Difference Quotient Problem

The evaluation of the limits of the piecewise function at the specified intervals are as follows;

[tex]\lim\limits_{x\to2^+}\frac{f(x) -1}{f(x + 4)} = -2[/tex]

[tex]\lim \limits_{x\to 1^-}f(f(x) + 1) = 11[/tex]

[tex]\lim\limits_{h\to 0}\frac{f(6+h) - f(6)}{h} = 2[/tex]

A piecewise function is a function with rules or definition of the function based on the interval of the function.

The equation representing the piece wise function at x → 2⁺, can be found as follows;

Points on the graph of the function are; (3, 2), and (4, 1)

Slope of the graph of the function = (1 - 2)/(4 - 3) = -1

Equation of the function is; y - 2 = -1·(x - 3) = 3 - x

y = 3 - x + 2 = 5 - x

y = 5 - x

y = f(x) = 5 - x

f(x) - 1 = 5 - x - 1 = 4 - x

f(x) - 1 = 4 - x

f(x + 4) = 5 - (x + 4) = 5 - x - 4 = 1 - x

f(x - 4) = 1 - x

[tex]\frac{f(x) - 1}{f(x + 4)} = \frac{4 - x}{1 - x}[/tex]

The coordinate points on the piecewise function at x → 1⁻ are; (0, 3), and (1, 4)

The slope of the graph is; (4 - 3)/(1 - 0) = 1

The equation is; y - 3 = x

y = f(x) = x + 3

f(f(x) + 1) = f((x + 3) + 1) = f((x + 3) + 3 + 1) = f(x + 7)

f(x + 7)  = (x + 7 + 3) = x + 10

The points on the piecewise function at x = 6 are; (5, 2), and (7, 6)

The slope is; (6 - 2)/(7 - 5) = 2

The equation is; y - 2 = 2·(x - 5) = 2·x - 10

y = 2·x - 10 + 2 = 2·x - 8

y = f(x) = 2·x - 8

f(6 + h) = 2·(6 + h) - 8 = 12 + 2·h - 8 = 4 + 2·h

f(6) = 2 × 6 - 8 = 4

f(6 + h) - f(6) = 4 + 2·h - 4 = 2·h

[tex]\frac{f(6 + h) - f(6)}{h} = \frac{2\cdot h}{h}= 2[/tex]

The limits of the function is therefore;

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Answer:

13/24

Step-by-step explanation:

13/11+13= 13/24

The probability that a randomly selected person is male given the person is left handed is 13/200.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes.

Here, total number of outcomes = 200

Number of favorable outcomes = 13

Now probability of an event = 13/200

Therefore, the probability that a randomly selected person is male given the person is left handed is 13/200.

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Answer:

2026 to maximize the present value function.

Step-by-step explanation:

To find the year in which the timber should be harvested to maximize the present value function, we need to find the time t that maximizes the present value function A(t). We can begin by finding A(t):

A(t) = V(t)e^(-0.09t)

A(t) = 140,000e^(0.72√t) * e^(-0.09t)

A(t) = 140,000e^(0.72√t - 0.09t)

To maximize A(t), we need to find the critical points of A(t). We can do this by taking the derivative of A(t) and setting it equal to zero:

A'(t) = 140,000(0.36/√t - 0.09)e^(0.72√t - 0.09t) = 0

Simplifying this equation, we get:

0.36/√t - 0.09 = 0

0.36/√t = 0.09

√t = 4

t = 16

Therefore, the critical point of A(t) occurs at t = 16 years.

We can check that this is a maximum by taking the second derivative of A(t) and evaluating it at t = 16:

A''(t) = 140,000(-0.648/t^3 - 0.243/√t + 0.081)e^(0.72√t - 0.09t)

A''(16) = 140,000(-0.648/16^3 - 0.243/4√16 + 0.081)e^(0.72√16 - 0.09(16))

A''(16) ≈ -4,980.4

Since the second derivative is negative, we can conclude that t = 16 years corresponds to a maximum for the present value function A(t).

Therefore, the timber should be harvested in the year 2010 + 16 = 2026 to maximize the present value function.

Answer:

Step-by-step explanation:

whatb you need

Answer: 7: 29 stickers for each friend

8: 412 seats in each section

Step-by-step explanation:

We will have to divide 145 (stickers) by 5 (friends) to get 29

We will have to divide 2472 by 6 to get 412

7.) 145 ÷ 5 = 29 → Each friend will get 29 stickers

8.) 2,472 ÷ 6 = 412 → There are 412 seats in each second.

I divided for both problems using the long division method and multiplied which is the inverse operation, to check my work.

The volume of the largest right circular cone that can be inscribed in a sphere of radius 18 cm is approximately 11622.85 cubic centimeters.

To find the largest right circular cone that can be inscribed in a sphere of radius 18 cm, we need to find the cone with the largest possible volume that can fit inside the sphere.

Let's assume that the apex of the cone is at the center of the sphere. Since the cone is a right circular cone, the base of the cone will lie on the surface of the sphere, forming a circle with radius equal to the radius of the sphere, which is 18 cm.

Let's call the height of the cone "h" and the radius of the base of the cone "r". Then we can use the Pythagorean theorem to relate the height, radius, and slant height (which is equal to the radius of the sphere):

r^2 + h^2 = (18 cm)^2

The volume of a cone can be expressed as V = (1/3)πr^2h. We want to maximize this expression subject to the constraint above. We can use the constraint to eliminate one of the variables in the volume expression, then differentiate with respect to the remaining variable and set the derivative equal to zero to find the maximum.

From the constraint, we can solve for h^2:

h^2 = (18 cm)^2 - r^2

Substituting into the volume expression, we get:

V = (1/3)πr^2(18 cm - sqrt(r^2 + h^2))

Simplifying this expression, we get:

V = (1/3)πr^2(18 cm - sqrt((18 cm)^2 - r^2))

Differentiating with respect to r, we get:

dV/dr = (1/3)π(36 cm)r - (1/3)πr^3/sqrt((18 cm)^2 - r^2)

Setting this equal to zero and solving for r, we get:

r = (18 cm)/sqrt(2)

Substituting this value of r back into the expression for h, we get:

h = (18 cm)/sqrt(2)

Finally, substituting these values of r and h into the expression for the volume, we get:

V = (1/3)π((18 cm)/sqrt(2))^2((18 cm) - sqrt(((18 cm)/sqrt(2))^2 + ((18 cm)/sqrt(2))^2))

Simplifying this expression, we get:

V ≈ 11622.85 cubic centimeters

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The mean number of employees who work on a given day is less than the median number of employees who work on a given day and option C is correct.

In statistics, we frequently choose a representative value that roughly describes the full collection to represent a set of data. The label "measure of central tendency" denotes that this representative value is one around which the data is centred. The mean, median, and mode are these central tendencies.

Mean = sum of all values/frequency

Mean = (20(0.01) + (21(0.1)) + (22(0.1)) + (23(0.4)) + (24(0.3))

mean is 22.7.

The median id the value of X for which P(X<x) is greater than or equal to 0.5 and P(X>x) is greater than or equal to 0.5. Therefore, median is 23.

Mode is the value of X having maximum probability. Therefore, mode is 23.

Hence, the mean number of employees who work on a given day is less than the median number of employees who work on a given day and option C is correct.

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The true statement on the height of the ball when thrown from a building, given the equation, can be found to be A. The ball reaches the ground in 3 seconds.

The equation given of the ball being thrown from a building, gives us the height when solved.

The height of the ball when it reaches the ground is 0 feet so we can insert this in the formula to find the time:
0 = - 5 ( t - 3 ) ( t + 5 )

We can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we can set each factor equal to zero and solve for t:

t - 3 = 0 or t + 5 = 0

Solving for t in the first equation, we get:

t - 3 = 0

t = 3

Seeing as t is 3 seconds the first option is correct.

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Answer:

Step-by-step explanation:

6 is the rate of change and 1/4 is the initial value

80 cubic inches is the volume of the given prism'

The given figure is made of triangular prisms. The formula for calculating the volume of rectangular prism is expressed as:

Volume = length * width * height

The given composite figure is made of 2 prisms. The volume is calculated as:

V = (5*1*6) + (2*5*5)

V = 30. + 50

V = 80 cubic inches

Hence the required volume of the figure is 80 cubic inches

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The standard deviation is 1.32

We can use the binomial distribution to calculate the probability of making a certain number of shots. The formula for the standard deviation of a binomial distribution is:

σ = √(np(1-p))

Where σ is the standard deviation, n is the number of trials (in this case, 10), p is the probability of success on each trial (0.39, or 39%), and (1-p) is the probability of failure on each trial (0.61, or 61%).

So, plugging in our values:

σ = √(10 * 0.39 * 0.61)

σ ≈ 1.32

In other words, if the player takes 10 free throws, we would expect her to make between 2 and 6 of them about 68% of the time.

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The equation of the midline for the function f(x) = 12sin(x) + 6 is y = 12. The midline of a periodic function is a horizontal line that represents the average value of the function over one period.

For a sinusoidal function, the midline is the line that passes through the center of the graph, or the average of the maximum and minimum values of the function.

To find the midline equation for the function f(x) = 12sin(x) + 6, we first need to find the maximum and minimum values of the function. The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. In this case, the amplitude is 12, so the maximum value is 12+6=18, and the minimum value is 6-12=-6.

The midline of the function is the line halfway between the maximum and minimum values, or at a height of (18 - 6)/2 + 6 = 12. Therefore, the equation of the midline is y = 12. This means that the function oscillates above and below this line by a maximum of 12 units.

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Sara need 9/4 ounces of chopped nuts

Sara wants to make dinner for herself
The recipe she will use calls for 13 1/2 ounces

The recipe feeds 6 people

The number of ounces of chopped nuts that Sara needs can be calculated as follows

13 1/2

= 27/2 ÷ 6

= 27/2 × 1/6

= 27/12

= 9/4

Hence Sara needs 9/4 ounces of chopped nuts for one serving

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The values of m for the given inequality should be in the interval

(-∞, -18/19).

What is meant by inequality?

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa. Literal inequalities are relationships between two algebraic expressions that are expressed using the inequality symbols. Typically, we use the "not equal" sign to indicate that two values are not equal. However several inequalities are utilised to compare the numbers, whether it is less than or higher than.

Given the inequality,

8m + 95 < -87m +5

We are asked to solve the inequality.

this means we have to find the value of m.

8m + 95 < -87m +5

8m + 87m < 5 - 95

95m < -90

m < -90/95

m< -18/19

Therefore the values of m for the given inequality should be in the interval (-∞, -18/19).

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Answer:

The trigonometric function that models the ocean tide for a period of 12 hours is f(x) = 6 sin (2πx/12).

Using the given ratio, we can see that there are 48 teachers.

Here we know that the ratio of teachers to male learners to female learners is:

1:7:8

So for each teacher there are 7 female learners and 8 male learners, for a total of 15.

So we can simplify this and say that there are 15 learners for each teacher.

Then if there are 720 learners, the number of teachers is:

720/15 = 48

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Answer:

Step-by-step explanation:

Part A:

To create the table, we substitute each of the given values of x into the equation and solve for y:

x y = -1025x + 30,750

0         30,750

5         25,375

8         21,050

10         18,725

30         -7,500

Part B:

The altitude after 5 minutes can be found by substituting x=5 into the equation:

y = -1025(5) + 30,750 = 25,375

The altitude after 30 minutes can be found by substituting x=30 into the equation:

y = -1025(30) + 30,750 = -7,500

After 5 minutes, the altitude of the airplane is 25,375 feet. After 30 minutes, the altitude of the airplane is -7,500 feet, which means the airplane is on the ground. This is because the altitude is decreasing by 1,025 feet every minute, and after 30 minutes, the altitude has decreased to 0 feet, which is the ground level.

Part C:

The ordered pair (0, 30,750) represents the initial value, where x=0. The initial value represents the altitude of the airplane before it started descending. In this problem, the initial value of 30,750 feet represents the altitude of the airplane when it was at cruising altitude.

Part D:

The rate of change in this equation is -1025. The rate of change represents the speed at which the altitude of the airplane is changing per minute. In this problem, the rate of change of -1025 feet per minute means that the altitude of the airplane is decreasing by 1025 feet every minute it descends.

A + A(9%) = A(1.09) is matching with the amount Jess owes Julie from borrowing A with 9% interest.

What is interest?

In the study of finance and economics, interest is the predetermined amount that a borrower or deposit-taking financial institution pays to a lender or depositor above and beyond the principal amount. It differs from any fees the borrower may be required to pay the lender or another entity.

Suppose here A is 100

We put the value of A in the equation

A + A(9%) = A(1.09)

100+ 100*9/100 = 100*1.09

Or,100+9=100*109/100

Hence the correct answer is D, amount Jess owes Julie from borrowing A with 9% interest.

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Answer:

Step-by-step explanation:

deuyfheuf

A quadratic equation in standard form to represent the data in the table is as follows;

y = 0.5x² - 4x + 9

In order to determine a quadratic equation in standard form for the line of best fit that models the data points described above, we would use a scatter plot (graphing calculator).

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display a quadratic equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the data set, a quadratic equation for the line of best fit is given by:

y = 0.5x² - 4x + 9

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The value of x is 2 or 10 and the value of length RA is 96

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length. . This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles.

If RA = MA

x²-4x = 8x -20

x²-4x-8x +20 = 0

x²-12x +20 = 0

x²-10x -2x +20 = 0

(x²-10x)(-2x+20) = 0

x( x -10) -2(x-10) = 0

(x-2)(x-10) = 0

x-2 = 0

or x-10 = 0

x = 2 or 10

therefore RA = x²-4

when x = 2

RA = 0

when x = 10

RA = 100-4

= 96

therefore the value of length of RA is 96

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Answer:

181

Step-by-step explanation:

To simplify this expression, we need to apply the order of operations (PEMDAS):

First, we evaluate the expression inside the parentheses: 9 ÷ 3 = 3.

Then, we add 6 + 3 = 9.

Finally, we divide 1,629 by 9 to get the answer:

1,629 ÷ 9 = 181.

Therefore, 1,629 ÷ (6 + 9 ÷ 3) simplifies to 181.

to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below.

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{70})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{100}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{100}-\stackrel{y1}{70}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{2}}} \implies \cfrac{ 30 }{ 2 } \implies 15[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{70}=\stackrel{m}{ 15}(x-\stackrel{x_1}{2}) \\\\\\ y-70=15x-30\implies {\Large \begin{array}{llll} y=15x+40 \end{array}}[/tex]

The expression into each letter space (a,b,c,d) to create an equation that is true for all values of x is X(x+2),X,(X+2),X^2.

Expression in mathematics is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

We are given that;

The equation with a,b,c,d=(3/x+2)+(4/x)+(2/x^2)=(3(a)+4(b)+2(c)/d)

Now,

=(3/x+2)+(4/x)+(2/x^2)

=3/x + 2x/2 + 4/x + 2/x^2

=(3+2x)/x+4/x+2/x^2

Therefore, the answer of the expression will be X(x+2),X,(X+2),X^2.

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The student's grade point average is 2.32.

The student will not be admitted to the business college.

It is the average value of the set given.

It is calculated as:

Mean = Sum of all the values of the set given / Number of values in the set

We have,

(a)

Total grade points earned

= (6 x 4) + (15 x 3) + (31 x 2) + (8 x 1)

= 24 + 45 + 62 + 8

= 139

Total credit hours.

= 6 + 15 + 31 + 8

= 60

Now,

The student's grade point average:

= Total grade points earned / Total credit hours taken

= 139 / 60

= 2.32

(b)

2.32 < 2.5

This means,

The student's grade point average is lower than the required 2.5 GPA for admission to the business college.

Thus,

The student's grade point average is 2.32.

The student will not be admitted to the business college.

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Answer:

D. [If D is saying "x = 4"]

Step-by-step explanation:

We can start solving the equation 216 = 62x - 1 by adding 1 to both sides to isolate the term with x:

216 + 1 = 62x

Simplifying the left side, we get:

217 = 62x

To solve for x, we can divide both sides by 62:

217/62 = x

Using a calculator, we can evaluate the quotient:

3.5 = x

Therefore, the solution to the equation 216 = 62x - 1 is x = 3.5.

Since none of the answer choices match this result exactly, we can round 3.5 to the nearest integer. Rounding 3.5 up to the nearest integer gives us x = 4.

The quadratic function has its solution to be x = 4/9 and x = -4/9

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

f(x) = 81x^2 - 16

Express the expression as difference of two squares

So, we have the following representation

f(x) = (9x)^2 - 4^2

Apply the difference of two squares rule

So we have

f(x) = (9x - 4)(9x + 4)

This gives

(9x - 4)(9x + 4) = 0

So, we have

9x = 4 and 9x = -4

Evaluate

x = 4/9 and x = -4/9

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A linear relationship between two quantities can be modeled using a mathematical function.

Function is the process or state of instruction that text inputs performance is specific tax and produce an output functional key components of programming language allowing the quarters to create complex commands with simple instruction for example a function can be used to add two numbers round together or to generate a random number function can also be combined to create more complex sequence of instruction.

The form of the function is generally expressed as y = mx + b, where y is the dependent variable, m is the slope of the line, x is the independent variable, and b is the y-intercept. The slope of the line, m, can be calculated from two points on the line using the formula m = (y2-y1)/(x2-x1). The y-intercept, b, can be calculated by substituting any point into the equation y = mx + b and solving for b.

Once the slope and y-intercept are known, the equation can be used to predict the value of the dependent variable given a certain value of the independent variable. For example, if the equation is y = 3x + 4, then for any given value of x, the corresponding value of y can be calculated. For x = 5, the corresponding value of y is 19 (5 x 3 + 4 = 19). This equation can also be used to graph the linear relationship between the two variables; by plotting the equation, a straight line is produced.

By using a function to model a linear relationship between two quantities, it is possible to calculate the relationship between them and predict values for either one given the other. This can be a useful tool for understanding the data and for making predictions about the data.

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The runners can finish in 362880 possible results

What is Permutation?

Permutation is the different number of arrangements that can be formed by taking r things from the n available things.

The formula  n! = 1 × 2 × 3 × 4 × .......× n.

To get the order the runners could possibly finish,

= 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

= 326880

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The description corresponds to an experiment.

Mathematics is the sciences and study of quality, structure, space, and change. Mathematician's are seek out patterns, formulate new conjectured, and establish truth by the  rigorous seduction from appropriately choden axioms and definitions.

An experiment is a type of scientific research method in which a researcher manipulates one or more variables and measures the effects of the manipulation on other variables. In this example, the researcher divided the thirty university students into two groups and manipulated one variable (free tutoring in mathematics) to measure the effects on another variable (final mathematical examination scores). Therefore, this is an experiment.

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Answer:

15.81 units

Step-by-step explanation:

Let the hypotenuse of the triangle be x.

Let us use the Pythagorean theorem to find x.

x² = 9² + 13²

x² = 81 + 169

x² = 250

x = √250

x = 15.81 units