10. A test consisting of 25 multiple-choice questions with 5 answer choices for each question is administered For each question; there is only | correct answer Let X be the number of correct answers if a student guesses randomly from the 5 choices for each of the 25 questions What is the probability distribution of X? This test, like many multiple-choice tests, is scored using penalty for guessing: The test score is determined by awarding point for each question answered correctly, deducting 0.25 point for each question answered incorrectly, and ignoring any question that is omitted. That is, the test score is calculated using the following formula Score = number of correct answers) (0.25 number of incorrect answers) + (0 number of omits) For example, the score for a student who answers 17 questions correctly, answers 3 questions incorrectly, and omits 5 questions is Score 17) - (0.25 * 3) + (0 x 5) = 16.25 Suppose student knows the correct answers for 18 questions, answers those 18 questions correctly, and chooses randomly from the 5 choices for each of the other 7 questions. Show that the expected value of the student'$ score is 18 when using the scoring formula above

1) The enrollment in 2016 will be 7500.

2) In 2026 year the enrollment reach 10,000.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The function f(x) represents the number of enrollment.

Defined as;

⇒ F(x) = 250x + 6000

Where x represents year since 2010.

(1) Now for finding the enrollment in 2016;

Put x = 2016 - 2010 = 6 in the function

⇒ F(6) = 250x6 + 6000

          = 7500

Thus, The required number of enrollment = 7500

(2) Now we have to find the year in which enrollment reach 10,000;

i.e f(x) = 10,000

=> 250x + 6000 = 10000

=> 250x = 4000

=> x = 16

Thus, The required year = 2010 + 16

                                     = 2026 answer.

1) The enrollment in 2016 will be 7500.

2) In 2026 year the enrollment reach 10,000.

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a) The event E, that the sum of the numbers showing face up is equal to 6, can be written as the set:

E = {11, 15, 24, 33, 42, 51}

b) The probability of E is 6/36 = 1/6.

c) The event F, that the sum of the numbers showing face up is at most 4, can be written as the set:

F = {11, 15, 22, 31, 33, 42, 44, 51}

d) The probability of F is 2/9.

Probability is a measure of the likelihood of an event occurring in a statistical experiment. It is a number between 0 and 1, where 0 indicates that the event will never happen and 1 indicates that the event will always happen. Probability can be used to make predictions about the outcome of statistical experiments, and it is a useful tool in many fields, including economics, finance, and science. It is also used in gambling and games of chance to determine the odds of winning.

The probability of event E is the number of outcomes in the event divided by the total number of outcomes in the sample space. There are 6 outcomes in the event E and 36 outcomes in the sample space.

So, the probability of E is 6/36 = 1/6.

The probability of event F is the number of outcomes in the event divided by the total number of outcomes in the sample space. There are 8 outcomes in the event F and 36 outcomes in the sample space.

So, the probability of F is 8/36 = 2/9.

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The probability of drawing 2 marbles of different colors would be 0.317.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is typically expressed as a fraction or a decimal between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur.

In the scenario you described, a bag contains 3 red, 6 blue, and 7 yellow marbles. If you draw 2 marbles out of the bag, the probability of drawing 2 marbles of different colors is:

Number of ways to draw 2 marbles of different colors: (3 red marbles) x (6 blue marbles) + (3 red marbles) x (7 yellow marbles) + (6 blue marbles) x (7 yellow marbles) = 18 + 21 + 42 = 81

Total number of ways to draw 2 marbles: 16 (4 marbles of each color)

Probability of drawing 2 marbles of different colors: 81/256 = 0.317

Hence, the probability of drawing 2 marbles of different colors would be 0.317.

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The probability of drawing 2 marbles of different colors would be 0.317.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is typically expressed as a fraction or a decimal between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur.

In the scenario you described, a bag contains 3 red, 6 blue, and 7 yellow marbles. If you draw 2 marbles out of the bag, the probability of drawing 2 marbles of different colors is:

Number of ways to draw 2 marbles of different colors: (3 red marbles) x (6 blue marbles) + (3 red marbles) x (7 yellow marbles) + (6 blue marbles) x (7 yellow marbles) = 18 + 21 + 42 = 81

Total number of ways to draw 2 marbles: 16 (4 marbles of each color)

Probability of drawing 2 marbles of different colors: 81/256 = 0.317

Hence, the probability of drawing 2 marbles of different colors would be 0.317.

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By solving the equation we get ,t = ± 7 using concept of square roots.

How to find the square roots ?

Square root, in mathematics, a factor of a number that, when multiplied by itself, gives the original number. For example, both 3 and –3 are square roots of 9.

For example, 2 is the square root of 4, and this is expressed as √4 = 2.

We know that every squared number has two square roots one is positive and another is negative.

In given que,

given condition is t squared minus 49 = 0

Form of equation is ax^ 2 = c.

given condition can also be written as t^2 - 49 = 0

                                                          i.e. t^2 = 49

where a = 1

x = t

c = 49

So, the square root are 7 and -7.

So, equation become t = ± 7.

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The volume of the largest right circular cylinder is [tex]=\frac{4\pi r^3}{3\sqrt{3} }[/tex] cu. unit

Now, According to the question:

The given sphere is of radius R.

Let h be the height and r be the radius of the cylinder inscribed in the sphere.

We know that:

Volume of cylinder

V = [tex]\pi R^2h[/tex]    .....(1)

In right Triangle OBA

[tex]AB^2 + OB^2 = OA^2[/tex]

[tex]R^2 + \frac{h^2}{4} = r^2[/tex]

So, [tex]R^2 = r^2 - \frac{h^2}{4}[/tex]

Putting the value of [tex]R^2[/tex] in equation (1), We get

V = [tex]\pi (r^2 - \frac{h^2}{4} )h[/tex]

V = [tex]\pi (r^2h - \frac{h^3}{4} )[/tex]   ....(2)

dV/dh = [tex]\pi (r^2 - \frac{3h^2}{4} )[/tex] .....(3)

For, Stationary point, dV/dh = 0

[tex]\pi (r^2 - \frac{3h^2}{4} )[/tex] = 0

[tex](r^2 - \frac{3h}{4} )[/tex]   => [tex]h^2 - \frac{4r^2}{3}[/tex] => [tex]h - \frac{2r}{\sqrt{3} }[/tex]

Now, [tex]\frac{d^2V}{dh^2} = \pi (-\frac{6}{4}h )[/tex]

[tex][\frac{d^2V}{dh^2}]_a_t_h_=_\frac{2r}{\sqrt{3} }[/tex]   = x[-3/2 , [tex]2r/\sqrt{3}[/tex]]< 0

Volume is maximum at h = 2r/[tex]\sqrt{3}[/tex]

Maximum volume is :

[tex]= \pi (r^2.\frac{2r}{\sqrt{3} }- \frac{1}{4}.\frac{8r^3}{3\sqrt{3} } )[/tex]

[tex]=\pi (\frac{2r^3}{\sqrt{3} }-\frac{2r^3}{3\sqrt{3} } )[/tex]

[tex]=\pi (\frac{6r^3-2r^3}{3\sqrt{3} } )[/tex]

[tex]=\frac{4\pi r^3}{3\sqrt{3} }[/tex] cu. unit

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It takes about 120 years for the graph to reach 100 million barrels per day whereas it takes 50 years to go from 100 million to 200 million.

As per the question the left axis indicates the amount of oil in the earth in trillions of barrels.

The right axis indicates the global consumption rate of oil in millions of barrels per day.

To the left of the red vertical line are model results that approximate reality whereas to the right are model-based predictions of the future.

The bottom axis is time in years.

The graph attached at the end of the solution.

Let the number of years of transition from using no oil to consuming 100 million barrels per day be a.

Let the number of years of transition from using 100 million to 200 million barrels per day be b.

From the given graph, we can see that initial consumption rate is low, and it takes 120 years for the graph to reach 100 million barrels.

But it takes 50 years to go from 100 million to 200 million.

This is known as exponential growth.

Therefore, the value of a is 120 and the value of b is 50.

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The correct answer is A) y = 9000x + 65,000.

The correct answer is A) y = 9000x + 65,000.

The house has increased in value by 54,000 ($119,000 - $65,000) over 6 years (change in x), so the slope must be 9000 (54,000/6).

The equation y = 9000x + 65,000

models the value of the house after x years, where y is the value of the house,

9000x is the slope of the equation,

and 65,000 is the y-intercept.

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Answer:

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

Given:

Find the missing sides:

Perimeter is:

The answers to find are 1) Δ CAB 2) Δ ABC 3) CDAB 4) 10 5) 6 and 6) 30

When two triangles having congruent corresponding angles and the corresponding sides are in equal ratio, then the triangles are said to be similar triangles.

Given are the similar triangles,

1) Δ HJG ~ Δ CAB

Scale factor = 3:1

2) Δ NPM ~ Δ ABC

Scale factor = 1:2

3) KJML ~ CDAB

Scale factor = 1:2

4) 40/x = 20/5

x = 10

5) 18/x = 21/7

x = 6

6) 6/3 = x/15

x = 30

Hence, the answers are: 1) Δ CAB 2) Δ ABC 3) CDAB 4) 10 5) 6 and 6) 30

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Answer: x = 9/10 - 7/10

Step-by-step explanation:

Answer:

  x = 15

Step-by-step explanation:

You have ∆UVW with angles U=(3x-10)°, V=(6x+5)°, and W=(4x-10)°, and you want to know the value of x.

The angle sum theorem tells you the sum of angles in a triangle is 180°.

  U +V +W = 180°

  (3x -10)° +(6x +5)° +(4x -10)° = 180°

  13x -15 = 180 . . . . . . . divide by °, collect terms

  13x = 195 . . . . . . . add 15

  x = 15 . . . . . . . divide by 13

The value of x is 15.

Answer:

0.25 − 3

=0.25 + −3

= −2.75

( a number line to help to)

Step-by-step explanation:

(a) The differential Equation that is satisfied by y is dy/dt = k × y × (1 - y)   ,

(b) Solution of the differential equation assuming y(0) = c is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .

In the question ,

Part (a)

let the fraction of people who heard the rumor is = y

So , the fraction who have not heard the rumor is = 1 - y .

the rate of rumor spread is ⇒ dy/dt = k×y(1 - y)

dy/y(1-y) = k.dt     ...where k is the constant of proportionality .

So , the differential equation is ..

dy/dt = k × y × (1 - y)

Part (b)

So , 1/y(1-y) = 1/y + 1/(1 - y)       ....equation(1)

integrating equation(1) , we get

∫dx/(1 + ax) = ㏑(1 + ax)/a     ,....where a is the constant

㏑y + ㏑(1-y)/(-1) = kt + d     ,.....where d is the constant

By using , ㏑a - ㏑b = ㏑(a/b) and taking exponential . we get ,

y/(1 - y) = c₁[tex]e^{k\times t}[/tex]

for t = 0 and y(0) = c

solving further , we get

c₁ = c/(1 - c)

So , y = (1-y)c₁[tex]e^{k\times t}[/tex]

y(1 + c₁[tex]e^{k\times t}[/tex]) = c₁[tex]e^{k\times t}[/tex]

y = c₁[tex]e^{k\times t}[/tex]/(1 + c₁[tex]e^{k\times t}[/tex])

taking c₁[tex]e^{k\times t}[/tex] common , and substituting the value of c₁ we get ,

the solution as , y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex]    .

Therefore , (a) the differential equation is dy/dt = k × y × (1 - y)   and

(b) the solution is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex]   .

The given question is incomplete , the complete question is

One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor.

(a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.)

(b) Solve the differential equation. Assume y(0) = C.

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Using a system of equations, the number of seats in the upper section of the racetrack is 3,600.

A system of equations, also called simultaneous equations, is two or more equations solved concurrently.

We can use any of the following methods to solve simultaneous equations:

In this situation, after forming the equations, we can use substitution and elimination methods to solve them.

Racetrack charge per lower seat = $85

Racetrack charge per upper seat = $60

Racetrack charge per field ticket = $35

Let lower seats = x

Let upper seats = 3x

Let field tickets = y

4x + y = 22,800 ... Equation 1

y = 22,800 - 4x ...Equation 3

85x + 60(3x) + 35y = 948,000

85x + 180x + 35y = 948,000 ... Equation 2

Substitute Equation 3 in Equation 2 to eliminate y:

85x + 180x + 35(22,800 - 4x) = 948,000

85x + 180x + 798,000 - 140x = 948,000

125x = 948,000 - 798,000

125x = 150,000

x = 1,200

Determining the number of seats:

Seats in the Lower section = 1,200

Seats in the Upper section = 3,600 (1,200 x 3)

Field tickets, y = 22,800 - 4x

y = 22,800 - 4(1,200)

= 18,000

Check:

85x + 180x + 35y = 948,000

85(1,200) + 180(1,200) + 35(18,000) = 948,000

102,000 + 216,000 + 630,000 = 948,000

948,000 = 948,000

Thus, based on simultaneous equations, there are 3,600 seats in the upper section of the racetrack.

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The Probability of, a ace, jack of clubs, King or queen of diamonds appears in drawing a single card from a well shuffled ordinary deck of 52 cards is 4/13.

As per the given data,

we need to find out the probability of, King, ace, jack of clubs or queen of diamonds appears in drawing a single card from a well shuffled ordinary deck of 52 cards.

We know that,

Probability(Event) =Number of Favorable Outcomes/Total number of

                                Outcomes = x/n.    

Total number of Outcomes is 52 (given)

So,  we will calculate Number of Favorable Outcomes as follows:

The total no. of King in deck of card is 4

The total no. of queen in deck of card is 4.

The total no. of ace in  deck of card is 4.

The total no. of jack in deck of card is 4.

Therefore, the number of favorable outcomes is 4+4+4+4= 16

Now, putting values in the above stated formula of probability, we get:

Probability= 16/52=4/13

Therefore, the probability of pulling a King, Ace, Jack of Clubs, or Queen of Diamonds from a 52-card standard deck that has been properly shuffled is 4/13.

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$10.12 will Cassie pay in all for the pair of binoculars after sales tax is included.

What is sales tax?

A sales tax is a fee that is paid to the government when specified products and services are sold. Typically, laws permit the vendor to charge the customer the tax at the time of purchase. Use taxes are often used to describe taxes on goods and services that consumers pay directly to a governing authority.

given data

we are told that the cost of binoculars is $155.75

and also the sales tax is 7%

Step two:

We want to first solve for the sales tax which is 7% of  $155.75

=7/100* 155.75

=0.070*155.75

=10.12

Cassie will pay a tax of $10.12

Also Cassie will pay a total amount of

= 155.75+10.12

=$165.87

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a) A discrete random variable is a variable that can take only a countable number of distinct values, such as 0, 1, 2, 3, 4, and so on.

b) Examples of discrete random variables are the number of children in a family, the number of people who go to the cinema on Friday nights, etc.

c) E(x) = 1/p

Discrete Random Variable:

A discrete random variable can be defined as a type of variable whose value depends on the numerical outcome of some random phenomenon. Also called a random variable. Discrete random variables are always easily countable integers. A probability mass function is used to describe the probability distribution of a discrete random variable.

Discrete random variables are used to quantify the results of random experiments. A discrete random variable takes on an infinite number of possible outcomes. In general, discrete random variables can be counted as 0, 1, 2, 3, 4, ...

Geometric Distribution :

The geometric variate is the variate that specifies the number of consecutive failures before the first success in Bernoulli trials. The probability of success of a Bernoulli trial is given by p and the probability of failure is 1 - p.

The Geometric Random Variable can be written as X ~ G(p).

The probability mass function is P(X = x) = (1  - p)ˣ⁻¹ p

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When the area of the poster is 256 square inches, the measurements are 32 inch and 8 inch.

The quantity area indicates the extent of a region on a planar or curved surface. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas area of a plane region or plane area refers to the area of a form or planar lamina. The total space occupied by a flat (2-D) surface or the form of an item is defined as its area. The area is the region defined by an object's form. The area of a form is the space covered by a figure or any two-dimensional geometric shape in a plane.

Here,

let length be l and width be w.

l=3w+8

l*w=256

(3w+8)*w=256

3w²+8w=256

3w²+32w-24w-256=0

3w(w-8)+32(w-8)=0

(3w+32)(w-8)=0

w=-32/3, 8

w=8 inch

l=3*8+8

l=32 inch

The dimensions for the poster are 32 inch and 8 inch when area of the poster is 256 square inches.

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Answer:

Step-by-step explanation:

The equation 5x-8+7y=y-6 is linear because it contains only terms with the variables x and y raised to the power of 1. In a linear equation, the highest power of any variable is 1. Nonlinear equations contain exponents that are higher than 1 on one or more variables.

Simple interest at a rate of 5% per year for five years on a principle of $5,000 has resulted in a total accrual of $6,250.00, which includes both the principal and interest.

To calculate simple interest, multiply the daily interest rate by the principle and the number of days between payments. Consumers that make on-time or early monthly loan payments benefit from simple interest. Most loans with simple interest rates are auto loans and short-term personal loans.

A = $6,250.00

I = A - P = $1,250.00

Formula: A = P(1 + rt)

First, convert R percent to r decimal, which is equal to 5%/100 or 0.05 per year.

Fixing our equation

A = 5000(1 + (0.05 × 5)) = 6250 \sA = $6,250.00

Simple interest at a rate of 5% per year for five years on a principle of $5,000 has resulted in a total accrual of $6,250.00, which includes both the principal and interest.

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The price of a ticket to the homecoming dance to maximize the council's profit is 4830.

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Answer:

$2 each ticket

Step-by-step explanation:

I don't really know how it's 2, but it's in the vertex of the graph. I just got my test back which the answer was 2. I'm sorry if you expected reasoning but I only got an answer. You can make a table for the equation and then see that 2 results in the highest profit though. 2 being substituted for t equates to the greatest profit.

Standard deviation will be √3.3516 .

To calculate the standard deviation for the given data first we have to calculate the total number of students , mid value , fiXi , fiXi².

After that , we have to calculate the Xbar by using

Xbar = ∑fiXi / N

for which we need the value of fi , Xi and N

N = 100

fiXi = 6478

we have calculated the values from the given data ,

Therefore ,

Xbar = 6478 / 100

= 64.78

Var(X) = αx² - ∑fiXi² / N - (Xbar²)

= 419980 / 100 - (64.78)²

= 4199.80 -4196.4484

=3.3516

Thus,

standard deviation ax = √var(X)

= √3.3516

Therefore , the standard deviation will be  √3.3516

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Answer: Susie is correct.

Step-by-step explanation:

Susie is correct, because the Pythagorean theorem does not work on this triangle. Since the Pythagorean theorem only works on right triangles, we can conclude that Susie is correct, and that this triangle is NOT a right triangle.

a²+b² = c²

a = 9

b = 12

c = 17

9² + 12² = 17²

==> 81 + 144 = 289

==> 225 = 289, This is not true

Therefore this triangle is not a right triangle.

Answer:

x1,y16 is the rate of change

Step-by-step explanatI subtracted 13-32 which gives me 16, then I added 16 to 32 to be sure and it gave me 48 so the change on the Y axis is going up by 16 and x axis is going up by 1

The solution of the equation written as an imaginary number is y =  3i or -3i

An imaginary number is a number of the form ai, where a is a real number and i is the square root of -1. Imaginary numbers are often denoted with the letter "i".

Given: y² + 11 = 2

y² + 11 = 2

y² = 2 - 11

y² = -9

y = ±√-9                      

y = ± 3i

y = 3i or -3i

(Note: √9 = 3 and √-9 = 3i)

Thus, the solution is y =  3i or -3i

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The equation of the function described as;  y = 7 sin ( 4x + π/2 ) + 3

The general equation of the sine curve can be written as;

y = a sin ( nx + α ) + b

where : a is the amplitude, n = 2π/period, b = shift in the direction of y

            α°= shift in the direction of x

We are Given period = π/2 the maximum value is 10, the minimum value is −4 and y-intercept of 10.

 Thus,

a = (maximum - minimum)/2 = (10 - -4)/2

a = 7

 n = 2π/period = 2π/(π/2)

n = 4

 b = maximum - a = 10 - 7

b= 3

To find α as y-intercept = 10

y = 10 at x = 0

Substitute in the general function;

y = a sin ( nx + α ) + b

10 = 7 sin ( 4*0 + α ) + 3

Thus, we have;

sin α = 1

α = π/2

So, the equation of the function described is;

 y = 7 sin ( 4x + π/2 ) + 3

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The ratio of rainy days to total days written as a percent will be 75%.

Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).

Ratio is used to compare two or more numbers. It is also used to indicate how big or small a quantity is when it is compared to another. It should be noted that in a ratio, two quantities are compared using division.

Since in the last 24 days, it rained 18 days.

Number of rainy days = 18.

Number of total days = 24

The ratio of rainy days to total days written as a percent will be:

= Number of rainy days / Total days × 100

= 18/24 × 100

= 3/4 × 100

= 75%

Therefore, the ratio is 3:4 which is 75%.

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Answer:

70

Step-by-step explanation:

The graph of the equation y = (1/3)x + 2 is passing through points  

(3, 3) and ( -3, 1).

The set of ordered pairings (x, y) where f(x) = y makes up the graph of a function.

These pairs are Cartesian coordinates of points in two-dimensional space and so constitute a subset of this plane in the general case when f(x) are real values.

Given, A linear equation y = (1/3)x + 2.

Now to graph the equation we'll simply put some arbitrary values of x which will correspond to some values of y and plot them then we'll join them with a straightedge.

When x = 3, y = 3 ⇒ (3, 3).

When x = - 3, y = 1 ⇒( -3, 1).

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