Find f(2) for f(x)=2(3)^x-1

a) A linear equation that describes the relationship between the number of buses and the number of vans needed for 120 students is 24x + 8y = 120.

b) If the teachers rent 4 buses, the number of vans they will need to transport all the students is 3.

c) If the teachers rent 6 vans, the number of buses they will need to transport all the students is 3.

d) If the teachers rent only vans, the number of vans needed will be 15.

e) Suppose the teachers rented only buses, the number of buses needed will be 5.

A linear equation is an algebraic equation of the form y = mx+b.

A linear equation involves a constant and a first-order (linear) term, with m as the slope and b as the y-intercept.

The total number of students on the field trip = 120

The number of students a bus can seat = 24

The number of students a van can seat = 8

Let the number of buses = x and the number of vans = y

24x + 8y = 120

b) Renting 4 buses, the number of vans needed:

24(4) + 8y = 120

96 + 8y = 120

8y = 24

y = 3

c) Renting 6 vans, the number of buses will be:

24x + 8(6) = 120

24x = 48 = 120

24x = 72

x = 3

d) Renting only vans, the number needed will be:

24(0) + 8y = 120

8y = 120

y = 15

e) Renting only buses, the number needed will be:

24x + 8(0) = 120

24x = 120

x = 5

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Answer:

You will pay $19.6 for a item with original price of $24.50 when discounted 20%. In this example, if you buy an item at $24.50 with 20% discount, you will pay 24.50 - 4.9 = 19.6 dollars.

Step-by-step explanation:

The probability that an arriving student will find at least one other student waiting in line is approximately 0.167.

To solve this problem, we'll use the M/M/1 queueing model with Poisson arrivals and exponential service times. Let's calculate the required values: (a) Percentage of time Judy is idle: The utilization of the system (ρ) is the ratio of the average service time to the average interarrival time. In this case, the average service time is 10 minutes, and the average interarrival time is 12 minutes. Utilization (ρ) = Average service time / Average interarrival time = 10 / 12 = 5/6 ≈ 0.8333

The percentage of time Judy is idle is given by (1 - ρ) multiplied by 100: Idle percentage = (1 - 0.8333) * 100 ≈ 16.67%. Therefore, Judy is idle approximately 16.67% of the time. (b) Average waiting time for a student:

The average waiting time in a queue (Wq) can be calculated using Little's Law: Wq = Lq / λ, where Lq is the average number of customers in the queue and λ is the arrival rate. In this case, λ (arrival rate) = 1 customer per 12 minutes, and Lq can be calculated using the queuing formula: Lq = ρ^2 / (1 - ρ). Plugging in the values: Lq = (5/6)^2 / (1 - 5/6) = 25/6 ≈ 4.17 customers Wq = Lq / λ = 4.17 / (1/12) = 50 minutes. Therefore, on average, a student spends approximately 50 minutes waiting in line.

(c) Average length of the line: The average number of customers in the system (L) can be calculated using Little's Law: L = λ * W, where W is the average time a customer spends in the system. In this case, λ (arrival rate) = 1 customer per 12 minutes, and W can be calculated as W = Wq + 1/μ, where μ is the service rate (1/10 customers per minute). Plugging in the values: W = 50 + 1/ (1/10) = 50 + 10 = 60 minutes. L = λ * W = (1/12) * 60 = 5 customers. Therefore, on average, the line consists of approximately 5 customers.

(d) Probability of finding at least one student waiting in line: The probability that an arriving student finds at least one other student waiting in line is equal to the probability that the system is not empty. The probability that the system is not empty (P0) can be calculated using the formula: P0 = 1 - ρ, where ρ is the utilization. Plugging in the values:

P0 = 1 - 0.8333 ≈ 0.1667. Therefore, the probability that an arriving student will find at least one other student waiting in line is approximately 0.167.

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The points that he average on the last two tests will be 93 points.

A mean is the average of the set of numbers that are given.

On the first five tests that he took, he averaged 86 points. The total points will be:

= 86 × 5

= 430 points

He averaged 88 points on all seven tests. The total will be:

= (88 × 7)

= 616 points

The point average on the last two tests will be:

= (616 - 430)/2

= 186/2

= 93 points

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Step-by-step explanation:

The function is increasing and decreasing over the same intervals as the function in the graph is y = | x+ 2|. The correct option is B.

The absolute value or modulus of a real number x, denoted |x| in mathematics, is the non-negative value of x regardless of its sign. Specifically,|x|=x.

In the given image the function is represented by the equation y = | x - 2 |. Now the similar function which is increasing and decreasing over the same intervals as the function in the graph is y = | x+ 2|.

Therefore, the correct option for the function is B.

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Answer:

129

Step-by-step explanation:

278 - 149 = 129

Answer:

the answer is 129

Step-by-step explanation:

278-149=129

Answer:

8.4

Step-by-step explanation:

How many times 20 goes into 168 = 168/20

                                                         = 8.4

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The slope of the line forming the smallest right triangle, when a line is drawn through the point (1, 2), is 2.

The slope of the line forming the smallest right triangle with the positive x and y axes, when a line is drawn through the point (1, 2), can be determined as follows.

First, let's consider the two axes as the legs of the right triangle, and the line drawn through (1, 2) as the hypotenuse. The slope of the hypotenuse can be calculated by finding the difference in y-coordinates divided by the difference in x-coordinates between the two endpoints.

Since the x-coordinate of the point where the line intersects the x-axis is 0 (positive x-axis), and the y-coordinate of the point where the line intersects the y-axis is 0 (positive y-axis), the difference in y-coordinates is 0 - 2 = -2, and the difference in x-coordinates is 0 - 1 = -1.

Therefore, the slope of the line forming the smallest right triangle is -2/-1 = 2.

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Three cards are drawn from a standard deck. Thus, the probability to get an ace, followed by a king, and then a queen, is  16/34,425

The formula for conditional probability is:

P(A∩B) = P(B|A) . P(A)

In the given problem:

P(ace) = 4 / 54

After an ace was drawn, the number of cards now is 51.

Hence,

P(king | ace) = 4/51

After the second drawn, the remaining cards is 50. Hence,

P(queen | (king | ace)) = 4/50

Therefore,

P(ace,king,queen) = 4/54 x 4/51 x 4/50 = 64 / 137,700 = 16/34,425

Thus, the probability to get an ace, followed by a king, and then a queen, is  16/34,425

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Answer:

27

Step-by-step explanation:

If  \(f(x) = \frac{1}{2x^2} - \frac{1}{4x +3}\),  then the value of  \(f(8)\) is  \(f(8) = \frac{-93}{4480}\).

A quadratic equation is an algebraic equation in the variable of the second degree. The quadratic equation is in the form of \(ax^{2} + bx + c = 0\), where \(a,b,c\)  are the real numbers.

We have

\(f(x) = \frac{1}{2x^2} - \frac{1}{4x +3}\)

According to Quadratic Equation,

\(f(x)=ax^{2} +bx+c\)

And we have,

\(f(x) = \frac{1}{2x^2} - \frac{1}{4x +3}\)

So,

\(f(8) = \frac{1}{2x^2} - \frac{1}{4x +3}\)

i.e. putting \(x=8\),

\(f(8) = \frac{1}{2(8)^2} - \frac{1}{4*8 +3}\)

\(f(8) = \frac{1}{128} - \frac{1}{35}\)

Solving the above part,

\(f(8) = \frac{ 35-128 }{ 128*35 }\)

\(f(8) = \frac{-93}{4480}\)

Hence, we can say that value of  \(f(8) = \frac{-93}{4480}\) .

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There are four possible three-letter arrangements that Amy can form using the letters from the word "dice" without repeating any letters. To form a three-letter arrangement using only letters from the word "dice" without repeating any letters, we can use the formula for combinations.

The formula is given by nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be chosen.
In this case, n = 4 (the total number of available letters) and r = 3 (the number of letters to be chosen for the arrangement).
5. Substituting these values into the formula, we get \(4C_{3}\) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 * 3 * 2) / (3 * 2 * 1) = 4.

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Answer:

-9/8

Step-by-step explanation:

We find the slope using the slope formula and get 8/9.

The perpendicular slope is the negative flipped number so we get -9/8.

Keisha sold 48 gallons of basic paint and 92 gallons of premium paint last weekend.

Let's denote the number of premium paint gallons sold by "x" and the number of standard paint gallons sold by "y".

According to the problem statement, we have:

\(x = 2y - 4\) (the number of standard paint gallons sold was four less than twice as many premium gallons sold)

The total sales revenue from paint sales can be expressed as:

\(40.5x + 24.5y = 4914\)

We can substitute x in the second equation with 2y - 4:

40.5(2y - 4) + 24.5y = 4914

Simplifying and solving for y, we get:

\(81y - 162 + 24.5y = 4914\\105.5y = 5076\\y = 48\)

Substituting y in the equation \(x = 2y - 4\), we get:

\(x = 2(48) - 4\\x = 92\)

Therefore, last weekend, Keisha sold 48 gallons of normal paint and 92 gallons of premium paint.

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There are three left cosets of H in Z, given by:

H = {3k | k ∈ Z}

[1] + H = {3k + 1 | k ∈ Z}

[2] + H = {3k + 2 | k ∈ Z}

To find all the left cosets of H in Z, we need to find the equivalence classes of the relation x ~ y if and only if x - y ∈ H. That is, two integers are equivalent if their difference is a multiple of 3.

To find the left cosets, we choose a representative for each equivalence class. We can choose any integer in the class as the representative. Then, we add H to the representative to obtain the left coset.

For example, the equivalence class [0] consists of all integers that are multiples of 3. We can choose 0 as the representative. Then, the left coset is

[0] + H = {0 + 3k | k ∈ Z} = {3k | k ∈ Z}

Similarly, we can choose 1, 2 as the representatives for [1], [2], respectively, and obtain the left cosets:

[1] + H = {1 + 3k | k ∈ Z} = {3k + 1 | k ∈ Z}

[2] + H = {2 + 3k | k ∈ Z} = {3k + 2 | k ∈ Z}

We can continue this process to find all the left cosets. In general, there are three left cosets of H in Z, given by:

H = {3k | k ∈ Z}

[1] + H = {3k + 1 | k ∈ Z}

[2] + H = {3k + 2 | k ∈ Z}

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Answer:

1018 cm

Step-by-step explanation:

A=piR^2

To find the radius from circumference, we divide by pi and then divide by two.  

C=36pi

36pi/pi=36

36/2=18

R=18

if we plug 18 into

A=piR^2,

we get

18pi^2=A

18pi^2=1017.88

round this to get 1018

A=1018

Answer:

\( \frac{5}{6} \: - \frac{5}{6} \)

Step-by-step explanation:

\( \sqrt{ \frac{25}{36} } = \frac{ \sqrt{25} }{ \sqrt{36} } = \frac{5}{6} \)

(a) dxdy = (cos(x) - 2sin(x))tan(x) + (sin(x) + 2cos(x))(sec^2(x)).

(b) dxdy = -sec^2(x)(2 - x) + (2 - x)(2sec^2(x)tan(x)).

(c) dxdy = (3e^(3x) + 2xtan(x+1)^2)dx + (2x + 1)(e^(3x) + x^2tan(x+1)^2)sec^2(x+1)dx.

(d) dxdy = 6(8x + 5 - x^2)^5(8 - 1/x^2)dx.

To find dxdy for each given function, we differentiate the function with respect to x using the product rule or chain rule where necessary, and keep dy/dx as a separate term. We don't simplify the answer to maintain the form in terms of the given variables.

(a) For y = (sin(x) + 2cos(x))tan(x), using the product rule, we differentiate each term with respect to x. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Applying the chain rule, the derivative of tan(x) is sec^2(x). Therefore, dxdy = (cos(x) - 2sin(x))tan(x) + (sin(x) + 2cos(x))(sec^2(x)).

(b) For y = (2 - x)sec^2(x), we differentiate each term with respect to x. The derivative of (2 - x) is -1, and the derivative of sec^2(x) is 2sec^2(x)tan(x). Therefore, dxdy = -sec^2(x)(2 - x) + (2 - x)(2sec^2(x)tan(x)).

(c) For y = e^(3x) + x^2tan(x+1)^3, we differentiate each term with respect to x. The derivative of e^(3x) is 3e^(3x), and the derivative of x^2tan(x+1)^3 involves both the chain rule and product rule. The derivative of x^2 is 2x, and the derivative of tan(x+1)^3 is 3tan(x+1)^2sec^2(x+1). Therefore, dxdy = (3e^(3x) + 2xtan(x+1)^2)dx + (2x + 1)(e^(3x) + x^2tan(x+1)^2)sec^2(x+1)dx.

(d) For y = (4x^2 + 5x - 1/x)^6, we differentiate the function using the chain rule and power rule. The derivative of (4x^2 + 5x - 1/x) is (8x + 5 - x^2)/x^2. Applying the power rule, we multiply the derivative by 6(4x^2 + 5x - 1/x)^5. Therefore, dxdy = 6(8x + 5 - x^2)^5(8 - 1/x^2)dx.

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In each scenario, the choice between mean and median as a representative measure of central tendency depends on the nature of the data and the specific context..

1. Scenario: Income distribution of a population

- If the income distribution is skewed or contains extreme values (outliers), the median would be a better representation of the central tendency. This is because the median is not influenced by outliers and provides a more robust estimate of the "typical" income level. However, if the income distribution is approximately symmetric without outliers, the mean can also be an appropriate measure.

2. Scenario: Exam scores in a class

- If the exam scores are normally distributed without significant outliers, the mean would be a suitable measure as it takes into account the value of each score. However, if there are extreme scores that deviate from the majority of the data, the median may be a better representation. This is especially true if the outliers are indicative of errors or exceptional circumstances.

3. Scenario: Housing prices in a city

- In this case, the median would be a more appropriate measure to represent the central tendency of housing prices. This is because the housing market often exhibits a skewed distribution with a few high-priced properties (outliers). The median, being the middle value when the data is sorted, is not influenced by these extreme values and provides a better understanding of the typical housing price in the city.

Ultimately, the choice between mean and median depends on the specific characteristics of the data and the objective of the analysis. It is important to consider the distribution, presence of outliers, and the context in which the data is being interpreted.

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Answer:

x=3

y=4

z=1

Step-by-step explanation:

4x-2y+6z=10

-(4x+12y+4z=64)

= -14y+2z=-54

= -7y+z=-27

-7y+z=-27

+(5y-z=19​)

= -2y=-8

y=4

5(4)-z=19

20-z=19

z=1

x+12+1=16

x=3

When something decreases by 15.7%, the growth factor is about 1.182 and when something decreases by 0.12%, the growth factor is about 1.00012.

In math, what is the definition of multiplying?

Multiplication is a mathematical process that shows the amount of times a number has been added to itself. It is represented by the multiplication symbols (x) or (*). Division is a mathematical process that shows how many equal amounts add up to a given quantity.

If something decreases by 15.7%, the increase in the factor is 100 / (100 - 15.7) Equals 1.182.

As a result, when something decreases by 15.7%, the growth factor is about 1.182.

When something falls by 0.12%, the expansion factor is 100 / (100 - 0.12) = 1.00012.

As a result, when something decreases by 0.12%, the growth factor is about 1.00012.

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Calculate what is 15% of $66 equal to. Then. substract that quantity from $66 to find the sale price:

Therefore, the sale price of the pair of shoes is $56.1

Answer:

3x=96

Step-by-step explanation:

From the information given, you can write the following:

a+x=96 (1)

a=2x (2)

where a is the number of CDs that Sue has and x is the number of CDs that Gwen has.

You can replace 2 in 1 and you would have the following:

2x+x=96

3x=96

From this, you can isolate x to find its value:

x=96/3

x=32

According to this, the equation that could be used to find the number of CDs, x, that Gwen has is: 3x=96.

Isoquants for f(K, L) = 1/2 min{K, L} that produce exactly 10 goods and 25 goods can be graphed, with labeled points.

For function f(K, L) = 1/2 min{K, L}, we can graph the isoquants that represent the combinations of capital (K) and labor (L) that produce exactly 10 goods and 25 goods.

An isoquant represents all the combinations of inputs (K and L) that yield the same level of output. In this case, the isoquants will be curves that connect different combinations of K and L, where the output is constant at either 10 goods or 25 goods.

To graph the isoquant for 10 goods, we plot points where f(K, L) = 10. For example, if we let K = 4 and L = 4, the minimum of K and L is 4, so f(K, L) = 1/2 * 4 = 2 goods.

Since this is less than 10, we need to increase either K or L. We can try K = 8 and L = 4, where the minimum of K and L is still 4, and f(K, L) = 1/2 * 4 = 2 goods. Again, this is less than 10, so we increase K or L further.

Continuing this process, we can plot a few more points and connect them to obtain the isoquant for 10 goods.

Similarly, we can graph the isoquant for 25 goods by plotting points where f(K, L) = 25. For each point, we determine the minimum of K and L, multiply it by 1/2, and check if the result is equal to 25. By plotting and connecting several such points, we obtain the isoquant for 25 goods.

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volume of the square pyramid = 245.33

given that

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C4v symmetry. If all edge lengths are equal, it is an equilateral square pyramid

A square pyramid characterized by a square base is a three-dimensional shape having five faces, thus called a pentahedron. The most famous example of such a square pyramid is the Great Pyramid of Giza. A pyramid is a polyhedron that has a base and 3 or greater triangular faces that meet at a point above the base.

a square pyramid has a base = 8 inches on each side

the height of the pyramid = 11.5 inches

in square pyramids all sides are equal so, length and width are equal

length(l) = 8 inches

width (w)= 8 inches

height(h) = 11.5 inches

volume = lwh/3

volume = (8)(8)(11.5)/3

volume = 245.33

volume of the square pyramid = 245.33

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Answer: its 1.6667.

Answer:

Step-by-step explanation

3*X=5

X=5/3

X=1.67

Answer:

From greatest to least: AB, AC, BC