) Rewrite as an exponential equation.
log8 1/64 =2

(b) Rewrite as a logarithmic equation.
3 0=1

Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

Taking the first and second derivatives of y_p, we get:

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

(r + 2)(r - 1) = 0

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

Taking the first and second derivatives of y_p, we get:

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

Taking the first and second derivatives of y_p, we get:

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

Answer:

a) To find the exponential growth function, we can use the formula:

P(t) = P0 * e^(rt)

Where:

P(t) = the population at time t

P0 = the initial population (in this case, 5.51 million)

e = the mathematical constant e (approximately 2.71828)

r = the annual growth rate (in decimal form)

t = the number of years

Substituting the given values, we have:

P(t) = 5.51 * e^(0.0382t)

b) To estimate the population of the city in 2018, we can substitute t = 6 (since 2018 is 6 years after 2012) into the exponential growth function:

P(6) = 5.51 * e^(0.0382*6) ≈ 6.93 million

Therefore, the estimated population of the city in 2018 is approximately 6.93 million.

c) To find when the population of the city will be 10 million, we can set P(t) = 10 and solve for t:

10 = 5.51 * e^(0.0382t)

e^(0.0382t) = 10/5.51

0.0382t = ln(10/5.51)

t ≈ 11.7 years

Therefore, the population of the city will be 10 million in approximately 11.7 years from 2012, or around the year 2023.

d) To find the doubling time, we can use the formula:

T = ln(2) / r

Where:

T = the doubling time

ln = the natural logarithm

2 = the factor by which the population grows (i.e., doubling)

r = the annual growth rate (in decimal form)

Substituting the given value of r, we have:

T = ln(2) / 0.0382 ≈ 18.1 years

Therefore, the doubling time for the population of the city is approximately 18.1 years.

f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).

An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.

According to the given information:

To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.

First, we find the first derivative of f(x):

f'(x) = (14x)/(7x²+6)²

Then, we find the second derivative of f(x):

f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4

Simplifying the expression, we get:

f''(x) = 28(42x² - 72) / (7x^2+6)³

To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.

Setting f''(x) = 0, we get:

42x² - 72 = 0

Solving for x, we get:

x = ±√(6/7)

These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.

We can use a sign chart to determine the sign of f''(x) on each interval.

Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.

Here is the sign chart for f''(x):

x  |   -∞   | -√(6/7) | √(6/7) |   ∞

f''(x)|   -    |     +      |     -     |   +

From the sign chart, we can see that:

a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).

b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).

c) The inflection points occur at x = -√(6/7) and x = √(6/7).

Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).

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The f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).

What is inflection Point?

An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.

According to the given information:

To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.

First, we find the first derivative of f(x):

f'(x) = (14x)/(7x²+6)²

Then, we find the second derivative of f(x):

f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4

Simplifying the expression, we get:

f''(x) = 28(4x² - 72) / (7x^2+6)³

To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.

Setting f''(x) = 0, we get:

42x² - 72 = 0

Solving for x, we get:

x = ±√(6/7)

These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.

We can use a sign chart to determine the sign of f''(x) on each interval.

Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.

Here is the sign chart for f''(x):

x  |   -∞   | -√(6/7) | √(6/7) |   ∞

f''(x)|   -    |     +      |     -     |   +

From the sign chart, we can see that:

a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).

b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).

c) The inflection points occur at x = -√(6/7) and x = √(6/7).

Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).

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Answer:

...............................

The next entry on the long division would be 0.054, and 0.0054

Long division is a method of dividing two numbers using a step-by-step process. Here's how to perform long division:

Step 1: Write the dividend (the number being divided) and the divisor (the number you're dividing by) in the long division format, with the dividend inside the division symbol and the divisor outside.

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Answer:

Step-by-step explanation:

The distance formula is

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

x1 is a,

x2 is 0,

y1 is 0 and

y2 is b. Fitting those into the formula where they belong:

[tex]d=\sqrt{(0-a)^2+(b-0)^2}[/tex] and

[tex]d=\sqrt{(-a)^2+(b)^2}[/tex]

Since a negative squared is a positive, then

[tex]d=\sqrt{a^2+b^2}[/tex]

which is the second choice down.

Answer: a.     c(n) = 15 + 0.75(n - 10)

b.   15 + 0.75(n - 10) = 2.50(n - 10)=

Simplifying and solving for n, we get:

n = 50

c.  n > 50

Step-by-step explanation:

a. The cost c of the credit union account in terms of the number of checks written n can be expressed as:

c(n) = 15 + 0.75(n - 10)

The first term, 15, represents the monthly cost of the account, and the second term represents the additional cost per check beyond the first 10 free checks.

The cost c of the bank account in terms of the number of checks written n can be expressed as:

c(n) = 2.50(n - 10)

The first term, 0, represents the monthly cost of the account, and the second term represents the additional cost per check beyond the first 10 free checks.

b. We want to find the number of checks for which the bank account is more expensive than the credit union account. Let x be the number of checks that makes the cost of the two accounts equal. Then we have:

15 + 0.75(n - 10) = 2.50(n - 10)

Simplifying and solving for n, we get:

n = 50

So if the number of checks written in a month is greater than 50, the bank account will be more expensive than the credit union account.

c. The solution to the inequality is:

n > 50

This means that the number of checks written in a month must be greater than 50 for the bank account to be more expensive than the credit union account.

The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).

The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).

Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.

To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:

g(x) = 0{-10≤x≤10}

Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].

However, if the formula is something like:

g(x) = 2 * 0{-10≤x+3≤10}

Then we can describe the transformations as follows:

Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.

Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.

Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.

To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

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The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).

The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).

Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.

To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:

g(x) = 0{-10≤x≤10}

Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].

However, if the formula is something like:

g(x) = 2 * 0{-10≤x+3≤10}

Then we can describe the transformations as follows:

Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.

Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.

Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.

To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

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The value of Tanθ is 20/21.

What is Pythagorean Theorem?

A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.

Here, we have

Given: sinθ = -20/29 and that angle terminates in quadrant III.

We have to find the value of tanθ.

Using the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

Sinθ = Perpendicular/hypotenuse

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

Adjacent = - √Hypotenuse² - perpenducualr²

Replace the known values in the equation.

Adjacent = -√29² - (-20)²

Adjacent = -21

Find the value of tangent.

Tanθ = Perpendicular/base

Tanθ = -20/(-21)

Hence, the value of Tanθ is 20/21.

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40% of Canadians read at least 10 books a year.

A percentage is a way of expressing a portion or a part of a whole as a fraction of 100. It is represented by the symbol "%". Percentages are often used to compare different quantities or to describe how much of a total is made up by a specific amount or group. For example, if you score 80 out of 100 on a test, your score can be expressed as 80%, meaning you got 80 out of 100 possible points.

If two out of every five Canadians read at least 10 books a year, then we can write this as a fraction:

2/5

We can multiply this fraction by 100 to turn it to a percentage:

(2/5) x 100 = 40%

Therefore, 40% of Canadians read at least 10 books a year.

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Answer:

They are collinear if they are on the same line

Answer: We can use the two given points to find the equation of the line and then plug in 39 for the calling time to find the corresponding monthly cost.

Let x be the calling time (in minutes) and y be the monthly cost (in dollars). Then we have the following two points:

(x1, y1) = (35, 16.83)

(x2, y2) = (52, 18.87)

The slope of the line passing through these two points is:

m = (y2 - y1) / (x2 - x1) = (18.87 - 16.83) / (52 - 35) = 0.27

Using point-slope form with the first point, we get:

y - y1 = m(x - x1)

y - 16.83 = 0.27(x - 35)

Simplifying, we get:

y = 0.27x + 7.74

Therefore, the monthly cost for 39 minutes of calls is:

y = 0.27(39) + 7.74 = $18.21

Step-by-step explanation:

Therefore, the answer is (A) 7x-9 when it is given that function F(x) = 4x - 8 and g(x) = -3x + 1.

In mathematics, a function is a relationship between two sets of values, where each input (or domain element) is associated with a unique output (or range element). In other words, a function is a rule or a process that takes an input (or inputs) and produces a corresponding output. Functions can be expressed using various mathematical notations, such as algebraic formulas, graphs, tables, or even verbal descriptions. They are widely used in many fields of mathematics, science, engineering, economics, and computer science, to model and solve problems that involve relationships between variables or quantities.

Here,

To find (f - g)(x), we need to subtract g(x) from f(x), so we get:

(f - g)(x) = f(x) - g(x)

Substituting the given functions, we get:

(f - g)(x) = (4x - 8) - (-3x + 1)

Simplifying the expression by distributing the negative sign, we get:

(f - g)(x) = 4x - 8 + 3x - 1

Combining like terms, we get:

(f - g)(x) = 7x - 9

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Answer:

3.14 cm^2

Step-by-step explanation:

1. Find radius:

If diameter is 2, divide it by 2 to get radius = 1

2. Find formula:

A=πr^2

3. Plug in:

A = π(1)^2

4. Solve (multiply):

A = π(1)^2:

3.14159265359

Or

3.14 cm^2

Answer:

3.14 cm^2

Step-by-step explanation:

A=[tex]\pi[/tex]r^2

r=2

2/2=1

A=[tex]\pi[/tex](1)^2

=[tex]\pi[/tex]1

≈3.14x1

≈3.14cm^2

Answer:

1. Using the kinematic equation h(t) = -16t^2 + v0t + h0, where h0 is the initial height, v0 is the initial velocity, and t is time, we have:

h(t) = -16t^2 + 72t + 4

To find the maximum height, we need to find the vertex of the parabolic function h(t). The t-coordinate of the vertex is given by t = -b/2a, where a = -16 and b = 72:

t = -b/2a = -72/(2(-16)) = 2.25 seconds

To find the maximum height, we substitute t = 2.25 seconds into the equation for h(t):

h(2.25) = -16(2.25)^2 + 72(2.25) + 4 = 82 feet

The range of the function h(t) is [4, 82], since the T-shirt starts at a height of 4 feet and reaches a maximum height of 82 feet before falling back to the ground.

2. Using the same kinematic equation as before, we have:

h(t) = -16t^2 + 80t + 3

To find the maximum height, we again need to find the vertex of the parabolic function h(t). The t-coordinate of the vertex is given by t = -b/2a, where a = -16 and b = 80:

t = -b/2a = -80/(2(-16)) = 2.5 seconds

To find the maximum height, we substitute t = 2.5 seconds into the equation for h(t):

h(2.5) = -16(2.5)^2 + 80(2.5) + 3 = 80 feet

The range of the function h(t) is [3, 80], since the T-shirt starts at a height of 3 feet and reaches a maximum height of 80 feet before falling back to the ground.

Step-by-step explanation:

The degree measure of the angles are;

1. 5π/3 = 300°

2 3π/4 = 135°

3. 5π/6 = 150°

4. -3π/2 = 90°

A degree is a unit of measurement which is used to measure circles, spheres, and angles while a radian is also a unit of measurement which is used to measure angles.

A circle has 360 degrees which are its full area while its radian is only half of it which is 180 degrees or one pi radian.

therefore π = 180°

1. 5π/ 3 = 5×180/3 = 300°

2. 3π/4 = 3× 180/4 = 540/4 = 135°

3. 5π/6 = 5×180/6 = 150°

4. - 3π/2 = -3 × 180/2 = -270° = 90°

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The truth table for (A ⋁ B) ⋀ ~(A ⋀ B) is:

A   B   (A ⋁ B) ⋀ ~(A ⋀ B)

0   0                0

0    1                0

1     0               0

1     1                0

A truth table is a table that displays all possible combinations of truth values (true or false) for one or more propositions or logical expressions, as well as the truth value of the resulting compound proposition or expression that is created by combining them using logical operators like AND, OR, NOT, IMPLIES, etc.

The columns of a truth table reflect the propositions or expressions themselves as well as the compound expressions created by applying logical operators to them. The rows of a truth table correspond to the various possible combinations of truth values for the propositions or expressions.

To complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B), we need to consider all possible combinations of truth values for A and B.

A   B   A ⋁ B   A ⋀ B   ~(A ⋀ B)   (A ⋁ B) ⋀ ~(A ⋀ B)

0   0      0            0             1                      0

0    1       1            0             1                      0

1    0       1            0             1                      0

1    1        1             1             0                     0

So, the only case where the expression is true is when both A and B are true, and for all other cases it is false.

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The range is the best measure of variability for this data, and its value is 4.

The line plot displays the number of roses purchased per day at a grocery store, with the data values ranging from 0 to 4 (since there are no dots above 4).

The best measure of variability for this data is the range, which is the difference between the maximum and minimum values in the data set. In this case, the minimum value is 0 and the maximum value is 4, so the range is:

Range = Maximum value - Minimum value = 4 - 0 = 4

Therefore, the range is the best measure of variability for this data, and its value is 4.

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An observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.

To determine the outliers in a dataset using the five-number summary, we need to calculate the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1).

IQR = Q3 - Q1

Where

Q1 = 52

Q3 = 63

So, we have

IQR = 63 - 52

IQR = 11

An observation is considered an outlier if it is:

Below Q1 - 1.5 × IQR

Above Q3 + 1.5 × IQR

Substituting the values, we get:

Below 52 - 1.5 × 11 = 35.5

Above 63 + 1.5 × 11 = 79.5

Therefore, an observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.

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a. TRUE, b. TRUE, c. TRUE, d. TRUE, e. TRUE, f. FALSE, g. TRUE

a. TRUE: The probability of an event can never be negative, and can at most be equal to 1, which represents certainty.

b. TRUE: The sample space is the set of all possible outcomes of an experiment, and the probability of the sample space is always equal to 1, since one of the outcomes must occur.

c. TRUE: If the events in a sequence are disjoint, then they have no outcomes in common, so the probability of the union of the events is the sum of the probabilities of the individual events.

d. TRUE: If one event is a subset of another event, then the probability of the subset is less than or equal to the probability of the superset. This follows from the fact that the subset contains fewer outcomes than the superset.

e. TRUE: The probability of the union of two events is the probability of the first event plus the probability of the second event, minus the probability of the intersection of the events, which is the probability of both events occurring together. This is known as the inclusion-exclusion principle.

f. FALSE: The formula (P(A ∩ B) = P(A)P(B)) only applies to independent events, but not all independent events are mutually exclusive. For example, if A is the event of rolling a 4 on a die, and B is the event of rolling an even number, then A and B are independent, but not mutually exclusive.

g. TRUE: If two events are mutually exclusive, then they have no outcomes in common, so the occurrence of one event tells us that the other event cannot occur. This dependence means that the events are not independent.

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The control group for the experiment would be option C: some plants will receive no sunlight.

The control group in an experiment is the group that does not receive the treatment or intervention being tested, so that the effects of the treatment can be compared to a baseline or reference point. In this experiment, the treatment being tested is the provision of sunlight as an energy source for plant growth.

Therefore, the control group should not receive sunlight, so that the effects of sunlight can be compared to the baseline of plant growth without sunlight.

Therefore, the control group for the experiment would be option C: some plants will receive no sunlight.

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After 6 fluid ounces , Naya uses 4 glasses as a result.

A unit of weight is an ounce. There are various kinds of ounces, including avoirdupois, troy, and fluid ounces. One sixteenth of a pound is equivalent to one avoirdupois ounce .  A troy ounce, often known as an apothecaries' measure, is equivalent to 480 grains or one-twelfth of a pound. A volume unit is a fluid ounce. 1/8 of a cup, 2 tablespoons, or 6 teaspoons make to one fluid ounce

In Naya's pitcher, there are three glasses of salted lassi.

She fills each glass with six fluid ounces of lassi.

By translating cups to fluid ounces and dividing the entire amount of lassi by the amount put into each glass, we can determine how many glasses Naya uses if she consumes all of the lassi.

8 fluid ounces make constitute a cup.

Consequently, 3 cups equal 24 fluid ounces (3 x 8).

24 divided by 6 results in:

4 glasses are equal to 24/6.

Naya uses four glasses as a result.

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The polynomial in factor form is y(x) = - (1 / 6) · (x + 3) · (x + 1) · (x - 2) · (x - 3).

In this problem we find a representation of polynomial set on Cartesian plane, whose expression is described by the following formula in factor form:

y(x) = a · (x - r₁) · (x - r₂) · (x - r₃) · (x - r₄)

Where:

Then, by direct inspection we get the following information:

y(0) = - 3, r₁ = - 3, r₂ = - 1, r₃ = 2, r₄ = 3

First, determine the lead coefficient:

- 3 = a · (0 + 3) · (0 + 1) · (0 - 2) · (0 - 3)

- 3 = a · 3 · 1 · (- 2) · (- 3)

- 3 = 18 · a  

a = - 1 / 6

Second, write the complete expression:

y(x) = - (1 / 6) · (x + 3) · (x + 1) · (x - 2) · (x - 3)

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The two consecutive natural numbers whose sum of squares is 181 are 9 and 10

Let's assume that the two consecutive natural numbers are x and x+1. Then, we can write an equation based on the given information:

x² + (x+1)² = 181

Expanding the equation:

x² + x² + 2x + 1 = 181

Combining like terms:

2x² + 2x - 180 = 0

Dividing both sides by 2:

x² + x - 90 = 0

Now, we can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = 1, and c = -90

x = (-1 ± √(1 + 360)) / 2

x = (-1 ± √(361)) / 2

x = (-1 ± 19) / 2

We discard the negative value, as it does not correspond to a natural number:

x = 9

Therefore, the two consecutive natural numbers are 9 and 10, and their sum of squares is 81 + 100 = 181.

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The range is expressed in interval notation as (-1, ∞)

To find the linear function f(x), let us use the table given.

A linear function with the following equation that passes through the points (a, g(a)) and (b, g(b)):

[tex]g(x) - g(a) = \frac{g(b)-g(a)}{b-a} (x-a)[/tex]

Because the g(x) line crosses through points (-6, 14) and (-3, 8), we have:

a = -6, g(a) = 16, b = -3 and, g(b) = 10

Therefore g(x)

[tex]g(x) - (16) = \frac{10-16}{-3-(-6)} (x--(6))\\g(x) - 16 = \frac{10-16}{-3+6} (x+6)\\g(x) - 16 = \frac{-6}{3} (x+6)\\g(x) - 16 = -2(x +6)\\g(x) = -2x -12+16\\g(x) = -2x+4[/tex]

now find the (f+g)(x).

[tex](f+g)(x) = f(x) + g(x) = x^{2} + 2x -5 -2x + 4\\(f+g)(x) = f(x) + g(x) = x^{2} - 1\\[/tex]

(f+g)(x) = (x-1)(x+1), therefore we get the values x = 1 and x = -1

The parabola's vertice has x-coordinate 0 (the midway between the roots). At x = 0, we get:

[tex](f +g)(x) = 0^{2} - 1 = -1[/tex]

Furthermore, because the coefficient of [tex]x^{2}[/tex] is 1, which is positive, this function indicates a parabola that has been opened upwards.

As a result, the function's minimal value is y = -1. As a result, the function's range includes all real numbers equal to or greater than -1.

The range is expressed in interval notation as (-1, ∞)

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For the given equation we have horizontal length of 5 and vertical width of 4 units. The graph that depicts this hyperbola is: Option B.

A hyperbola is a particular kind of conic section in mathematics, which is a curve created by the intersection of a cone and a plane. The collection of all points in a plane that have a constant difference between them and two fixed points, known as the foci, is referred to as a hyperbola. The distance between the foci is referred to as this consistent difference and is represented by the symbol 2a.

Two separate branches that are mirror reflections of one another make up a hyperbola. The vertices are the two locations where the two branches of a hyperbola are joined.

For the given equation of the parabola, the value of a and b = 5 and 4.

Thus, we have horizontal length of 5 and vertical width of 4 units

The graph that depicts this hyperbola is: Option B.

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The complete question is:

Therefore , the solution of the given problem of probability comes out to be a)1/78 ,b)65/624 ,c)1/4 ,d)0 and e)12/13.

The basic goal of any considerations technique is to assess the probability that a statement is accurate or that a specific incident will occur. Chance can be represented by any number range between 0 and 1, where 0 normally indicates a percentage but 1 typically indicates the level of certainty. An illustration of probability displays how probable it is that a specific event will take place.

Here,

a.

P(Bert gets a Jack and Ernie rolls a five) = P(Bert gets a Jack) * P(Ernie rolls a five)

= (4/52) * (1/12)

= 1/78

b.

P(Bert gets a heart and Ernie rolls a number less than six) = P(Bert gets a heart) * P(Ernie rolls a number less than six)

= (13/52) * (5/12)

= 65/624

c.

P(Bert gets a face card and Ernie rolls an even number) = P(Bert gets a face card) * P(Ernie rolls an even number)

= (12/52) * (6/12)

= 1/4

d.

P(Bert gets a red card and Ernie rolls a fifteen) = 0

e.

Ernie rolls a number that is not twelve, and Bert draws a card that is not a Jack:

A regular 52-card deck contains 48 cards that are not Jacks,

so the likelihood that Bert will draw one of those cards is 48/52, or 12/13.

On a 12-sided dice with 11 possible outcomes,

Ernie rolls a non-12th-number (1, 2, 3, etc.).

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