(a) From Latoya's results, compute the experimental probability of rolling a 1 or 6.
(b)Assuming that the cube is fair, compute the theoretical probability of rolling a 1 or 6.
(c)Assuming that the cube is fair, choose the statement below that is true.

A. The experimental and theoretical probabilities must always be equal.

B. As the number of rolls increases, we expect the experimental and theoretical probabilities to
become closer, though they might not be equal.

C. As the number of rolls increases, we expect the experimental and theoretical probabilities to
become farther apart.

To determine how the mean changes when a value of 98° is added to the data, we need to calculate the mean before and after the addition.

Before adding 98°, the given data set has 12 values. We can calculate the mean by summing all the values and dividing by the total number of values:

Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12

Mean ≈ 83.67°

After adding 98° to the data set, the total number of values becomes 13. To calculate the new mean, we sum all the values, including the added 98°, and divide by the total number of values:

New Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57 + 98) / 13

New Mean ≈ 86.15°

Therefore, the mean increases by approximately 2.48° when a value of 98° is added to the data. None of the provided answer choices accurately reflects this change, as they all mention different values (8.2° and 1.4°) that do not correspond to the actual change in the mean.

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(a)The experimental probability of rolling an odd number is 3/5.

(b) The theoretical probability of rolling an odd number is 1/2.

(c)The true statement is:

With a small number of rolls, it is not surprising when the experimental probability is much greater than the theoretical probability.

Probability is the likelihood of a desired event happening.

Experimental probability is a probability that relies mainly on a series of experiments.

Theoretical probability is the theory behind probability. To find the probability of an event, an experiment is not required. Instead, we should know about the situation to find the probability of an event occurring.

We have:

Kala rolled a number cube 20 times.

(a) From Kala's results, odd number (1, 3 and 5) appearance is:

4 + 4 + 4 = 12

Thus, the experimental probability of rolling an odd number will be:

12/20 = 3/5

(b) If the cube is fair, we have the numbers have equal chance of appearance.

Theoretically, an odd numbers (1, 3 and 5) will have the probability of 3 out of 6 as illustrated with the numbers below:

1, 2, 3, 4, 5, 6

The theoretical probability of rolling an odd number will be:

3/6 = 1/2

(c)The true statement is:

With a small number of rolls, it is not surprising when the experimental probability is much greater than the theoretical probability.

Because when conducting a small number of rolls with a fair cube, the experimental probability may not necessarily align closely with the theoretical probability.

This difference is expected due to the limited sample size. As the number of rolls increases, the experimental probability tends to converge towards the theoretical probability.

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Answer: The expression, 3v^4, means that the variable, v, is multiplied by itself 4 times (v * v * v * v) and then is multiplied after by the coefficient 3.

Step-by-step explanation:

Exponent indicates the number of times a number needs to be multiplied by itself.

For example if your given x^2

that simply means that the variable, x, is being multiplied by itself twice:

x * x

if your given 2z^9

that means that the variable, z, is first multiplied by itself nine times and after that it is multiplied by the coefficient, two.

Answer:

$2,500 or 25%

Step-by-step explanation:

Let's use the same example:

Cost of Equipment = $10,000

Useful Life = 4 years

Depreciation Rate = (Annual Depreciation / Cost of Equipment) * 100

Annual Depreciation = Cost of Equipment / Useful Life

Annual Depreciation = $10,000 / 4 years

Annual Depreciation = $2,500

Depreciation Rate = ($2,500 / $10,000) * 100

Depreciation Rate = 0.25 * 100

Depreciation Rate = 25%

The calculated value of the amplitude of the function is 3

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

y = 3sin[2(θ - 90)] + 2

A sinusoidal function is represented as

f(x) = Asin(B(x + C)) + D

Where

Amplitude = A

using the above as a guide, we have the following:

Amplitude = A = 3

Hence, the amplitude of the function is 3

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Question

What is the amplitude?

y = 3sin[2(θ - 90)] + 2

To calculate the interest earned in each case, we can use the formula for compound interest:

A = P(1 + r/n)^(nt) - P

where:

A is the final amount (including principal and interest)

P is the principal amount (initial deposit)

r is the interest rate (in decimal form)

n is the number of times interest is compounded per year

t is the number of years

Given:

P = $21,000

r = 4% = 0.04

(a) If interest is compounded semiannually, n = 2 (twice a year), and t = 2 years. Plugging these values into the formula, we get:

A = 21000(1 + 0.04/2)^(2*2) - 21000

Calculate A to find the final amount and subtract the principal to get the interest earned.

(b) If interest is compounded quarterly, n = 4 (four times a year), and t = 2 years. Plugging these values into the formula, we get:

A = 21000(1 + 0.04/4)^(4*2) - 21000

Calculate A to find the final amount and subtract the principal to get the interest earned.

(c) To find the difference in interest earned when compounded quarterly, subtract the interest earned in part (a) from the interest earned in part (b).

By calculating the interest earned and the difference in interest, we can determine the specific amounts.

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Answer:

(9,16)

Step-by-step explanation:

We Know

The equation y = 3x - 2

Find the y coordinate (9, ?)

We just simply put 9 in for x and solve for y

y = 3(9) - 2

y = 18 - 2

y = 16

So, the coordinate are (9,16)

Answer:

$2100

Step-by-step explanation:

40% of 1500 is 600

1500 + 600 = 2100

The selling price will be $2100

To construct a frequency histogram based on the given temperature data, we will use the temperature ranges as the x-axis and the number of days as the y-axis.

The temperature ranges and their corresponding frequencies are as follows:

30-39: 2 days

40-49: 26 days

50-59: 28 days

60-69: 8 days

70-79: 2 days

To create the histogram, we will represent each temperature range as a bar and the height of each bar will correspond to the frequency of days.

Using an appropriate scale, we can label the x-axis with the temperature ranges (30-39, 40-49, 50-59, 60-69, 70-79) and the y-axis with the frequency values.

Now, we can draw rectangles (bars) on the graph, with the base of each rectangle corresponding to the temperature range and the height representing the frequency of days. The height of each bar will be determined by the corresponding frequency value.

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The value of side length x (DF) in the triangle is 4.

The figures in the image is that of two similar triangle.

Triangle ABC is similar to triangle DEF.

From the diagram:

Leg 1 of the smaller triangle DE = 5

Leg 2 of the smaller triangle DF = x

Leg 1 of the larger triangle AB = 30

Leg 2 of the larger triangle AC = 24

To find the value of x, we take the ratio of the sides of the two triangle since they similar:

Hence:

Leg DE : Leg DF = Leg AB : Leg AC

Plug in the values:

5 : x = 30 : 24

5/x = 30/24

Cross multiplying, we get:

30x = 5 × 24

30x = 120

x = 120/30

x = 4

Therefore, the value of x is 4.

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Dot plots are graphical displays of data that use dots to represent the frequency or count of a specific value in a data set. The ages of five children in each of two families are represented in the dot plots above.

Both dot plots have a similar range of ages, but family A has a higher median age and a smaller spread of ages than family B.

For family A, the median age is 10 years old because the middle dot in the plot is located at 10. Family A has a small range of ages because all the dots are concentrated in the 8-12 age range.

In contrast, family B has a wider range of ages, from 6 to 15 years old, and a larger spread because the dots are scattered across the plot.

The median age for family B is approximately 11 years old.

In conclusion, dot plots are useful for comparing data sets and analyzing their characteristics such as median age and spread.

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Answer: 2.8

Step-by-step explanation:

2x^2+6=10

2x^2=10-6

x^2=16 divided by 2

x= square root of 8

x=2.8

The implicit derivative is given as follows:

dx/dt(x = 4)  = 1/12.

The function in this problem is given as follows:

y = 3x² + 1.

The implicit derivative, relative to the variable t, is given as follows:

dy/dt = 6x dx/dt.

(the derivative of the constant 1 is of zero).

The parameters for this problem are given as follows:

x = 4, dy/dt = 2.

Hence the derivative is obtained as follows:

2 = 6(4) dx/dt

dx/dt = 2/24

dx/dt = 1/12.

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Answer:

  (a)  Use the distance formula to find the length of each side, and then add the lengths.

Step-by-step explanation:

You want to know a suitable strategy for finding the perimeter of a quadrilateral when the coordinates of its vertices are known.

The perimeter of a figure is the sum of its side lengths. When the coordinates of the ends of a side segment are known, the length of that segment can be found using the distance formula.

This should make it obvious that a suitable strategy for finding the perimeter is to find the length of each side and add the lengths.

__

Additional comment

The slope is irrelevant when the perimeter is what you want.

The slope can be relevant if you're trying to prove a quadrilateral is a parallelogram or rectangle (or something else with parallel sides). (There are easier ways than using the slope.)

Multiplying side lengths will be relevant if you want the area of a rectangle. The product has no relation to the perimeter.

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The distance between A and B based on the diagram is approximately 1,685.2 cm.

Hypotenuse = 1,920 cm

Adjacent = 920 cm

A and B = opposite

Hypotenuse² = adjacent² + opposite²

1920² = 920² + AB²

3,686,400 = 846,400 + AB²

3,686,400 - 846,400 = AB²

2,840,000 = AB²

find the square root of both sides

AB = √2,840,000

AB = 1685.229954635271

Approximately,

AB = 1,685.2 cm

Hence, distance between A and B is 1,685.2 cm.

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Answer:

We cannot determine if the triangles are similar.

Step-by-step explanation:

The diagram shows two triangles, ΔABC and ΔDEF.

The tick marks on the sides of the triangles show that two corresponding sides are congruent:

In similar triangles:

As the diagram does not include any information about the interior angles or the length of the third side, we cannot determine if the triangles are similar.

To prove the triangles are similar by SAS similarity, we would need the measure of the included angles (angles B and E).

To prove the triangles are similar by SSS similarity, we would need the measure of the third side.

To prove the triangles are similar by AA similarity, we would need the measures of two corresponding angles.

The true statement about the set is n.

The correct answer choice is option A.

The intersection of two sets for instance, set A and B, denoted (n) is the set containing all elements of set A that also belongs to set B.

{6, 8, 10, 12} _ {5, 6, 7, 8, 9} = {6, 8}

Let

{6, 8, 10, 12} = set A

{5, 6, 7, 8, 9} = set B

A n B = {6, 8}

Therefore, the intersection of set A and set B, that is, the elements of set A that are also contained in B are {6, 8}

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Answer: D. 18

Step-by-step explanation:

      We will use the order of operations (sometimes known as PEMDAS).

Given:

  2(4 - 1)²

Subtraction:

  2(3)²

Square:

  2(9)

Multiply:

  18

             D. 18

The general solution of the partial differential equation is:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

The partial differential equation (PDE) is given as:

Uxx = (1/2)Ut with the boundary and initial conditions as 0< X <3 with U(0,t) = U(3, t)=0, U(0, t) = 5sin(4πx)

Assume that the solution can be written as a product of two functions:

U(x, t) = X(x)T(t)

Substituting this into the PDE, we have:

X''(x)T(t) = (1/2)X(x)T'(t)

Dividing both sides by X(x)T(t), we get:

(X''(x))/X(x) = (1/2)(T'(t))/T(t)

Since the left side only depends on x and the right side only depends on t, both sides must be equal to a constant, denoted as -λ²:

(X''(x))/X(x) = -λ²

(1/2)(T'(t))/T(t) = -λ²

Simplifying the second equation, we have:

T'(t)/T(t) = -2λ²

Solving the second equation, we find:

T(t) = Ce^(-2λ²t)

Applying the boundary condition U(0, t) = 0, we have:

U(0, t) = X(0)T(t) = 0

Since T(t) ≠ 0, we must have X(0) = 0.

Applying the boundary condition U(3, t) = 0, we have:

U(3, t) = X(3)T(t) = 0

Again, since T(t) ≠ 0, we must have X(3) = 0.

Therefore, we can conclude that X(x) must satisfy the following boundary value problem:

X''(x)/X(x) = -λ²

X(0) = 0

X(3) = 0

The general solution to this ordinary differential equation is given by:

X(x) = Asin(λx) + Bcos(λx)

Applying the initial condition U(x, 0) = 5*sin(4πx), we have:

U(x, 0) = X(x)T(0) = X(x)C

Comparing this with the given initial condition, we can conclude that T(0) = C = 5.

Therefore, the complete solution for U(x, t) is given by:

U(x, t) = Σ [Aₙsin(λₙx) + Bₙcos(λₙx)]*e^(-2(λₙ)²t)

where:

Σ represents the summation over all values of n

λₙ are the eigenvalues obtained from solving the boundary value problem for X(x).

To find the eigenvalues λₙ, we substitute the boundary conditions into the general solution for X(x):

X(0) = 0: Aₙsin(0) + Bₙcos(0) = 0

X(3) = 0: Aₙsin(3λₙ) + Bₙcos(3λₙ) = 0

From the first equation, we have Bₙ = 0.

From the second equation, we have Aₙ*sin(3λₙ) = 0. Since Aₙ ≠ 0, we must have sin(3λₙ) = 0.

This implies that 3λₙ = nπ, where n is an integer.

Therefore, λₙ = (nπ)/3.

Substituting the eigenvalues into the general solution, we have:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

where Aₙ are the coefficients that can be determined from the initial condition.

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Answer: [tex]10x^3-19x^2+20x-21[/tex]

Step-by-step explanation:

This problem is a binomial being multiplied by a trinomial.

We can solve it by multiply each term in the first expression by each term in the second expression and combine like terms.

[tex](2x - 3)(5x^2 -2x + 7)[/tex]

First lets multiply the 2x to the trinomial, (5x^2 - 2x + 7)

2x * 5x^2 = 10x^3

2x * -2x = -4x^2

2x * 7 = 14x

Next lets multiply the -3 to the trinomial (5x^2 - 2x + 7)

-3 * 5x^2 = -15x^2

-3 * -2x = 6x

-3 * 7 = -21

Now put all in order by highest power (Exponent) to lowest (to lowest exponent/ or constant)

[tex]10x^3-4x^2-15x^2+14x+6x-21[/tex]

And lastly, combine like terms:

[tex]10x^3-19x^2+20x-21[/tex]

Your final answer is [tex]10x^3-19x^2+20x-21[/tex]

The slope is represented by the number 1473 in the equation T = 1473c + 5495. The slope denotes the rate of change, or the amount by which the overall cost (T) rises with each new class attended (c).

In this scenario, the total cost of the semester increases by $1473 for each extra class attended at Michigan State University. The slope reflects the linear relationship between the total cos and the number of classes taken.

The number 5495 represents the y-intercept in the equation. When the number of classes (c) is 0, the y-intercept is the value of T. The y-intercept of $5495 shows the total cost for a semester at Michigan State University when no classes are taken. It includes a one-time accommodation and board price.

The y-intercept can be understood as the university's fixed cost component, which is independent of the number of classes taken. It includes expenditures such as housing and food plans that are incurred regardless of how many classes a student enrolls in.

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The speed of the car is 108 kilometers per hour and the distance covered in 5 hours is 540 kilometers.

Speed is simply referred to as distance traveled per unit time.

It is expressed as;

Speed = Distance ÷ time.

Given that the car is traveling at a rate of 30 meters per second.

First, convert the car's speed from meters per second to kilometers per hour using the conversion factor.

1 kilometer = 1000 meters

1 hour = 3600 seconds

Hence;

Speed = 30m/s = ( 30 × 3600/1000 )kmh

Speed = 108 kmh

Next, the distance covered in 5 hours will be:

Speed = Distance / time

Distance = speed × time

Distance = 108 kmh × 5 h

Distance = 540 km

Therefore, the disatnce covered is 540 kilometers.

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The six terms of the series 247 are 2, 2, 2, 2, 2, and 2.

To find the six terms of a series, we need to first understand what a series is. A series is defined as the sum of the terms of a sequence. A sequence is a set of numbers in a specific order.

Therefore, a series is a sum of terms from a sequence. There are different types of series, and they have different formulas for calculating their terms.In this case, we are required to find the six terms of the series 247. Since this is a finite series, we can use a formula to calculate the nth term of a finite series.

The formula is given as follows:Tn = a + (n - 1) dWhere Tn is the nth term of the series, a is the first term of the series, n is the number of terms in the series, and d is the common difference between the terms of the series.To find the six terms of the series 247, we need to know the value of a and d. In this case, we can see that the first term of the series is 2. Therefore, a = 2.

Since this is a constant series, we can see that the common difference between the terms is 0. Therefore, d = 0.Substituting these values in the formula, we get:T1 = a + (1 - 1) dT1 = 2 + 0T1 = 2T2 = a + (2 - 1) dT2 = 2 + 0T2 = 2T3 = a + (3 - 1) dT3 = 2 + 0T3 = 2T4 = a + (4 - 1) dT4 = 2 + 0T4 = 2T5 = a + (5 - 1) dT5 = 2 + 0T5 = 2T6 = a + (6 - 1) dT6 = 2 + 0T6 = 2.

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The vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

To find the vertex of the parabola, we can use the formula:

x = -b / (2a)

In this case, the function is p(d) = -4d² + 200d - 100, which can be rewritten in the form of ax² + bx + c.

Comparing it with the standard form ax² + bx + c, we can see that a = -4, b = 200, and c = -100.

Now, let's substitute these values into the formula to find the vertex:

d = -200 / (2 * -4)

d = -200 / -8

d = 25

The x-coordinate of the vertex is 25. To find the corresponding y-coordinate, we substitute this value back into the original function:

p(25) = -4(25)² + 200(25) - 100

p(25) = -4(625) + 5000 - 100

p(25) = -2500 + 5000 - 100

p(25) = 2400

The y-coordinate of the vertex is 2400.

Therefore, the vertex of the parabola is (25, 2400).

The vertex reveals important information about the situation. In this case, it represents the optimal point for maximizing profit. The x-coordinate (25) represents the price at which the theater should set the tickets to maximize their profit. The y-coordinate (2400) represents the maximum profit achievable at that price.

Additionally, since the coefficient of the quadratic term (a) is negative (-4), it indicates that the parabola opens downwards, forming a concave shape. This means that as the ticket price increases or decreases from the optimal price, the profit will decrease. Therefore, setting the tickets at a price other than the one corresponding to the vertex would result in a lower profit for the theater.

In summary, the vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

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To find three points on the circle defined by the equation (x+67)2+(y+4)2=494(x+76​)2+(y+4)2=449​, we can manipulate the equation to extract the center and radius information.

Expanding the equation, we have:

x2+127x+3649+y2+8y+16=494x2+712​x+4936​+y2+8y+16=449​

Combining like terms, we get:

x2+y2+127x+8y+1149=0x2+y2+712​x+8y+4911​=0

Comparing this equation to the standard form of a circle, (x−a)2+(y−b)2=r2(x−a)2+(y−b)2=r2, we can identify the center (a,b)(a,b) as −67,−4−76​,−4 and the radius rr as 7227​.

Now we can find three points on the circle by substituting different angles into the equation. For example:

   At 00 degrees: (−6/7+72),−4(−6/7+27​),−4 or (11/14,−4)(11/14,−4)

   At 9090 degrees: −6/7,−4+72−6/7,−4+27​ or −6/7,1/2−6/7,1/2

   At 180180 degrees: (−6/7−72),−4(−6/7−27​),−4 or (−55/14,−4)(−55/14,−4)

Therefore, three points on the circle are (11/14,−4)(11/14,−4), (−6/7,1/2)(−6/7,1/2), and (−55/14,−4)(−55/14,−4).

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Answer:

Yes; TSU ~ PJM

Step-by-step explanation:

Because the angle in between the sides are the same, we can see if each corresponding side is proportional to each other. If they are, then they are similar. (This is SAS similarity theorem)

First we can divide 10 by 15 and see if 14 divided by 21 is the same:

10/15 = 0.6666....

14/21 = 0.6666....

Yes, the two triangles are similar

Answer:

Δ STU  [tex]\bold{\sim}[/tex]  ΔPJM similar

Step-by-step explanation:

Similar triangles are two or more triangles that have the same shape, but their sides are in proportion. In other words, if you divide the length of any side of one triangle by the corresponding side of another triangle, you will get the same number.

For Question:

In Δ STU and ΔPJM

TS:US =10:14=5:7

PJ: JM=15:21= 5:7

Since the length of any side of one triangle is by the corresponding side of another triangle, you will get the same number.

Therefore,

Δ STU  [tex]\bold{\sim}[/tex]  ΔPJM is similar.

Hence Proved:

The rest of the amount, represented by R, is $2000 in this case.

If the total amount is $3000 and a certain amount, c, is deposited in one investment, we can represent the remaining amount in dollars by subtracting the deposited amount from the total.

Let's denote the remaining amount as R.

R = Total amount - Deposit amount

R = $3000 - c

Here, R represents the rest of the amount in dollars after the deposit of c dollars. By subtracting the deposit amount from the total amount, we can determine the value of the remaining amount.

For example, if $1000 is deposited in one investment (c = $1000), we can find the remaining amount as follows:

R = $3000 - $1000

R = $2000

Therefore, the rest of the amount, represented by R, is $2000 in this case.

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If the bottle is completely full at 11:40 a.m., the statements that help explain this result include:

Exponential growth involves a pattern or process that increases the initial quantity or value over time at a constant rate.

In an exponential growth situation, the rate or speed of growth remains proportional to the initial population.

The reasons for choosing Options A, B, and D follow:

Option A: at 11:35 a.m., the bottle was 1/2 full.  If at 11:40 a.m. it is completely full, it implies that it takes 5 minutes to become full (11:40 a.m. - 11:35 a.m.)

Option B: At 11:30 a.m., the bottle was 1/4 full and 10 minutes later, 11:40 a.m. it was full, meaning that it doubled its volume after 10 minutes.

Option D: Exponential growth involves a constant rate of growth and can be expressed as a(1+r)ˣ, where a is the initial value or quantity, r is the growth rate, 1+r is the growth factor,, and x is the exponent in time.

Thus, in this situation, the correct options are A, B, and D.

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A) The bottle was 1/2 full at 11:35 a.m. and doubled again after 5 minutes means that at 11:40 a.m. the bottle is completely full.

B) The bottle was 1/4 full at 11:30 a.m. and doubled twice after 10 minutes means that at 11:40 a.m. the bottle is completely full.

C) The bottle was 1/2 full at 11:35 a.m. and more bacteria were added to fill the bottle means that the bottle is completely full but with bacteria present in it.

D) Exponential growth involves a constant multiplicative rate of change means that bottle level rises with a constant multiplicative rate of change.

E) Exponential growth involves a constant additive rate of change means that bottle level rises with a constant additive rate of change.

Answer:

If you earn $50,000 a year and you max out your contribution to a 401(k) plan, you would contribute $19,500 (assuming you are under 50 years old). This means you would save $19,500 per year towards your retirement.

Therefore, the range of the function g(x) = -1/2(x - 6) + 1 is all real numbers less than or equal to 1, written as (-∞, 1].

To determine the range of the function g(x) = -1/2(x - 6) + 1, we need to find the set of all possible output values or the range of the function.

The given function is in the form of a linear equation, specifically in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. In this case, the slope is -1/2 and the y-intercept is 1.

Since the slope is negative, it means the function is decreasing. The range of the function would be the set of all possible y-values that the function can produce. Since the function is decreasing, the maximum value it can reach is its initial y-intercept, which is 1.

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