Need help with these problems

Answer:

[tex]\frac{8}{3}[/tex]

Step-by-step explanation:

We know that

Line q's slope = [tex]-\frac{3}{8}[/tex]

AND we know that line q is perpendicular to line p.

When a line is perpendicular to another line, this means that the slope is the opposite reciprocal of the other slope.

This means that we simply have to flip our fraction and change the sign.

Thus, the slope is [tex]\frac{8}{3}[/tex]

Expression B has the least value if  g and h are whole numbers, and

g < h.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Given that,

g < h

To determine which expression has the least value, we need to simplify each expression as much as possible and compare the results.

First, let's simplify expression A:

A = (h + g) * (h - g)

 = h*h - g*g

Next, let's simplify expression B:

B = h*h- 2hg + g*g

Finally, let's simplify expression C:

C = h*h + 2hg + g*g

Now we can compare the expressions. We know that g < h, so g*g < h*h. Therefore, the smallest value will be produced by the expression with the smallest coefficient for the h*h term.

A has a coefficient of 1 for the h*h term, while B and C have coefficients of -2h and 2h, respectively. Since h is positive, -2h is the smallest coefficient, so expression B has the smallest value.

Therefore, expression B has the least value.

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In this case, it means that the company fails to identify components that absorb too much humidity, leading to a potential safety hazard for customers.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

The consequence of a Type II error in this context is that the company believes the humidity absorbed is not more than 3%, when in fact it is more than 3%.

This means that the company will fail to detect components that absorb a dangerous level of humidity, and these components will be sold to customers, potentially causing a safety hazard. This is because the null hypothesis in this case is that the humidity level absorbed is 3%, and the alternative hypothesis is that it is more than 3%. A Type II error occurs when the null hypothesis is not rejected, even though it is false, and the alternative hypothesis is true.

Therefore,  In this case, it means that the company fails to identify components that absorb too much humidity, leading to a potential safety hazard for customers.

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1. If there were two people dividing the cost of the gift, each person would spend $180. If there were three people dividing the cost, each person would spend $120. If there were five people dividing the cost, each person would spend $72. If there were ten people dividing the cost, each person would spend $36. If there were one hundred people dividing the cost, each person would spend $3.60.

2. The function that could be used to model the amount each person would spend depending on the number of people contributing to the gift is:

cost per person = total cost / number of people

3. The table will be:

Number of people Amount per person

2. $180

3 $120

5. $72

10. $36

100. $3.60

The graph of the function would be a straight line passing through the points (2, $180), (3, $120), (5, $72), (10, $36), and (100, $3.60).

4. The domain of the function is all positive integers greater than zero, since you cannot have a fractional or negative number of people contributing.

The range of the function is all positive real numbers, since the cost per person can be any positive amount.

The function is decreasing, since the cost per person decreases as the number of people contributing increases.

There is no maximum or minimum value for the cost per person, since it can be any positive amount. The function is continuous, but the number of people contributing must be a discrete value (i.e., a whole number).

As the number of people contributing approaches infinity, the cost per person approaches zero. The y-intercept of the function is the cost of the gift, and the x-intercept is not applicable in this context.

There is a horizontal asymptote at y = 0, since the cost per person approaches zero as the number of people contributing approaches infinity.

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Chile is celebrating her Quinceañera. Hannah knows the perfect gift to buy Chile, but it costs $360. Hannah can't afford to pay for this on her own so thinks about asking some friends to join in and share the cost.

1. How much would each person spend if there were two people dividing the cost of the gift? How much would each person spend if there were three people dividing the cost? Five people? Ten? One hundred?

2. Determine the function that could be used to model the amount each person would spend depending on the number of people contributing to the gift.

3. Use multiple representations to show how the amount each person would contribute to the gift would change depending on the number of people contributing. Describe the connections between the representations.

4. Describe the features of the function based on the context (domain/range, increasing/decreasing, maxima/minima, discrete/continuous, end behavior, intercepts, asymptotes).

The rate of interest when compounded half-yearly is 4.45% and there are 14 conversion periods.

The formula for calculating the compound interest is:

A = P (1 + r/n)^(n*t)

Where:

In this case, P = Rs 5000, r = 9% p.a. and the interest is compounded half-yearly (i.e., n = 2).

To find the number of conversion periods, we need to multiply the number of years by the number of conversion periods per year:

Number of conversion periods = n*t = 2 * 7 = 14

So there are 14 conversion periods in 7 years.

To find the rate of interest when compounded half-yearly, we can rearrange the formula and solve for r:

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

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

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

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

r = n[(A/P)^(1/n*t) - 1]

Substituting the given values, we get:

r = 2[(5000*(1 + 0.09/2)^(27))^(1/(27)) - 1]

= 0.0445 or 4.45%

Therefore, the rate of interest when compounded half-yearly is 4.45%.

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Triangle JKL has a greater area than triangle XYZ, as it has an area of 98 square inches

To determine which triangle has the greater area, we need to use the formula for the area of a triangle, which is half the product of the base and height.

For triangle JKL, with a base of 14 inches and a height of 14 inches, we have:

Area = (14 x 14) / 2 = 98 square inches

For triangle XYZ, with a base of 20 inches and a height of 8 inches, we have:

Area = (20 x 8) / 2 = 80 square inches

Therefore, triangle JKL has a greater area than triangle XYZ, as it has an area of 98 square inches compared to 80 square inches for triangle XYZ. This is because the base and height of triangle JKL are both larger than those of triangle XYZ.

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Draw triangle​ JKL, with a height of 14 inches and a base of 14​inches, and triangle​ XYZ, with a height of 8 inches and a base of 20 inches. Which triangle has the greater​ area? in 100 words

If Sam purchases a new car for $29,500 and the car depreciates at a rate of 13. 25% per year, then the value of the car after 7 years is $11652.50

We can use the formula for exponential decay to find the value of the car after 7 years:

V = P × e^(-rt)

where:

V = value of the car after 7 years

P = initial price of the car

r = annual depreciation rate (as a decimal)

t = time in years

Plugging in the values we get:

V = 29500 × e^(-0.1325 × 7)

V = 29500 × e^(-0.9275)

V = 29500 × 0.395

V = $11652.50

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The polynomial 0 can be written in the form at^2 + bt + c as 0t^2 + 0t + 0. In this case, a, b, and c are all equal to 0.

Therefore, the equation of this line is: x = 8/7.

An equation is a mathematical statement that shows that two expressions are equal. It contains one or more variables, and the goal is to solve for the value(s) of the variable(s) that make the equation true.

Given by the question.

To find the equation of the line passing through points A and B, we need to determine the slope and the y-intercept of the line.

The slope of the line can be found using the formula:

slope = (change in y) / (change in x)

Using the coordinates of A and B, we have:

slope = (0 - 6) / (8 - 0) = -6/8 = -3/4

The y-intercept of the line can be found by substituting the coordinates of point A and the slope into the slope-intercept form of the equation of a line:

y = mx + b

where m is the slope and b are the y-intercept.

Using the coordinates of point, A and the slope we just calculated, we have:

6 = (-3/4) (0) + b

b = 6

Therefore, the equation of the line passing through points A and B is:

y = -3/4 x + 6

(ii) To find the coordinates of point M where the line y = x + 1 intersects the line AB, we need to solve the system of equations:

y = -3/4 x + 6 (equation of line AB)

y = x + 1 (equation of line y = x + 1)

Substituting y = x + 1 into the equation of line AB, we have:

x + 1 = -3/4 x + 6

Solving for x, we have:

x = 8/7

Substituting x = 8/7 into the equation of line y = x + 1, we have:

y = 8/7 + 1 = 15/7

Therefore, the coordinates of point M are:

M (8/7, 15/7)

(iii) The line passing through point M and parallel to the x-axis is a horizontal line with equation y = c, where c is the y-coordinate of point M. Therefore, the equation of this line is:

y = 15/7

(iv) The line passing through point M and parallel to the y-axis is a vertical line with equation x = c, where c is the x-coordinate of point M.

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Answer:

Step-by-step explanation:

The soccer team:

1 2 3 4, 1 2 3 4, 1 2 3 4, 1 2 3 4, 1 2 3 4

The basketball team:

1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5.

[ the bold number is the day of playing ]

Hope this helps.

The absolute maximum of the function f(x,y)=7x^2+2y^2 on the given domain is 28 and the absolute minimum is 0.

The absolute maximum and minimum of the function f(x,y)=7x^2+2y^2 on the closed triangular plate bounded by the lines x=0, y=0, y+2x=2 in the first quadrant can be found by using the method of Lagrange multipliers.

First, we need to find the critical points of the function on the interior of the triangular plate. The gradient of the function is given by ∇f = <14x, 4y>. Setting ∇f = 0, we get x = 0 and y = 0. However, these points are on the boundary of the triangular plate, so they are not critical points on the interior.

Next, we need to find the critical points on the boundary of the triangular plate. We can use the method of Lagrange multipliers to do this. The constraint equation is given by g(x,y) = y + 2x - 2 = 0. The gradient of the constraint equation is given by ∇g = <2, 1>. Setting ∇f = λ∇g, we get the following system of equations:

14x = 2λ
4y = λ
y + 2x - 2 = 0

Solving this system of equations, we get two critical points: (2/3, 2/3) and (2, 0).

Finally, we need to evaluate the function at the critical points and at the corners of the triangular plate to find the absolute maximum and minimum. The function values at these points are:

f(0,0) = 0
f(2/3, 2/3) = 14/3
f(2,0) = 28
f(0,2) = 8

The absolute maximum is 28 and the absolute minimum is 0.

Therefore, the absolute maximum of the function on the given domain is 28 and the absolute minimum is 0.

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From the given information provided, the given sentences in the form of algebraic expressions are as follows:

1. 4 + x

2. 2D

3. 6n + 41

4. x + 17

5. Mary's height - Frank's height

6. Iquan's age / 4

7. 50Arielle's age

8. 75 + x

9. 400 - 2x

10. 11 + x

11. 2d

12. 2x + 10

13. 3v - 40

14. 2t - 60

15. x/15 - 3

16. 5 + x

17. x - 33

18. 2S - 15

19. 60 - 2x

20. 4D

Question - 1. Four plus a number 2. Twice Dharia's age 3. Six times a number plus forty-one 4. The sum of a number and 17 5. The difference between Mary's height and Frank's height 6. The quotient of Aquaman's age and 4 7. The product of Arielle's age and 50 8. Seventy-five increased by a number 9. Four hundred decreased by twice a number Eleven ples more than a number 10. 11. Twice as many dogs 41X 12. A number doubled plus ten 13. A variable tripled less 40. 14. Twice the temperature minus 60 degrees 15. A number divided by fifteen less than 3 16. Five more than a number 17. Thirty-three less than a number 18. Twice Solomon's weight less fifteen pounds 19. The difference between sixty and twice a number 20. The factor of a variable and the coefficient four. Translate the following into algebraic expression.

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Answer:

4 bc if u go by the side where a block counts u count it till that side ends and it will be 4

As given the conversion unit: 1 cup = 0.237 liters. Sameer drinks 0.71 liters of coffee.

Any physical quantity can be measured by comparing it to a recognised standard, and the magnitude is almost always expressed in terms of the reference standard known as a unit.

The FPS system, which measures length, mass, plus time in feet, pounds, and seconds, is one of the three systems that also was utilised for the measurement. The MKS system, which stands for metre, kilogramme, and seconds, has replaced the CGS system in centimetre, gramme, and seconds as the one that is widely used.

Units of Capacity are given as:

System Units      Metric System Units

1 gallon       -       3.79 liters

1 quart        -       0.95 liters

1 cup           -       0.237 liters

Coffee Intake of Sameer:  3 cups

1 cup           -       0.237 liters

Multiply both sides by 3.

1*3 cup           -       0.237*3 liters

3 cup           -        0.711  liters

Thus, Sameer drinks 0.71 liters of coffee (rounded off nearest tenth).

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Complete question:

Units of Capacity are given as:

System Units      Metric System Units

1 gallon       -       3.79 liters

1 quart        -       0.95 liters

1 cup           -       0.237 liters

Sameer usually drinks 3 cups of coffee in the morning. How many liters of coffee does he drink? Round your answer to the nearest tenth.

The missing angle measures are 50 degrees, 100 degrees, and 80 degrees.

x = 50, we can substitute this value into the expressions for the other two angles:

2x = 2(50) = 100

x + 30 = 50 + 30 = 80

When two rays are united at a common point, an angle is created. The two rays are referred to as the arms of the angle, while the common point is referred to as the node or vertex. The symbol stands for the angle. Angle is a derivative of the Latin word "Angulus."

The construction of an angle is a type of geometric shape made by connecting two rays at their termini. Three letters that make up the shape of the angle can alternatively be used to symbolize the angle, with the middle letter indicating the location of the angle (i.e.its vertex).

From the question:

The total of the measures of the angles in any triangle is always 180 degrees, as shown by the characteristics of triangle angles.

This knowledge allows us to construct an equation to account for the missing angle measurements:

x + 2x + 30 = 180

Combining like terms, we get:

3x + 30 = 180

Subtracting 30 from both sides, we get:

3x = 150

Dividing both sides by 3, we get:

x = 50

Knowing that x = 50, we can change the formulas for the remaining two angles to reflect this value:

2x = 2(50) = 100

x + 30 = 50 + 30 = 80

As a result, the missing angle measurements are 50, 100, and 80 degrees.

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The value οf f(-2) fοr this functiοn is 12.

A functiοn is a mathematical rule οr prοcess that assigns a unique οutput value tο each input value. In οrder tο evaluate a functiοn at a specific input value, we need tο knοw the functiοn itself. The functiοn can be given in different fοrms, such as an equatiοn οr a graph, and we use these fοrms tο determine the οutput value fοr a given input value.

Fοr example, if we are given the equatiοn οf a functiοn[tex]f(x) = x^2 - 3x + 2,[/tex] we can evaluate f(-2) by substituting -2 for x in the equation:

[tex]f(-2) = (-2)^2 - 3(-2) + 2 = 12[/tex]

Therefοre, the value οf f(-2) fοr this functiοn is 12. Hοwever, if we are nοt given the equatiοn οr any οther infοrmatiοn abοut the functiοn, we cannοt determine its οutput at a specific input value.

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Complete Qustion:

How to find f (- 2 for a function?

The probability that at most 350 of the 500 Irvine residents selected will be over the age of 60 is 0.9292. The probability that at least 350 of the 500 residents in the sample were over the age of 60 is 0.0708. The probability that between 400 and 475 of the residents were over the age of 60 is 5.88.

The probability that at most 350 of the 500 Irvine residents selected will be over the age of 60 can be found by calculating the z-score for 350 and finding the corresponding probability from a normal distribution table. The z-score for 350 is (350-375)/17 = -1.47. The corresponding probability from a normal distribution table is 0.0708.

The sampling distribution of the sample proportion for the sample size of 500 is a normal distribution with a mean of 0.75 and a standard deviation of  [tex]√[(0.75)(0.25)/500] = 0.017.[/tex]

The probability that at least 350 of the 500 residents in the sample were over the age of 60 can be found by calculating the z-score for 350 and finding the corresponding probability from a normal distribution table. The z-score for 350 is (350-375)/17 = -1.47. The corresponding probability from a normal distribution table is 1 - 0.0708 = 0.9292.

The probability that between 400 and 475 of the residents were over the age of 60 can be found by calculating the z-scores for 400 and 475 and finding the corresponding probabilities from a normal distribution table. The z-score for 400 is (400-375)/17 = 1.47 and the z-score for 475 is (475-375)/17 = 5.88.

The corresponding probabilities from a normal distribution table are 0.9292 and 1.0000, respectively. The probability that between 400 and 475 of the residents were over the age of 60 is 1.0000 - 0.9292 = 0.0708.

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Jermaine bought 9 large shirts for $12, and 12 small shirts for $8 for his 21 employees.

Let's represent the number of large shirts Jermaine bought as "L" and the number of small shirts as "S". We can set up a system of equations based on the information given:

L + S = 21 (equation 1, the total number of shirts is 21)

12L + 8S = 204 (equation 2, the total cost of the shirts is $204)

To solve for L, we need to eliminate S. We can do this by multiplying equation 1 by 8 and subtracting it from equation 2:

12L + 8S = 204

8L + 8S = 168 (multiply equation 1 by 8)

4L = 36

Dividing both sides by 4, we get:

L = 9

Therefore, Jermaine bought 9 large shirts for his 21 employees. We can find the number of small shirts by substituting L = 9 into equation 1:

L + S = 21

9 + S = 21

S = 12

So, by using linear equation system we find that Jermaine bought 9 large shirts and 12 small shirts for his 21 employees.

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The length of the side a , to the nearest 10th of a centimeter is 75cm

It is important to note that the sum triangle theorem states that the sum of the interior angles of a triangle is equal to 180 degrees.

Then, we have;

m< A + m< B+ m < C = 180

substitute the values, we have;

m < A = 180 - 154 - 13

subtract the values

m < A = 13 degrees

Using the sine rule, we have that;

sin A/a = sin B/b = sin C/c

Where; the capital letters are the angles and the small letters are the sides.

We have;

sin A/a = sin C/c

substitute the values

sin 13/a = sin 13/75

cross multiply

a = sin 13 × 75/sin 13

a = 75cm

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Answer:    y=1x+6

Step-by-step explanation:

The equation of a line is y=mx+c

m is the gradient and c is the y intercept.

on the graph the line intercepts the y axis at 6- the y intercept!

The gradient is the difference in y divide by the difference in x.

(0,6) and (4,10)

10-6=4

4-0=

4

4/4 is 1! so the equation is

y=1x+6

Answer:

y = x + 6

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (0, 6) and (x₂, y₂ ) = (4, 10) ← 2 points on the line

m = [tex]\frac{10-6}{4-0}[/tex] = [tex]\frac{4}{4}[/tex] = 1

the line crosses the y- axis at (0, 6 ) ⇒ c = 6

y = x + 6 ← equation of line

Check the picture below.

[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ S(\stackrel{x_1}{-2}~,~\stackrel{y_1}{7})\qquad R(\stackrel{x_2}{-1}~,~\stackrel{y_2}{3}) ~\hfill SR=\sqrt{(~~ -1- (-2)~~)^2 + (~~ 3- 7~~)^2} \\\\\\ ~\hfill SR=\sqrt{( 1 )^2 + ( -4)^2} \implies \boxed{SR=\sqrt{ 17}}[/tex]

[tex]R(\stackrel{x_1}{-1}~,~\stackrel{y_1}{3})\qquad U(\stackrel{x_2}{-1}~,~\stackrel{y_2}{5}) ~\hfill RU=\sqrt{(~~ -1- (-1)~~)^2 + (~~ 5- 3 ~~)^2} \\\\\\ ~\hfill RU=\sqrt{( 0)^2 + ( 2)^2} \implies RU=\sqrt{ 4}\implies \boxed{RU=2} \\\\\\ U(\stackrel{x_1}{-1}~,~\stackrel{y_1}{5})\qquad T(\stackrel{x_2}{2}~,~\stackrel{y_2}{5}) ~\hfill UT=\sqrt{(~~ 2- (-1)~~)^2 + (~~ 5- 5~~)^2} \\\\\\ ~\hfill UT=\sqrt{( 3)^2 + ( 0)^2} \implies UT=\sqrt{ 9}\implies \boxed{UT=3}[/tex]

[tex]T(\stackrel{x_1}{2}~,~\stackrel{y_1}{5})\qquad S(\stackrel{x_2}{-2}~,~\stackrel{y_2}{7}) ~\hfill TS=\sqrt{(~~ -2- 2~~)^2 + (~~ 7- 5~~)^2} \\\\\\ ~\hfill TS=\sqrt{( -4)^2 + ( 2)^2} \implies \boxed{TS=\sqrt{ 20}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{\LARGE Perimeter} }{\sqrt{17}+2+3+\sqrt{20} }~~ \approx ~~ \text{\LARGE 13.6}[/tex]

In 3-dimensional affine Euclidean space R³, a regular 2-surface is a surface that has a well-defined tangent plane at each point on the surface, and a ruled 2-surface is a surface that can be generated by moving a straight line (the generator) along a curve (the directrix) on the surface.

For the quadric surfaces, we have:

Regarding the statement by M. Audin, she means that if we allow lines to be imaginary, then any quadric surface can be generated by moving a straight line (even if it is an imaginary line) along a curve on the surface.

In other words, every quadric surface can be considered a ruled surface if we allow imaginary lines. This is a consequence of the fact that any two points on a quadric surface can be connected by at least one real or imaginary line.

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Answer:

adult ticket £17

child ticket £9

Step-by-step explanation:

Let the cost of an adult ticket be X and child ticket be Y

make two simultaneous equations

2X+3Y=61

3X+5Y=96

solve for each variable by either substitution or elimination method

2X+3Y=61

X=(61-3Y)/2

3[(61-3Y)/2]+5Y=96

(183-9Y)/2 + 5Y=96

183-9Y+10Y=192

Y=9

X=(61-3(9))/2

X=17

The measure of angle NSR.

The situation in the picture is when two chords intersect in a circle in this case the angle NSR is calculated using the formula

angle NSR = 1/2 (arc NR + arc QP)

Plugging in the values

angle NSR = 1/2 (176 + 50)

angle NSR = 1/2 (226)

angle NSR = 113 degrees

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It will take 10.4 years for the value of the car to depreciate to 5500 dollars.

The equation for calculating the number of years until the value of the car is 5500 dollars is:

y = (19700 - 5500) / 0.0925

y = 10.4 years

In order to calculate the number of years until the value of the car is 5500 dollars, we can use the equation y = (19700 - 5500) / 0.0925. This equation uses the initial value of the car (19700) and the desired value (5500), as well as the depreciation rate of 9.25% per year. By solving this equation, we can determine that it will take 10.4 years for the value of the car to depreciate to 5500 dollars.

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The cοrrect statement is "The greater the value οf the base, the greater the functiοn's rate οf grοwth"

An expοnential functiοn is a mathematical functiοn οf fοrm f(x) = abˣ,

where a and b are cοnstants, and x is the variable.

The base b is a pοsitive cοnstant and is typically greater than 1. The expοnent x represents the degree οf grοwth οr decay οf the functiοn.

Expοnential functiοns are cοmmοnly used tο mοdel grοwth οr decay in variοus real-wοrld phenοmena, such as pοpulatiοn grοwth, cοmpοund interest, radiοactive decay, and the spread οf disease.

When b is greater than 1, the functiοn represents expοnential grοwth. As x increases, the functiοn grοws at an increasing rate.

When b is between 0 and 1, the functiοn represents expοnential decay. As x increases, the functiοn decays at a decreasing rate.

Therefοre,

The cοrrect statement is "The greater the value οf the base, the greater the functiοn's rate οf grοwth".

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18 responses would the company get over the course of the first 8 days after the magazine was published.

What is a geometric sequence?

A geometric progression, often referred to as a geometric sequence, is a series of non-zero values where each term following the first is obtained by multiplying the preceding value by a constant, non-zero number known as the common ratio.

Here, we have

Given: A large company put out an advertisement in a magazine for a job opening. On the first day, the magazine was published the company got 125 responses, but the responses were declining by 24% each day.

We apply here geometric sequence.

aₙ = arⁿ⁻¹

where

aₙ = n^{th} term of the sequence

r = is the common ratio

a = the first term of the sequence

a = 125

r = 100% - 24% = 76% = 76/100 = 0.76

aₙ = (125)(0.76)⁸⁻¹

aₙ = 125(0.76)⁷

aₙ = 18

Hence, 18 responses would the company get over the course of the first 8 days after the magazine was published.

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Answer:

463 (to the nearest whole number)

Step-by-step explanation:

We can model the given scenario as a geometric sequence.

The first term, a, is the number of responses the company got on the first day:

The common ratio is the number you multiply by at each stage of the sequence. As the responses are declining by 24% each day, then each day the responses are 76% of the previous day's responses, since 100% - 24% = 76%. Therefore, the common ratio, r, is:

To calculate the total responses the company would get over the course of the first 8 days after the magazine was published, use the Geometric Series formula.

[tex]\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}[/tex]

Substitute a = 125, r = 0.76 and n = 8 into the formula and solve for S:

[tex]\implies S_8=\dfrac{125(1-0.76^8)}{1-0.76}[/tex]

[tex]\implies S_8=\dfrac{125(1-0.111303478...)}{0.24}[/tex]

[tex]\implies S_8=\dfrac{125(0.888696521...)}{0.24}[/tex]

[tex]\implies S_8=\dfrac{111.087065...}{0.24}[/tex]

[tex]\implies S_8=462.862771...[/tex]

[tex]\implies S_8=463[/tex]

Therefore, the total number of responses the company would get over the course of the first 8 days after the magazine was published is 463 to the nearest whole number.

The probability that 11 or more of these consumers have called an 800 or 900 telephone number for information about some product is approximately 0.326.

The number of successes in a certain number of independent trials that all have the same probability of success are described by the discrete probability distribution known as the binomial probability distribution. The probability of success is constant during all trials, and it is used in statistics to describe situations where there are two alternative outcomes (success or failure).

The binomial probability distribution is given as:

P(X ≥ k) = 1 - P(X < k)

Here, p = 0.60, and the sample size is n = 20.

Thus,

P(X ≥ 11) = 1 - P(X < 11)

= 1 - ∑(20 choose x) (0.6)ˣ (0.4)⁽²⁰⁻ˣ⁾ for x from 0 to 10

P(X ≥ 11) = 0.326

Hence, the probability that 11 or more of these consumers have called an 800 or 900 telephone number for information about some product is approximately 0.326.

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The complete question is:

If the student correctly answers a question, the probability that the student really knew the correct answer is  0.941.

The probability that the student knows the answer to the question and correctly answers it is 0.8 x 1 = 0.8. The probability that the student guesses and correctly answers the question is 0.2 x 0.25 = 0.05. The probability that the student correctly answers the question is 0.8 + 0.05 = 0.85.

The probability that the student really knew the correct answer given that they correctly answered the question is 0.8 / 0.85 = 0.941. Therefore, the probability that the student really knew the correct answer is approximately 94.1%.

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