Wanahton purified a portion of water with
900
900900 grams of contaminants. Each hour, a third of the contaminants was filtered out.
Let
�
(
�
)
g(n)g, left parenthesis, n, right parenthesis be the amount of contaminants (in grams) that remained by the beginning of the
�
th
n
th
n, start superscript, start text, t, h, end text, end superscript hour.
�
gg is a sequence. What kind of sequence is it?

The equation of the tangent line at (3, 324) is y = 288x - 540.

To find y' (the derivative of y), we can differentiate the given function y = [tex](x^2 + 2x + 3)^2[/tex] using the chain rule.

y' = [tex]2(x^2 + 2x + 3) \times (2x + 2) = 4(x^2 + 2x + 3)(x + 1)[/tex]

Now, to find the slope of the tangent line at (3, 324), we substitute x = 3 into y' and evaluate it:

y' = [tex]4(3^2 + 2(3) + 3)(3 + 1) = 4(9 + 6 + 3)(4) = 4(18)(4) = 288[/tex]

So, the slope of the tangent line at (3, 324) is 288.

To find the y-intercept of the tangent line, we substitute the coordinates (3, 324) and the slope (288) into the point-slope form equation: y - y1 = m(x - x1).

Using (x1, y1) = (3, 324) and m = 288, we have:

y - 324 = 288(x - 3)

y - 324 = 288x - 864

y = 288x - 540

Therefore, the equation of the tangent line at (3, 324) is y = 288x - 540.

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The probability that the 5 starting members of the basketball team are lined up from shortest to tallest is 1/120. Option A.

To calculate the probability that the 5 starting members of the basketball team are lined up from shortest to tallest, we need to determine the number of favorable outcomes (arrangements where the players are lined up from shortest to tallest) and divide it by the total number of possible outcomes.

Since the order matters and we want the players to be arranged from shortest to tallest, there is only one favorable outcome. The shortest player must be in the first position, followed by the second shortest, and so on, until the tallest player is in the last position.

The total number of possible outcomes is given by the number of permutations of 5 players, which is denoted as 5!.

Therefore, the probability is:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Probability = 1 / 5!

Probability = 1 / (5 * 4 * 3 * 2 * 1)

Probability = 1 / 120 Option A is correct.

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We need to apply a translation of 3 units up.-

We can see that both of them have almost the same slope, but the line d starts at y = 0, while the data starts at y = 3.

Then we could apply a translation up of 3 units, so if the line is:

d = f(x)

Then the translated line is:

d = f(x) + 3

And that translates the line 3 units upwards.

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The trigonometric ratios for this problem are given as follows:

The three trigonometric ratios are the sine, the cosine and the tangent of an angle, and they are obtained according to the formulas presented as follows:

The hypotenuse for this problem is of 34, and the sides are given as follows:

Hence the trigonometric ratios are given as follows:

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The graph with the upward parabola plotted in quadrant 1 of a coordinate plane, with the vertex at (3, -1), and left and right slopes at (2, 1) and (4, 1), matches the quadratic function f(x) = (x - 3)^2 - 1.T

To match the graph with the quadratic function it represents, we need to analyze the given information about the graph.
The graph shows an upward parabola plotted in quadrant 1 of a coordinate plane. This means that the parabola opens upwards.
The vertex of the parabola is given as (3, -1), which means that the axis of symmetry is a vertical line passing through x = 3, and the vertex is the lowest point on the parabola.
The left slope of the parabola is given as (2, 1), which means that the slope on the left side of the vertex is positive and equal to 1.
The right slope of the parabola is given as (4, 1), which means that the slope on the right side of the vertex is also positive and equal to 1.
Based on this information, we can determine the quadratic function that matches the graph.
The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
In this case, the vertex is (3, -1), so the equation can be written as f(x) = a(x - 3)^2 - 1.
Since the parabola opens upwards, the coefficient of a must be positive.
The left and right slopes of the parabola are both equal to 1, which means that the coefficient a must be equal to 1.
Therefore, the quadratic function that matches the graph is f(x) = (x - 3)^2 - 1.

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Answer:

x = 42

y = 21

Step-by-step explanation:

Since its a right triangle, we have:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

[tex]sin (60) = \frac{21\sqrt{3} }{x} \\\\x= \frac{21\sqrt{3} }{sin (60)} \\\\x= \frac{21\sqrt{3} }{\frac{\sqrt{3} }{2} } \\\\x = \frac{21\sqrt{3} *2 }{\sqrt{3} } \\\\x = 21*2\\\\x = 42[/tex]

[tex]cos (60) = \frac{y}{x}\\\\cos (60) = \frac{y}{42}\\\\y = cos (60)*42 \\\\y = \frac{1}{2} * 42\\\\y = 21[/tex]

Answer:

1,4,9,16,25,36,49,64,81,100

Answer:

Step-by-step explanation:

Interval        Tally            Frequency

0-1                1111 111          8

2-3               1111 1              6

4-5               11                  2

6-7               11                  2

8-9              1111               5

where the bold and underlined ones showed the tally of the 5th number whose line will cut the first four lines

The factored form of 3x^2 - 5x - 18x + 30 is (3x - 5)(x - 6).

To factor the expression3x^2 - 5x - 18x + 30, we can group the terms and then factor out common factors.

Step 1: Group the terms.

3x^2 - 5x - 18x + 30,

Step 2: Factor out the common factors from each group.

x(3x - 5) - 6(3x - 5)

Step 3: Notice that we now have a common binomial factor of (3x - 5).

(3x - 5)(x - 6)

Therefore, the factored form of the expression 3x^2 - 5x - 18x + 30 is (3x - 5)(x - 6).

In this factored form, (3x - 5) and (x - 6) are the factors of the expression. To check if the factoring is correct, you can multiply the factors back together to see if you get the original expression:

(3x - 5)(x - 6) = 3x(x) - 3x(6) - 5(x) + 5(6)

=3x^2 - 18x - 5x + 30

= 3x^2 - 23x + 30  

The result matches the original expression, so the factoring is correct.

Therefore, the factored form of 3x^2 - 5x - 18x + 30 is (3x - 5)(x - 6).

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Answer:

An arithmetic sequence with a common difference of −2 is 20,18,16,14,12..

Step-by-step explanation:

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is same. Here, the common difference is -2, which means that each term in the sequence is obtained by subtracting 2 from the previous term.

To find the arithmetic sequence with a common difference of -2, you can start with an first term and then subtract 2 successively to find the subsequent terms.

Let the initial term is 20. Subtracting 2 from 20, we get 18. Subtracting 2 from 18, we get 16. Continuing this pattern, we subtract 2 from each subsequent term to generate the sequence. The arithmetic sequence with a common difference of -2 starting from 20 is

20,18,16,14,12

In this sequence, each term is obtained by subtracting 2 from the previous term, resulting in a common difference of -2.

The sequence of transformations that maps ABC to A’B’C is a reflection across the line y = x followed by a translation of 10 units right and 4 units up.

The equivalent vertices C and C' are given as follows:

Considering the change in the orientation of the figure, it was reflected over the line y = x, for which the rule is given as follows:

(x,y) -> (y,x).

Hence:

(-4,2) -> (2,-4)

To go from (2,-4) to (12,0), the translation rule is given as follows:

(x,y) -> (x + 10, y + 4)

Hence the translation is defined as follows:

10 units right and 4 units up.

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Answer:

An ROE, or Record of Employment, is a form that employers fill out for their employees who receive insurable earnings. This form provides proof of work and allows employees to collect unemployment insurance if they experience an interruption of earnings. The purpose of an ROE is to inform Service Canada whether the employee is eligible to receive Employment Insurance (EI).

Step-by-step explanation:

The ROE (Record of Employment) serves as an important document that provides information about an individual's employment history and earnings when their employment is terminated. The ROE contains details such as the reason for termination, the dates of employment, and the insurable earnings during the period of employment. This information is used by the government to determine eligibility and calculate the amount of EI benefits an individual may receive. The ROE acts as proof of employment and earnings for individuals who may need to demonstrate their work history and income when applying for loans, mortgages, or other financial services. It is often required by banks, financial institutions, or other organizations to verify employment and income information. The information on the ROE is also used for tax purposes. It helps individuals accurately report their employment income and related deductions when filing their annual tax returns. Employers submit copies of the ROE to the tax authorities to ensure accurate tax reporting. The ROE serves as an official record of an individual's employment history. It documents the start and end dates of employment, the reason for termination, and other important details that may be useful for future reference or potential future employers.

The expression (3 - 11/5) ÷ (3 / -9/5) simplifies to -12/25.

To simplify the expression (3 - 11/5) ÷ (3 / -9/5), we can follow the order of operations, which is commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division - from left to right, Addition and Subtraction - from left to right).

First, let's simplify the numerator:

3 - 11/5 = 15/5 - 11/5 = 4/5

Next, let's simplify the denominator:

3 / -9/5 = 3 * -5/9 = -15/9 = -5/3

Now we have:

(4/5) ÷ (-5/3)

To divide fractions, we can multiply by the reciprocal of the divisor. So, we have:

(4/5) * (-3/5)

Multiplying across the numerators and denominators, we get:

(4 * -3) / (5 * 5) = -12/25

The simplified expression is -12/25.

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The measure of the length VT from the diagram is 5.4

From the given diagram, we can see that the triangle TSR and TUV are similar, hence the correct expressed needed to determine the value of x is given as:

6/x+2 = 14/4x-1

Cross multiply to have
6(4x - 1) = 14(x + 2)

24x - 6 = 14x + 28

24x - 14x = 28 + 6

Simplify to have:

10x = 34

x = 3.4

Determine the measure of VT

VT = x + 2

VT = 3.4 + 2

VT = 5.4

Hence the measure of VT from the diagram is 5.4

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The number of cars in the parking lot at 5 pm is 106.1.

The formula for the number of cars in the parking lot n hours after 3 pm is:

[tex]A_{n} = 92(1.09)^n[/tex]

where:

[tex]A_{n}[/tex] is the number of cars in the parking lot n hours after 3 pm

92 is the initial number of cars in the parking lot

1.09 is the growth rate of the number of cars in the parking lot

n is the number of hours after 3 pm

For example, if it is 5 pm, then n = 2. Plugging this value into the formula, we get:

[tex]A_{5} = 92(1.09)^2[/tex]

[tex]A_{5} = 106.1[/tex]

Therefore, the number of cars in the parking lot at 5 pm is 106.1.

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x = 4√3

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

6x² - 6x + 5 is the algebraic expression that is a polynomial with a degree of 2.

The algebraic expression that is a polynomial with a degree of 2 is:

6x² - 6x + 5

A polynomial expression is a sum of terms where each term consists of a coefficient multiplied by one or more variables raised to non-negative integer exponents. The degree of a polynomial is determined by the highest exponent of the variable in any term of the polynomial.

In the expression 6x² - 6x + 5, the highest exponent of the variable x is 2 in the term 6x². The other terms have lower exponents, with -6x having an exponent of 1 and 5 having an exponent of 0. Therefore, the degree of the polynomial is 2, which is the highest exponent in the expression.

Hence, 6x² - 6x + 5 is the algebraic expression that is a polynomial with a degree of 2.

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They can buy 140 vans, 70 small trucks, and 70 large trucks.

The truck company plans to spend 14 million on 280 vehicles. Each commercial van cost 55,000 dollars. Each small truck 20,000 dollars and each large truck 70,000 dollars. They needed twice as many van as small truck

Therefore,

let

s = number of small truck

number of van = v

let

l = number of large truck

v + s + l = 280

55,000(v) + 20,000(s) + 70,000(l) = 14,000, 000

11v + 4s + 14l = 2800

Hence,

v = 2s

So,

26s+ 14l = 2800

26s + 14l = 2800

3s + l = 280

multiply equation(ii) by 14

26s + 14l = 2800

42s + 14l = 3920

subtract the equation

16s = 1120

s = 70

Therefore, they can buy the following:

number of small truck = 70

number of van = 140

number of large truck = 280 - 140 - 70 = 70

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The solution to the equation 60 = 9p − 3 + 7p is 3.94.

To solve the equation 60 = 9p - 3 + 7p, we need to simplify and combine like terms in order to isolate the variable "p" on one side of the equation.

First, let's combine the "p" terms by adding the coefficients of "p":

60 = 9p + 7p - 3

This simplifies to:

60 = 16p - 3

To isolate the variable "p", we want to get rid of the constant term (-3) on the right side of the equation. We can do this by adding 3 to both sides:

60 + 3 = 16p - 3 + 3

63 = 16p

Now we have the equation 63 = 16p. To solve for "p", we need to get rid of the coefficient 16. We can do this by dividing both sides of the equation by 16:

63/16 = 16p/16

3.9375 = p

Therefore, the solution to the equation is p ≈ 3.9375, or approximately p ≈ 3.94.

In summary, by simplifying and combining like terms, we obtained the equation 63 = 16p. By dividing both sides by 16, we found the value of "p" to be approximately 3.9375.

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Answer:

slope (m) = 2/3

Step-by-step explanation:

slope = change in x /change in y

Also, slope is y2 - y1 / x2 -x1. That is what I apply for this activity, hence:

slope = 3 - (-1) / 0 - (-6)

         = 3 + 1 / 0 + 6

         = 4 / 6

         = 2/3

∴ slope(m) = 2/3

The expression that is equivalent to StartRoot [tex]8 x^7 y^8[/tex] EndRoot is ([tex]2 x^3 y^4[/tex] StartRoot 2 x EndRoot)^2.

To understand why this is the case, let's break down each expression and simplify them step by step:

StartRoot [tex]8 x^7 y^8[/tex] EndRoot:

We can rewrite 8 as [tex]2^3[/tex], and since the square root can be split over multiplication, we have StartRoot [tex](2^3) x^7 y^8[/tex] EndRoot. Applying the exponent rule for square roots, we get StartRoot [tex]2^3[/tex] EndRoot StartRoot [tex]x^7[/tex] EndRoot StartRoot [tex]y^8[/tex] EndRoot.

Simplifying further, we have 2 StartRoot [tex]2 x^3 y^4[/tex] EndRoot StartRoot [tex]2^2[/tex] EndRoot StartRoot [tex]x^2[/tex] EndRoot StartRoot [tex]y^4[/tex] EndRoot. Finally, we obtain 2 [tex]x^3 y^4[/tex] StartRoot 2 x EndRoot, which is the expression in question.

([tex]2 x y^2[/tex] StartRoot 8 x^3 EndRoot)^2:

Expanding the expression inside the parentheses, we have [tex]2 x y^2[/tex]StartRoot [tex](2^3) x^3[/tex] EndRoot. Applying the exponent rule for square roots, we get [tex]2 x y^2[/tex] StartRoot [tex]2^3[/tex] EndRoot StartRoot [tex]x^3[/tex] EndRoot.

Simplifying further, we have [tex]2 x y^2[/tex] StartRoot 2 x EndRoot. Squaring the entire expression, we obtain ([tex]2 x y^2[/tex] StartRoot 2 x EndRoot)^2.

Therefore, the expression ([tex]2 x^3 y^4[/tex] StartRoot 2 x EndRoot)^2 is equivalent to StartRoot [tex]8 x^7 y^8[/tex] EndRoot.

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When summarizing the plot, it is important to pay attention to the key ideas and how the conflict develops over time. The given pattern suggests a narrative structure where the plot unfolds gradually, revealing information and building tension.

The plot starts with an introduction (line 1) and then introduces the conflict or problem (lines 2 and 3). As the plot progresses, the conflict continues to develop, possibly reaching a climax or a turning point (line 4). Afterward, the plot moves towards a resolution or conclusion (lines 5, 6, and 7).

The pattern presented in the given example showcases the basic elements of a plot, including the exposition, rising action, climax, falling action, and resolution. The plot begins by setting the stage and introducing the characters and setting (line 1). Then, the conflict is introduced or developed, creating tension and driving the narrative forward (lines 2 and 3).

The plot further progresses with the development of the conflict, possibly reaching a high point or turning point (line 4). Finally, the plot moves towards its resolution or conclusion, providing closure to the story (lines 5, 6, and 7). By understanding these key ideas and the progression of the conflict, one can effectively summarize the plot and capture its essential elements.

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The enough evidence to support the claim that the wait time to see an IRS representative is more than 50 minutes at a 0.100 level of significance.

The null hypothesis is, H0:

µ ≤ 50 minutes.

The alternative hypothesis is,

Ha: µ > 50 minutes.

The significance level is α = 0.100Sample size = 300

Sample mean (x) = 52 minutes

Sample standard deviation (s) = 4 minutes

We have to find whether there is enough evidence to support the claim that the waiting time to see an IRS representative is more than 50 minutes or not.

For that, we perform a one-sample t-test.

The test statistic is given by:

t = (x - µ) / (s/√n)t

= (52 - 50) / (4/√300)t

= 7.905

Critical t-value = t 0.100,299 = 1.646

Since the calculated t-value (7.905) is greater than the critical t-value (1.646), we can reject the null hypothesis.

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Answer:

1. Geometric Sequence Equation for Table A:

[tex]\boxed{\tt a_n = a_1 \times r^{(n-1)}}[/tex]

2. Arithmetic Sequence Equation for Table B:

[tex]\boxed{\tt a_n = a_1 + (n-1) \times d}[/tex]

3.

Step-by-step explanation:

1. Geometric Sequence Equation for Table A:

The geometric sequence equation is given by the formula:

[tex]\tt \[a_n = a_1 \times r^{(n-1)}\][/tex]

where:

For Table A, since we don't have the actual values, we can represent the equation as:

[tex]\boxed{\tt a_n = a_1 \times r^{(n-1)}}[/tex]

[tex]\hrulefill[/tex]

2. Arithmetic Sequence Equation for Table B:

The arithmetic sequence equation is given by the formula:

[tex]\tt a_n = a_1 + (n-1) \times d[/tex]

where:

For Table B, since we don't have the actual values, we can represent the equation as:

[tex]\boxed{\tt a_n = a_1 + (n-1) \times d}[/tex]

[tex]\hrulefill[/tex]

3. Finding Term 20 for both sequences:

In order to find the 20th term for both sequences, we need the actual values for [tex]\tt \(a_1\), \(r\)[/tex] and d.

in the case of Table A

[tex]\tt a_{20} = a_1 \times r^{(20-1)}[/tex]

[tex]\boxed{\tt a_{20} = a_1 \times r^{19}}[/tex]

in the case of Table B.

[tex]\tt a_{20} = a_1 + (20-1) \times d[/tex]

[tex]\boxed{\tt a_{20} = a_1 + 19\times d}[/tex]

By using this formula, we can easily fill up the box.

Answer:

[tex]\frac{\sqrt{2}}{2}[/tex]

Step-by-step explanation:

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

The graph rainbow which is in the shape of a parabola and the selected points indicates that we get;

1. (a) The table can be presented as follows;

x; -10, -4, 2, 8

f(x); 10, 20, 30, 40

2. Domain; [-6, 6], range; [0, 36]

Yes, the values make sense

3. The piecewise function is; f(x) = x² + 15, when 0 ≤ x ≤ 5 and y = x - 1 when 5 < x ≤ ∞

Please find attached the graph of the piecewise function created with MS Excel

A parabola is the shape of the graph of a quadratic function.

1. (a) The points on the graph of the parabola which are on a straight line are;

(-4, 20), and (2, 30)

The table of values are therefore;

x            [tex]{}[/tex] f(x)

-10 [tex]{}[/tex]          10

-4[tex]{}[/tex]            20

2[tex]{}[/tex]             30

8 [tex]{}[/tex]            40

2. The domain is the set of the possible x-values of the data, therefore;

The domain of the rainbow is; [-6, 6]

The range is the set of the possible x-values of the data, therefore;

The range of the rainbow is; [0, 36]

The domain represents the length of the rainbow at ground level

The range represents the height of the rainbow

The values are reasonable, where the axis of symmetry for the rainbow is the y-axis

The x-intercepts are; (-6, 0), and (6, 0), which are the points the rainbow reaches the ground

The y-intercept is the point (0, 36), which is the highest point on the rainbow

The slope of the linear function is; (30 - 20)/(2 - (-4)) = 5/3

The positive slope indicates that the linear function is positive

i(a) The solutions of the system of equation are the point of intersection of the airplane and the rainbow, which are; (-4, 20), and (2, 30)

The point of intersection represents the intersection of the airplane and the rainbow

3. The piece wise function that can be created can be presented as follows;

f(x) = x² + 15 Where; 0 ≤ x ≤ 5

f(x) = x - 1 Where; 5 < x < ∞

Please find attached the graph of the piecewise function created with MS Excel

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There can be multiple outputs (monthly charges) for a given input (number of credit cards), violating the definition of a function. hence, Situation 3 does not represent a function.

Situation 3: the total amount of money charged monthly on credit cards to the number of credit cards owned does not represent a function.

In a function, each input (or x-value) should have a unique output (or y-value). However, in this situation, the total amount of money charged monthly on credit cards depends on the number of credit cards owned. It is possible to have different credit card numbers but still have the same total amount of money charged monthly.

For example, two people could own different numbers of credit cards but have the same monthly charges.

As a result, the definition of a function can be violated by having many outputs (monthly charges) for a single input (number of credit cards).

Because of this, case 3 does not represent a function.

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The standard error represents the average deviation of the sample means from the true population mean. Rounding this value to four decimal places, the standard error of X¯¯¯ is approximtely 0.6500 mpg.

A smaller standard error indicates that the sample means are more likely to be close to the population mean.To calculate the standard error of X¯¯¯, the mean from a random sample of 25 fill-ups by one driver, we can use the formula:

Standard Error (SE) = σ / sqrt(n),

where σ is the standard deviation and n is the sample size.

In this case, the standard deviation (σ) is given as 3.25 mpg, and the sample size (n) is 25.

Plugging in these values into the formula, we have:

SE = 3.25 / sqrt(25).

Calculating the square root of 25, we get:

SE = 3.25 / 5.

Performing the division, we find:

SE ≈ 0.65.

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The measure of angle Q is given as follows:

m < Q = 70º.

Two angles are defined as supplementary angles when the sum of their measures is of 180º.

In a parallelogram, we have that consecutive interior angles are supplementary, hence the value of x is obtained as follows:

6x + 4 + 10x = 180

16x = 176

x = 176/16

x = 11.

Then the measure of angle Q is given as follows:

m < Q = 6 x 11 + 4

m < Q = 70º.

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