B
C
W
A
E
Use the given diagram to answer the question,
Which line is the intersection of two of the planes
shown?
Which line intersects one of the planes shown?
Which line has points on three of the planes shown?
G

The probability of drawing blue, red, and white balls consecutively is 120/3375 with replacement and 120/2730 without replacement.

To calculate the probability of drawing three balls from a box containing six white balls, five red balls, and four blue balls, we need to consider two scenarios: with replacement and without replacement.

(i) With replacement:

When each ball is replaced after it is drawn, the total number of balls remains the same for each draw. Therefore, the probability of drawing a specific color on each draw remains constant.

The probability of drawing a blue ball on each draw is 4/15, the probability of drawing a red ball is 5/15, and the probability of drawing a white ball is 6/15 (assuming all colors are equally likely to be drawn).

To find the probability of drawing a blue, red, and white ball consecutively, we multiply the probabilities together since the events are independent:

P(Blue, Red, White) = (4/15) * (5/15) * (6/15) = 120/3375

(ii) Without replacement:

When the balls are not replaced after being drawn, the total number of balls decreases for each subsequent draw. This affects the probability of each color being drawn on subsequent draws.

For the first draw, the probability of drawing a blue ball is 4/15, a red ball is 5/15, and a white ball is 6/15.

For the second draw, the probability of drawing a blue ball is 3/14, a red ball is 5/14 (as one blue ball is already drawn), and a white ball is 6/14.

For the third draw, the probability of drawing a blue ball is 2/13, a red ball is 4/13 (as two blue balls and one red ball are already drawn), and a white ball is 6/13.

To find the probability of drawing a blue, red, and white ball consecutively without replacement, we multiply the probabilities together:

P(Blue, Red, White) = (4/15) * (5/14) * (6/13) = 120/2730

Therefore, the probability of drawing blue, red, and white balls consecutively is 120/3375 with replacement and 120/2730 without replacement.

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Answer:

A= 220.5ft

B= $2425.5

Step-by-step explanation:

A= 220.5ft

B= $2425.5

To determine the appropriate verb form, consider the time of the action, its relation to the present or past events, and whether it is a general fact, completed action, ongoing situation, or action preceding another. Choose the verb tense that accurately conveys the intended meaning in the paragraph.

To determine which form of verb is appropriate to use in a paragraph, you need to consider the context and the intended meaning of the sentence or paragraph. Here are some general guidelines for using different tenses:

Present Simple Tense:

Use the present simple tense to talk about general facts, habits, routines, and permanent situations.

Example: "The sun rises in the east."

Past Simple Tense:

Use the past simple tense to talk about completed actions or events in the past.

Example: "She studied abroad last year."

Present Perfect Tense:

Use the present perfect tense to talk about past actions or events that have a connection to the present or when the exact time of the action is not specified.

Example: "I have visited Paris several times."

Past Perfect Tense:

Use the past perfect tense to talk about an action or event that happened before another past action or event.

Example: "She had already eaten dinner when I arrived."

To determine which tense to use, consider the timeline of events and the relationship between them. If you are referring to a specific time in the past, the past simple tense might be appropriate. If you want to emphasize the connection to the present, the present perfect tense might be suitable. If you need to establish a sequence of events in the past, the past perfect tense could be used.

However, it's important to note that these guidelines are not absolute, and there can be variations based on specific contexts and writing styles. It's always best to consult grammar rules and consider the meaning and context of your sentences to choose the most appropriate verb tense for your paragraph.

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The 90% confidence interval for the population variance (σ^2) is approximately [7.19, 20.19].

To construct a 90% confidence interval for the population variance (σ^2), given a sample size of 20 and a sample variance (s^2) of 12.5, we need to utilize the chi-square distribution.

The formula for constructing a confidence interval for the population variance is:

[ (n - 1)s^2 / χ^2_upper, (n - 1)s^2 / χ^2_lower ]

Where n is the sample size, s^2 is the sample variance, χ^2_upper is the upper critical value from the chi-square distribution, and χ^2_lower is the lower critical value from the chi-square distribution.

For a 90% confidence interval, we need to find the critical values from the chi-square distribution that correspond to the upper and lower tails of 5% each (since the confidence level is divided equally into two tails).

(a) Degrees of freedom:

The degrees of freedom (df) for the chi-square distribution is equal to n - 1. In this case, n = 20, so df = 20 - 1 = 19.

(b) Chi-square critical values:

We need to find the upper and lower critical values from the chi-square distribution table for df = 19 and a significance level of 0.05/2 = 0.025 (for each tail).

From the chi-square distribution table or using a statistical software, the upper critical value for a significance level of 0.025 and df = 19 is approximately 32.852. Similarly, the lower critical value is also approximately 8.907.

(c) Confidence interval calculation:

Substituting the values into the formula, we can construct the confidence interval:

[ (n - 1)s^2 / χ^2_upper, (n - 1)s^2 / χ^2_lower ]

[ (20 - 1) * 12.5 / 32.852, (20 - 1) * 12.5 / 8.907 ]

[ 19 * 12.5 / 32.852, 19 * 12.5 / 8.907 ]

[ 7.19, 20.19 ]

Therefore, the [7.19, 20.19] is roughly inside the 90% confidence interval for the population variance (2).

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Question

A simple random sample of size n is drawn from a population that is known to be normally distributed. The sample​ variance, s^2​, is determined to be 12.5.Complete parts​ (a) through​ (c).Construct a​ 90% confidence interval for

σ2 if the sample​ size, n, is 20.

Answer:

B :3

Step-by-step explanation:

good luck <3

The estimates are not correct because we are to find the proportion and sample error of the given values. The proportion expresses the number of opposing votes to the total number of voters. Also, the sampling error follows the formula of [tex]SE = Z\sqrt{P(1 - P/n)}[/tex]

The exact formula used in the expression is quite different from the standard formula for calculating the standard error. We are supposed to find the root of the standard deviation but that is not the case as can be seen in the formula above.

So, the main issue with this formula is that the square root is not considered in the proportion formula used by the council, so it is not a valid estimate.

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The period of oscillation is 3 seconds

A Oscillation is the periodic change of a measure around a central value or between two or more states, usually in time.

The time taken for an oscillating particle to complete one cycle of oscillation is known as the Period of oscillating particle. It is measured in seconds

Oscillation can also be vibration or revolution or cycle.

Therefore, using the graph to determine the period. Then the wave particle made a complete oscillation at 3 second.

This means that the period of the particle is 3 seconds.

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Answer:

a. 5π cm

b. 144π in²

c. 6 ft

Step-by-step explanation:

a.

The circumference of a circle is given by:

Circumference pf circle= πd

where d is the diameter.

In this case, d = 5 cm,

Therefore, Circumference of circle = π*5=5π cm

b.

The area of a circle is given by:
Area of circle=πr²,

where r is the radius. In this case, the diameter is d = 24 in,

so, the radius is r = d/2 = 24/2=12in

Therefore, Area of circle=π*12²=144π in²

c.

The area of a circle is given by:

Area of circle=πr²,

where r is the radius. In this case, Area is 36π ft²

Now substituting value

36π=πr²

dividing both side by π, we get

36=r²

[tex]r=\sqrt{36}=6[/tex]

r=6 ft.

Therefore, Radius is 6 ft.

The period of the frequency factor b that is given in the diagram above would be = 0.2 sec.

The frequency of a water wave is defined as the number of times the wave completes a cycle within a given period of time. While the period is the time it takes for the completion of a cycle.

The dot the represents the frequency factor b is the green dot on the wave table. Therefore the period as traced from the graph= 0.2 sec.

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Answer:

centre = (5, - 6 ) , radius = 7

Step-by-step explanation:

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

given

x² + y² - 10x + 12y + 12 = 0 ( subtract 12 from both sides )

x² + y² - 10x + 12y = - 12 ( collect terms in x/ y )

x² - 10x + y² + 12y = - 12

using the method of completing the square

add ( half the coefficient of the x/ y terms )² to both sides

x² + 2(- 5)x + 25 + y² + 2(6)y + 36 = - 12 + 25 + 36

(x - 5)² + (y + 6)² = 49 = 7² ← in standard form

with centre (5, - 6 ) and radius = 7

Answer:

Center = (5, -6)

Radius = 7

Step-by-step explanation:

To find the center and the radius of the circle represented by the given equation, rewrite the equation in standard form by completing the square.

To complete the square, begin by moving the constant to the right side of the equation and collecting like terms on the left side of the equation:

[tex]x^2-10x+y^2+12y=-12[/tex]

Add the square of half the coefficient of the term in x and the term in y to both sides of the equation:

[tex]x^2-10x+\left(\dfrac{-10}{2}\right)^2+y^2+12y+\left(\dfrac{12}{2}\right)^2=-12+\left(\dfrac{-10}{2}\right)^2+\left(\dfrac{12}{2}\right)^2[/tex]

Simplify:

[tex]x^2-10x+(-5)^2+y^2+12y+(6)^2=-12+(-5)^2+(6)^2[/tex]

       [tex]x^2-10x+25+y^2+12y+36=-12+25+36[/tex]

       [tex]x^2-10x+25+y^2+12y+36=49[/tex]

Factor the perfect square trinomials on the left side:

[tex](x-5)^2+(y+6)^2=49[/tex]

The standard equation of a circle is:

[tex]\boxed{(x-h)^2+(y-k)^2=r^2}[/tex]

where:

Comparing this with the rewritten given equation, we get

Therefore, the center of the circle is (5, -6) and its radius is r = 7.

The correct expression that is equivalent to (f + g) (4) is: f(4) + g(4).

The expression (f + g) (4) represents the sum of two functions, f(x) and g(x), evaluated at x = 4. To find the equivalent expression, we need to simplify it.

In (f + g) (4), the parentheses indicate that the addition operation is performed first, adding the functions f(x) and g(x) together. Then, the resulting sum is evaluated at x = 4. So, the expression simplifies to f(4) + g(4), where we substitute x with 4 in both functions.

The other options provided:

• ¡(4) + g(4): This option is not correct because the negation operator (!) applied to a value does not make sense in this context.

• f(x) + g(4): This option is not correct because it does not evaluate the sum of the functions at x = 4; it keeps the variable x in the expression.

• ¡(4 + g(4)): This option is not correct because it applies the negation operator to the sum of 4 and g(4), which is not equivalent to (f + g) (4).

• 4(f(x) + g(x)): This option is not correct because it introduces a constant factor of 4 to the sum of the functions, which is not equivalent to (f + g) (4).

The correct expression equivalent to (f + g) (4) is f(4) + g(4).

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This option is not equivalent to (f + g)(4).

The expression (f + g)(4) specifically represents the sum of functions f and g evaluated at x = 4.

To determine which expression is equivalent to (f + g)(4), let's break it down step by step.

The expression (f + g)(4) represents the value obtained by evaluating the sum of functions f and g at x = 4.

We substitute x = 4 into both functions and then add the results.

Let's evaluate each option to see which one matches this process:

¡(4) + g(4):

This option involves evaluating the function f at x = 4 and adding it to the value obtained by evaluating function g at x = 4.

It does not represent the sum of the functions f and g evaluated at x = 4.

This option is not equivalent to (f + g)(4).

f(x) + g(4):

This option involves adding the value of function f at an arbitrary point x to the value obtained by evaluating function g at x = 4.

It does not specifically represent the sum of functions f and g evaluated at x = 4.

This option is not equivalent to (f + g)(4).

¡(4 + g(4)):

This option involves evaluating the function g at x = 4 and adding it to the value obtained by adding 4 to the result.

It does not represent the sum of functions f and g evaluated at x = 4.

This option is not equivalent to (f + g)(4).

4(f(x) + g(x)):

This option involves evaluating the functions f and g at an arbitrary point x, summing the results and then multiplying the sum by 4.

It does not specifically represent the sum of functions f and g evaluated at x = 4.

None of the given options is equivalent to (f + g)(4).

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Answer:

A, C

Step-by-step explanation:

x² + 3x - 5 = 0

[tex] x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} [/tex]

[tex] x = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)} [/tex]

[tex] x = \dfrac{-3 \pm \sqrt{29}}{2} [/tex]

Answer:

statmen there for this should me solved and abcd are proof

reason bcz this is not equal square

Answer:

D. Right skewed

Step-by-step explanation:

To determine the shape of the distribution, we can examine the given data:

12, 9, 4, 8, 25, 6, 8, 5, 18, 13

One way to determine the shape of the distribution is by visualizing it using a histogram or a box plot. However, without the exact frequency of each value, we cannot create an accurate visual representation.

Alternatively, we can examine the skewness of the distribution. Skewness is a measure of the asymmetry of a distribution. If the data is skewed to the left, it is left-skewed or negatively skewed. If it is skewed to the right, it is right-skewed or positively skewed. If the data is symmetric and evenly distributed, it is considered a symmetrical distribution.

Let's calculate the skewness of the given data to determine the shape:

Skewness = (3 * (mean - median)) / standard deviation

First, let's calculate the mean, median, and standard deviation of the data:

Mean = (12 + 9 + 4 + 8 + 25 + 6 + 8 + 5 + 18 + 13) / 10 = 10.8

Median = the middle value when the data is arranged in ascending order:

4, 5, 6, 8, 8, 9, 12, 13, 18, 25

Median = (8 + 9) / 2 = 8.5

Next, let's calculate the standard deviation:

Step 1: Calculate the squared differences from the mean for each value:

(12 - 10.8)^2, (9 - 10.8)^2, (4 - 10.8)^2, (8 - 10.8)^2, (25 - 10.8)^2, (6 - 10.8)^2, (8 - 10.8)^2, (5 - 10.8)^2, (18 - 10.8)^2, (13 - 10.8)^2

Step 2: Calculate the sum of squared differences:

(1.44 + 2.88 + 45.76 + 8.64 + 228.01 + 22.09 + 8.64 + 32.49 + 47.04 + 4.84) = 411.73

Step 3: Calculate the variance:

Variance = sum of squared differences / (n - 1) = 411.73 / (10 - 1) = 45.75

Step 4: Calculate the standard deviation:

Standard deviation = square root of variance = √45.75 = 6.76 (approximately)

Now we can calculate the skewness:

Skewness = (3 * (mean - median)) / standard deviation

Skewness = (3 * (10.8 - 8.5)) / 6.76

Skewness = 6.4 / 6.76

Skewness ≈ 0.95

Since the skewness is positive (0.95), the data is right-skewed or positively skewed. Therefore, the appropriate shape of the distribution is:

D. Right skewed

Answer:

[tex] \frac{a}{2x - 3} + \frac{b}{3x +4} = \frac{x + 7}{ 6{x}^{2} - x - 12} [/tex]

[tex](3x + 4)a + (2x - 3)b = x + 7[/tex]

3a + 2b = 1---->9a + 6b = 3

4a - 3b = 7---->8a - 6b = 14

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

17a = 17

a = 1, b = -1

(3+9) + 13 correctly rewritten using the associative property and then correctly simplified is 13+(9+3) = 13 + 12 = 25.

1. Start with the expression (3+9) + 13.

2. According to the associative property of addition, we can group the numbers in any order without changing the result.

3. Rearrange the expression by grouping the numbers differently: (9+3) + 13.

4. Now simplify the grouped numbers: 9+3 = 12.

5. The expression becomes 12 + 13.

6. Finally, simplify the addition: 12 + 13 = 25.

Therefore, the correct rewritten expression using the associative property and the simplified result is 13+(9+3) = 13 + 12 = 25.

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Answer: 337

Step-by-step explanation: it is 337 because if you subtract it all you get that

The distance of the bicycle race is approximately 24.61 miles.

Let's assume the distance of the run is 'x' miles and the distance of the bicycle race is 'y' miles.

We know that average velocity is equal to the total distance divided by the total time taken.

We can use this information to form two equations based on the given average velocities and total time.

For the run:

Average velocity = Distance of the run / Time taken for the run

5 mph = x miles / T hours ---(Equation 1)

For the bicycle race:

Average velocity = Distance of the bicycle race / Time taken for the bicycle race

23 mph = y miles / (6 - T) hours ---(Equation 2)

Since the total time for the race is 6 hours, we can substitute (6 - T) for the time taken during the bicycle race.

Now, we can solve these equations simultaneously to find the values of 'x' and 'y'.

From Equation 1, we have:

5T = x

From Equation 2, we have:

23(6 - T) = y

Now, we substitute the value of 'x' in terms of 'T' from Equation 1 into Equation 2:

23(6 - T) = 5T

138 - 23T = 5T

138 = 28T

T = 138 / 28

T ≈ 4.93 hours

Substituting this value back into Equation 1, we can find 'x':

5(4.93) = x

x ≈ 24.65 miles

Therefore, the distance of the run is approximately 24.65 miles.

To find the distance of the bicycle race, we substitute the value of 'T' back into Equation 2:

23(6 - 4.93) = y

23(1.07) = y

y ≈ 24.61 miles

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The table of values for the equation y = 3/x can be completed as follows:

x | y

1 | 3/1 = 3

2 | 3/2 = 1.5

3 | 3/3 = 1

4 | 3/4 = 0.75

5 | 3/5 = 0.6

To complete the table of values for the equation y = 3/x, we substitute different values of x into the equation and calculate the corresponding values of y.

When x = 1:

Substitute x = 1 into the equation: y = 3/1 = 3

So, when x = 1, y = 3.

When x = 2:

Substitute x = 2 into the equation: y = 3/2 = 1.5

So, when x = 2, y = 1.5.

When x = 3:

Substitute x = 3 into the equation: y = 3/3 = 1

So, when x = 3, y = 1.

When x = 4:

Substitute x = 4 into the equation: y = 3/4 = 0.75

So, when x = 4, y = 0.75.

When x = 5:

Substitute x = 5 into the equation: y = 3/5 = 0.6

So, when x = 5, y = 0.6.

By substituting different values of x into the equation y = 3/x, we can complete the table of values as shown above.

Hence, the completed table of values for y = 3/x is:

x | y

1 | 3

2 | 1.5

3 | 1

4 | 0.75

5 | 0.6

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Using the given operations and nine 9s, you can make 1001.

To make 1001 using nine 9s, you can use the following mathematical operations:

1. (9 + 9 + 9) * 9 * 9 - 9 - 9 - 9 = 1001

- Adding three 9s together: (9 + 9 + 9) = 27

- Multiplying the sum by two 9s: (27 * 9 * 9) = 2187

- Subtracting three 9s: (2187 - 9 - 9 - 9) = 2160

- Adding one 9: (2160 + 9) = 2169

- Adding 832 (which is (9 * 9 * 9 + 9 * 9 + 9)) to 2169: (2169 + 832) = 3001

- Subtracting two 9s: (3001 - 9 - 9) = 2983

- Adding nine 9s: (2983 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9) = 1001

Therefore, using the given operations and nine 9s, you can make 1001.

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The numbers to complete the pythagorean triple in the equation (n² - 1)² + k² = (n² + 1)² are B. 35 and 37

A pythagorean triple is a set of 3 numbers that obey the pythagorean theorem.

Given the equation

(n² - 1)² + k² = (n² + 1)², we need to find the remaining numbers generated by the equation when k = 12. So, we proceed as follows/

Since we have the equation

(n² - 1)² + k² = (n² + 1)²

Subtracting (n² + 1)², from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = (n² + 1)² - (n² + 1)²

(n² - 1)² - (n² + 1)² + k² = 0

Now, subtracting k from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = 0

(n² - 1)² - (n² + 1)² + k² - k² = 0 - k²

(n² - 1)² - (n² + 1)² + 0 = - k²

(n² - 1)² - (n² + 1)² = - k²

Using the difference of two squares a² - b² = (a + b)(a - b)

(n² - 1)² - (n² + 1)² = - k²

(n² - 1 + n² + 1)[n² - 1 - (n² + 1)] = - k²

(n² + n² - 1 + 1)[n² - n² - 1 - 1)] = - k²

(2n² + 0)[0 - 2)] = - k²

(2n²)(- 2) = - k²

-4n² = - k²

4n² = k²

n² = k²/4

Taking square root of both sides, we have that

n = √(k²/4)

n = k/2

Since k = 12, we have that

n = 12/2

n = 6

So, the first number is (n² - 1) = (6² - 1)

= 36 - 1

= 35

The second number is (n² + 1) = (6² + 1)

= 36 + 1

= 37

So, the numbers are B. 35 and 37

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Answer:

$3.3

Step-by-step explanation:

4x = 1.32

x = 0.33$ / ft

10x = 10 * 0.33 = 3.3$

Answer:

80

Step-by-step explanation:

Just average the two numbers to get (70+90)/2 = 160/2 = 80

Answer:

Step-by-step explanation:

To find the midpoint between two numbers, you add them together and divide the sum by 2.

In this case, the midpoint between 70 and 90 would be:

(70 + 90) / 2 = 160 / 2 = 80.

Therefore, the midpoint between 70 and 90 is 80.

Answer:

Step-by-step explanation:

Answer:

3.34 * 10^-21 g

Step-by-step explanation:

1.67*10^-24 * 2000 = 1.67 * 10^-24 * 2 *10^3 = 3.34 * 10^-21 g

The equation (-14) + x = 14 is solved by adding 14 to both sides of the equation, resulting in x = 28. This means that 28 is the value of x that satisfies the equation and makes it true.

To solve the equation (-14) + x = 14, we need to isolate the variable x on one side of the equation. Let's go through the steps:

Step 1: Add 14 to both sides of the equation to eliminate the -14 on the left side.

(-14) + x + 14 = 14 + 14

x = 28

The solution to the equation (-14) + x = 14 is x = 28.

In this equation, we start with (-14) on the left side, and we want to determine the value of x that makes the equation true. To do that, we need to isolate x. By adding 14 to both sides of the equation, we cancel out the -14 on the left side, leaving us with just x. On the right side, 14 + 14 simplifies to 28.

Therefore, the solution to the equation is x = 28. This means that if we substitute 28 for x in the original equation, (-14) + 28 will indeed equal 14. Let's verify this:

(-14) + 28 = 14

14 = 14

The left side of the equation simplifies to 14, and the right side is also 14. Since both sides are equal, it confirms that x = 28 is the correct solution to the equation.

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The function that matches the graph is of the form:

4cos((pi x)/7) + 1

Graphs of trigonometric functions are graphs used in representing trigonometric functions.

From these graphs, some basic properties such as Amplitude, phase difference, period and vertical shift can be deduced.

From the given graph in the question, it can be seen that the graph crosses the y-axis at it's amplitude (highest point), so its easier to use the cosine relation.

To calculate the midline M:

Use the formula,

M = (maximum + minimum)/2

= (5 + -3)/2 = 2/2 = 1

Vertical shift: It can be seen from the graph that there is a vertical upward shift of 1 unit. C = 1

Amplitude: Maximum value - vertical shift is:

A = 5 - 1 = 4

Period = spacing between repeating patterns. There are 14 units between each peak (peak when x = -14, next peak when x = 0).

k = 2pi/Period;

So: k = 2pi/14 = pi/7

Therefore y = 4cos(pix/7) + 1 is the function that matches the given graph.

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Answer:

The correct answer is A. As x increases, the rate of change of f(x) exceeds the rate of change of g(x)

Step-by-step explanation:

The population at 4:00 P.M., 27 minutes earlier, is[tex]3^3[/tex] times the initial population P.

To determine the population of a bacteria culture 27 minutes earlier, we need to calculate the population growth from 4:00 P.M. to 4:27 P.M. given that the bacteria culture triples every 5 minutes.

Let's break down the time period into intervals of 5 minutes:

From 4:00 P.M. to 4:05 P.M., the population triples once.

From 4:05 P.M. to 4:10 P.M., the population triples again.

From 4:10 P.M. to 4:15 P.M., the population triples for the third time.

From 4:15 P.M. to 4:20 P.M., the population triples for the fourth time.

From 4:20 P.M. to 4:25 P.M., the population triples for the fifth time.

From 4:25 P.M. to 4:27 P.M., the population undergoes partial growth.

Since the population triples every 5 minutes, we can express the population at 4:27 P.M. as 3^5 times the initial population at 4:00 P.M.

Let's denote the initial population at 4:00 P.M. as P. Then, the population at 4:27 P.M. is [tex]3^5[/tex] * P.

To find the population 27 minutes earlier, we need to reverse the growth from 4:27 P.M. to 4:00 P.M. Since the population triples every 5 minutes, we need to divide the population at 4:27 P.M. by [tex]3^{(27/5).[/tex]

Therefore, the population at 4:00 P.M., 27 minutes earlier, can be calculated as:

Population at 4:00 P.M. = (Population at 4:27 P.M.) / [tex]3^{(27/5)[/tex]

[tex]= (3^{5} * P) / 3^{(27/5)\\\\\\\\= 3^{(25/5)} * P\\= 3^5 * P / 3^2\\= 3^3 * P[/tex]

Hence, the population at 4:00 P.M., 27 minutes earlier, is 3^3 times the initial population P.

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