colored marbles and rolling one six-sided die.

This can be determined using the multiplication principle of counting.

The multiplication principle of counting can be used to find the number of ways two or more tasks can be performed together. The principle states that if there are m ways to perform task A and n ways to perform task B, then there are m × n ways to perform both tasks A and B.

To apply the multiplication principle of counting to this problem, we need to determine the number of ways the die can land and the number of ways a marble can be chosen. There are six possible outcomes of rolling a six-sided die, and there are 15 different colored marbles to choose from. Thus, the number of possible combinations of marble color and die number is: 15 different colored marbles × 6 possible die numbers= 90 possible combinations.

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87.6  is the difference between the mean of the warmest and coldest days recorded.

What is the mean called in statistics?

By summing all the numbers in a data collection and dividing by the total number of values in the set, one can determine the mean (average) of the data set. When a data collection is ranked from least to greatest, the median is the midpoint. 

WARMEST: 98, 97, 102, 98, 97

                = 97, 97 , 98 , 98 , 102

               = 97 + 97 + 98 + 98 + 102/5

                 = 492/5 = 98.4

COLDEST: 12, 14, 9, 7, 12

               = 7 , 9 , 12 ,12,  14

              =  54/5

               = 10.8

the difference between the mean of the warmest and coldest days recorded    =   98.4 - 10.8

                  =  87.6

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The length of the ladder can be calculated using trigonometry as 31/sin 78.

The length of the ladder can be calculated using trigonometry. First, use the given information to draw a diagram. The angle of 78 degrees forms the angle of the ladder relative to the ground, and the side opposite of this angle is the length of the ladder. The side adjacent to the angle is the height of the electric box, which is given as 31 feet.

Now, use the sine function to calculate the length of the ladder. The sine function relates the ratio of the opposite side to the hypotenuse, so the equation is 31/sin 78.

Plugging this equation into a calculator will give the length of the ladder as approximately 51.7 feet.

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The value of the system determinant is -2.

A system of equations is the given term of math for two or more equations with the same variables. The solution of these equations represents the point of the intersection.

You can solve a system of equations by substitution or adding methods. Or, applying a tool called determinant. Here, for solving this exercise you should apply the determinant since the question asks about its value. The example below shows the steps that you should do.

Example:

2x+y=10

-x-y=25

Firstly, you should identify the coefficients of the system.  The coefficients are the previous numbers before the each variable. Therefore, for the example, the coefficients are: 2, 1 ,-1 and -1.

After that, you should write these coefficients to form a matrix. See: [tex]\left[\begin{array}{ccc}2&1\\-1&-1\\\end{array}\right][/tex].

Finally, you have a matrix and you should calculate the determinant. The determinant is calculated as shown below.

[tex]|D|=\left[\begin{array}{ccc}a&b\\c&e\\\end{array}\right]\\ \\ |D|=a*e-b*c[/tex]

Now, you can get to solve the given exercise.

         1,1,1,-1

         [tex]\left[\begin{array}{ccc}1&1\\1&-1\\\end{array}\right][/tex]

       D= 1*(-1)-(1)*(1)

       D= -1-(1)

       D= -1-1=-2

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Answer:

Step-by-step explanation:

$250,000 per depositor per insured bank *fire emoji*

Maxima set is {(6, 9)}.

To find the maxima set from the given set of points, we need to identify the points with the highest y-values.

Step 1: Examine the y-values of the given points.

(7, 2), (3, 1), (9, 3), (4, 5), (1, 4), (6, 9), (2, 6), (5, 7), (8, 6)

Step 2: Identify the highest y-value among the points.

In this case, the highest y-value is 9.

Step 3: Find the points with the highest y-value.

There is one point with a y-value of 9: (6, 9)

Step 4: Create the maxima set.

The maxima set consists of the point(s) with the highest y-value: {(6, 9)}

So the maxima set is {(6, 9)}.

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By finding a common denominator, we can rewrite the sum as:

3/8 + 7/12 = 9/24 + 14/24 = 23/24

Here we want to find the sum between two fractions that have different denominators.

To sum them, we need to find a common denominator, so we need to find a common multiple between 8 and 12.

We know that:

8*3 = 24

12*2 = 24

Then we can multiply and divide the first fraction by 3:

3/8*(3/3) = 9/24

7/12*(2/2) = 14/24

Then the sum will give:

9/24 + 14/24 = 23/24

That is the outcome.

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You will need 0.75 cups of milk when quadrupling the recipe.

When quadrupling the recipe, you would need 12 tablespoons of milk. To make the calculations easier, it is more efficient to convert the tablespoons to cups. So, the question is how many cups of milk will you need? We know that there are 16 tablespoons in a cup. Therefore, we can find how many cups of milk you will need as follows:

12 tablespoons = (12/16) cups

12 tablespoons  =  0.75 cups

So, you would need 0.75 cups of milk when quadrupling the recipe.

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The pricing for purchasing apples from the orchard, stating that for the first 10 pounds, the price is $2 per pound, and for each additional pound, the price is $1 per pound.

What is the unit price?

In order to find the unit price of a certain item, we just need to divide the total price paid (or total cost) by the amount of items bought.

If Anne buys less than or equal to 10 pounds of apples from David's Apple Orchard, then she will pay $2 per pound.

If she buys more than 10 pounds, then she will pay $1 per pound for each additional pound after the first 10 pounds.

To buy apples from David's Apple Orchard, Anne has to follow these rules

For the first 10 pounds of apples, she will pay $2 per pound.

For each additional pound of apples after the first 10 pounds, she will pay $1 per pound.

Hence, the pricing for purchasing apples from the orchard, stating that for the first 10 pounds, the price is $2 per pound, and for each additional pound, the price is $1 per pound.

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Answer:

Step-by-step explanation: 160$

The inequality x/6.2 ≥ -2 is solved by multiplying both sides by 6.2 to get x ≥ -12.4, which means x is greater than or equal to 12.4.

To solve the inequality x/6.2 ≥ -2, we want to find all possible values of x that make the inequality true.

First, we can isolate x by multiplying both sides of the inequality by 6.2 (which is positive, so we do not need to flip the inequality sign):

x/6.2 * 6.2 ≥ -2 * 6.2

Simplifying, we get:

x ≥ -12.4

This means that any value of x that is greater than or equal to -12.4 will make the inequality true.

To see why this is the case, we can substitute a few values for x and see if they satisfy the inequality. For example, if we let x = 12.4, then:

x/6.2 = 12.4/6.2 = 2

And we can check that 2 is indeed greater than or equal to -2. Therefore, x = 12.4 is a valid solution.

Similarly, if we let x = 20, then:

x/6.2 = 20/6.2 ≈ 3.23

And we can check that 3.23 is also greater than or equal to -2. Therefore, x = 20 is also a valid solution.

However, if we let x = -15, then:

x/6.2 = -15/6.2 ≈ -2.42

And we can see that -2.42 is not greater than or equal to -2. Therefore, x = -15 is not a valid solution.

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The function F(x)'s valid domain is therefore (-, -3), (-3, -2), and (-2, ).

We enter each value of x into the function and simplify as follows in order to evaluate the function F(x) at the values of the supplied domain:

[tex]F(-3) = (2(-3) + 6)/((-3)^2 + 5(-3) + 6) = 0/0[/tex] (undefined) (undefined)

[tex]F(-2) = (2(-2) + 6)/((-2)^2 + 5(-2) + 6) = 0/0[/tex](undefined) (undefined)

[tex]F(0) = (2(0) + 6)/(0^2 + 5(0) + 6) = 1[/tex]

[tex]F(2) = (2(2) + 6)/(2^2 + 5(2) + 6) = 2/3[/tex]

[tex]F(3) = (2(3) + 6)/(3^2 + 5(3) + 6) = 2/4 = 1/2[/tex]

The function is undefined at x=-3 and x=-2 because the fraction's

denominator is zero at these points, and division by zero is undefined, as can be seen from the aforementioned evaluations. All real numbers, with the exception of -3 and -2, fall inside the given function's valid domain.

The function F(x)'s valid domain is therefore (-, -3), (-3, -2), and (-2, ).

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