A deck of cards is shuffled, if we want to know the chance that the top card is the ace of spades or the bottom card is the ace of spades, we will use: Neither of the rules Addition rule Multiplication rule & Either of the rules

a) The exact solutions on [0, 2π) are t = π/3, π, and 5π/3. b) The exact solution on the interval [0, π) is t = 2π/3. c) The solutions are θ = π/3 and θ = 5π/3. d) The exact solutions on the interval [0, 2π) are: t = arcsin[(-1 +  [tex]\sqrt{41}[/tex])/4] and t = 2π - arcsin[(-1 +  [tex]\sqrt{41}[/tex])/4] using trigonometric identities.

a) We can rewrite the equation as:

2[tex]cos^{2}t[/tex] + cos(t) - 1 = 0

Using the quadratic formula, we get:

cos(t) = [-1 ± [tex]\sqrt{1-4(2)(-1)}[/tex] ] / (4)

cos(t) = [-1 ± [tex]\sqrt{9}[/tex]]/4

cos(t) = [-1 ± 3]/4

Thus, we have two solutions:

cos(t) = 1/2, which gives us t = π/3 and t = 5π/3

cos(t) = -1, which gives us t = π

Therefore, the exact solutions on [0, 2π) are t = π/3, π, and 5π/3.

b) We can use the identity [tex]tan^{2}t[/tex] + 1 = [tex]sec^{2}t[/tex] to rewrite the equation as:

[tex]tan^{2}t[/tex] = - [tex](3/2)^{2}[/tex]

Taking the square root of both sides and remembering that tan(t) is negative in the third quadrant, we get:

tan(t) = -3/2

Using the identity tan(t) = sin(t)/cos(t), we can rewrite this as:

sin(t)/cos(t) = -3/2

Multiplying both sides by cos(t) and rearranging, we get:

sin(t) = -3cos(t)/2

Squaring both sides and using the identity [tex]sin^{2}t[/tex] + [tex]cos^{2}t[/tex] = 1, we get:

9[tex]cos^{2}t[/tex]/4 + [tex]cos^{2}t[/tex] = 1

Expanding and simplifying, we get:

13 [tex]cos^{2}t[/tex] /4 = 1

[tex]cos^{2}t[/tex]  = 4/13

Taking the square root of both sides and remembering that cos(t) is negative in the third quadrant, we get:

cos(t) = -2[tex]\sqrt{13}[/tex] /13

Using the identity  [tex]sin^{2}t[/tex] +  [tex]cos^{2}t[/tex]  = 1, we can solve for sin(t) as:

sin(t) = -3/2 cos(t) = 3[tex]\sqrt{13}[/tex] /13

Therefore, the exact solution on the interval [0, π) is t = 2π/3.

c) We can use the identity sin(2θ) = 2sin(θ)cos(θ) to rewrite the equation as:

sin(θ) = 2sin(θ)cos(θ)

Dividing both sides by sin(θ) (assuming sin(θ) is non-zero), we get:

1 = 2cos(θ)

cos(θ) = 1/2

Therefore, the solutions are θ = π/3 and θ = 5π/3.

d) We can use the identity cos(2t) = 1 - 2 [tex]sin^{2}t[/tex]  and rearrange the equation as:

2 [tex]sin^{2}t[/tex]  + sin(t) - 5 = 0

Solving this quadratic equation using the quadratic formula, we get:

sin(t) = [-1 ± [tex]\sqrt{41}[/tex]]/4

Since the sine function has a range of [-1, 1], only the positive solution is possible, so we have:

sin(t) = (-1 + [tex]\sqrt{41}[/tex])/4

Using the identity [tex]cos^{2}t[/tex] = 1 -  [tex]sin^{2}t[/tex] , we can solve for cos(t) as:

cos(t) =[tex]\sqrt{1-sin^{2}t}[/tex] = [tex]\sqrt{17-\sqrt{41} }/4[/tex]

Therefore, the exact solutions on the interval [0, 2π) are:

t = arcsin[(-1 +  [tex]\sqrt{41}[/tex])/4] and t = 2π - arcsin[(-1 +  [tex]\sqrt{41}[/tex])/4]

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Answer:

The formula for calculating the future value of an investment with compound interest is:

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

where A is the future value, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time period in years.

In this case, P = $2500, r = 0.0125 (1.25% as a decimal), n = 12 (since the interest is compounded monthly), and t = 4.

Plugging these values into the formula, we get:

A = 2500(1 + 0.0125/12)^(12*4) ≈ $2,628.11

Therefore, the answer is $2628.11.

Answer: $ 2628.11

Step-by-step explanation:

Compound interest is nasty.

the formula (which you need to memorize) is:

[tex]A = P(1+\frac{r}{n} )^t^n[/tex]

A is the final amount, P is the principle, (original amount of money invested), r is the rate of return (as a decimal NOT percent), n is the number of times compounded annually, and t is the time, (in years), invested.

With all this information, our formula becomes:

[tex]A = 2500(1+\frac{0.0125}{12} )^4^8[/tex]

solving, we get A ≅ 2628.11

This corresponds to answer choice C

If Y is directly proportional to the cube root of x, the percentage increase in y when x is increased by 700% is 100%.

If y is directly proportional to the cube root of x, we can write this relationship as y = [tex]kx^{(1/3)[/tex], where k is the constant of proportionality.

To find the percentage increase in y when x is increased by 700%, we first need to determine the new value of x. An increase of 700% means that x is multiplied by (1 + 700%) = 8. Therefore, the new value of x is 8 times the original value.

Substituting the new value of x into the equation for y, we get:

y = [tex]k(8x)^{(1/3)[/tex]

Simplifying this expression, we can write:

y = [tex]k(2^{3x)^{(1/3)[/tex]

y = k2x

So, we can see that when x is increased by 700%, y is increased by a factor of 2. This corresponds to a percentage increase of 100%.

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Based on the function f(x) = 3 - x², the value of f(-2) include the following: f(-2) = -1.

In Mathematics and Geometry, a function can be defined as a mathematical expression which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair in tables or relations.

In Mathematics and Geometry, a domain is sometimes referred to as input value and it can be defined as the set of all real numbers for which a particular function is defined.

When the domain (input value) of the given function f(x) is -2, the output value (range) is given by;

f(x) = 3 - x²

f(x) = 3 - (-2)²

f(x) = 3 - 4

f(x) = -1

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An argument can be invalid even if the premises are true and the conclusion is false.This statement is actually false.

Invalidity refers to the structure of the argument, not the truth value of the premises or conclusion. An invalid argument fails to establish the truth of its conclusion even if its premises are true. In contrast, a valid argument guarantees that the conclusion follows from the premises, regardless of the truth or falsity of the premises. Therefore, an argument can be valid with false premises, and an argument can be invalid with true premises.

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Conclude that there is no guarantee that the conclusion will be false when all the premises are false in an invalid

argument.

If an argument is invalid, then whenever the premises are all false, the conclusion must also be false.

An invalid argument is one where the conclusion does not necessarily follow from the premises, even if the premises

are all true.

In other words, it is possible for the premises to be true, but the conclusion to be false. However, in the scenario you

described, all the premises are false. In an invalid argument, there is no guarantee that the conclusion will also be false.

Identify the argument as invalid.

Recognize that all the premises are false.

Understand that in an invalid argument, the conclusion does not necessarily follow from the premises.

Conclude that there is no guarantee that the conclusion will be false when all the premises are false in an invalid

argument.

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Answer: B. Fixed Annuity

Step-by-step explanation: A Fixed Annuity is a type of retirement plan where you make regular contributions to an insurance company, and in return, they guarantee a fixed payout during your retirement years. This guaranteed benefit is typically paid out as a stream of income for a specific period or for the rest of your life, depending on the terms of the annuity contract.

Unlike other retirement plans such as a 401(k), Roth IRA, or Traditional IRA, which are investment-based and subject to market fluctuations, a Fixed Annuity provides a guaranteed benefit regardless of how the financial markets perform.  This can provide peace of mind for individuals who prioritize a stable and predictable income during retirement.

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The answer of this question is equation [-6x+4y=16].

An equation is a mathematical statement that shows the equality between two expressions, typically separated by an equal sign. It can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value of the variable that makes the equation true.

In mathematics, a system refers to a collection of mathematical objects or entities that are related or interconnected in some way.

System A

5x-y=-11

3x-2y=-8

System B

5x-y=-11

-2(3x-2y)=-8×(-2)

5x-y=-11

-6x+4y=16

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3⁴ is 3 times larger than 3³, 5³ is 5 times larger than 5², 7¹⁰ is 49 times larger than 7⁸ and 17⁶ is 289 times larger than 17⁴.

3⁴ / 3³ = (3 × 3 × 3 × 3) / (3 × 3 × 3) = 3

This means that 3⁴ is 3 times larger than 3³.

5³ is 5 times larger than 5².

5³ / 5² = (5 × 5 × 5) / (5× 5) = 5

7¹⁰ is 49 times larger than 7⁸.

7¹⁰/ 7⁸ = 7² =49

17⁶ is 289 times larger than 17⁴.

17⁶ /17⁴ = 289

Hence, 3⁴ is 3 times larger than 3³, 5³ is 5 times larger than 5², 7¹⁰ is 49 times larger than 7⁸ and 17⁶ is 289 times larger than 17⁴.

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Answer:

A 48 pound catfish would be approximately 45 inches long

Step-by-step explanation:

First find the slope of the line, m, using 2 points on the line, (20, 3) and (25, 12)

m = (12-3) / (25-20) = 9/5

Use the point (30, 21) and slope to find the equation (in point-slope form) of the line:

(y - 21) = 9/5(x - 30) = 9/5x - 270/5 = 9/5x - 54

y = 9/5x - 54 + 21 = 9/5x - 33

y = 9/5x - 33      now you have the equation of the line in y = mx + b form

Find x (catfish length) when y = 48 (catfish weight)

48 = 9/5x - 33

9/5x = 48 + 33 = 81

x = 81 · 5/9 = 45

x = 45 inches

3/4

If she ate part of the large cookie and was left with 1/4 of the cookie, that means only 3/4 of the cookie is left.

In 2010, Haiti was hit by a devastating earthquake, which registered at 7.0 on the Richter scale. The largest recorded earthquake since 1900 occurred in 1960 in Valdivia, Chile, and registered at 9.5 on the Richter scale. About 316 times greater was the intensity of the Chilean earthquake.

Hence, the correct option is D.

The Richter scale is a logarithmic scale that measures the magnitude of an earthquake. Each increase of one on the Richter scale represents a tenfold increase in the amplitude of the seismic waves.

To find out how many times greater the intensity of the Chilean earthquake was compared to the Haitian earthquake, we will have to find the difference in magnitude between the two earthquakes.

9.5 - 7.0 = 2.5

Since each increase of one on the Richter scale represents a tenfold increase in the amplitude of the seismic waves, a 2.5 increase represents

10^(2.5) = 316.2

Hence, the intensity of the Chilean earthquake was about 316 times greater than the intensity of the Haitian earthquake.

Hence, the correct option is D.

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Answer:d

Step-by-step explanation:

its d

The reason is 2) Vertical Angles Theorem.

Given is figure in which two triangles are joined,

We need to complete the proof of ΔBCA ≅ ΔDCE,

So,

BC = DC, AC = EC [given]

∠BCA = ∠DCE [Vertical Angles Theorem]

ΔBCA ≅ ΔDCE by SAS rule,

Hence, the reason is 2) Vertical Angles Theorem.

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The values of the missing sides and angles of the triangle using sine rule are;

DE = 18.3

<D = 84°

EF = 20.8

Sine rule is a method that is used to find the missing angles and sides of triangles.

Thus, it is expressed as;

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

Thus;

12/sin 35 = DE/sin 61

DE = 12(sin 61)/sin 35

DE = 18.3

Sum of angles in a triangle is 180°. Thus;

<D = 180 - (61 + 35)

<D = 84°

Similarly;

EF/sin 84 = 12/sin 35

EF = 12(sin 84)/sin 35

EF = 20.8

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From distance formula, the distance covered by a particle with position (x, y) as time, t is equals to the [tex] 18\sqrt{2}[/tex].

The distance d, traveled by the particle with position vector, r(t) = <x(t), y(t) > over the interval [a,b] is calculated by following

[tex]d = \int_{a}^{b} ( (\frac{dx}{dt})² +(\frac{dy}{dt})² )dt[/tex]. We have a particle with position value (x, y) as t varies in the time interval. The parametric equations are, x = 3 sin² (t), y = 3 cos²(t), on the

time interval [0,3π]. Now the derivative of above parametric equations with respect to time t are [tex]\frac{dx}{dt} = 6 sin(t)cos(t) [/tex] = 3 sin(2t)

[tex]\frac{dy}{dt} = -6 sin(t)cos(t) [/tex]

= - 3 sin(2t)

Therefore, the required distance travelled by particle in the interval [0,3π] is written as [tex]d = \int_{0}^{3π} \sqrt{( (3 sin(2t))² + (-3 sin(2t))²) dt} \\ [/tex]

[tex]= \int_{0}^{3π} \sqrt{ ( 9 sin²(2t) + 9sin²(2t))}dt \\ [/tex]

[tex]= \int_{0}^{3π} \sqrt{18sin²(2t)dt}[/tex]

[tex]= 3\sqrt{2}\int_{0}^{3π} |sin(2t)|dt[/tex]

[tex]= 3\sqrt{2}[ \int_{0}^{\frac{π}{2}} sin(2t)dt - \int_{\frac{π}{2}}^{π} sin(2t)dt + \int_{π}^{\frac{3π}{2}} sin(2t)dt - \int_{\frac{π}{2}}^{2π}sin(2t)dt + \int_{2π}^{\frac{5π}{2}}sin(2t)dt - \int_{\frac{5π}{2}}^{3π}sin(2t)dt \\ [/tex]

[tex]= 3\sqrt{2}([ \frac{cos(2t)}{2}]_{0}^{\frac{π}{2}} - [\frac{cos(2t)}{2}]_{\frac{π}{2}}^{π} + [\frac{cos(2t)}{2}]_{π}^{\frac{3π}{2}} - [\frac{cos(2t)}{2}]_{\frac{3π}{2}}^{2π} + [\frac{cos(2t)}{2}]_{2π}^{\frac{5π}{2} }- [ \frac{cos(2t)}{2}]_{\frac{5π}{2}}^{3π})\\ [/tex]

[tex]= 3\sqrt{2}([ \frac{1}{2} +{\frac{1}{2}] - [\frac{-1}{2}- \frac{1}{2}}] + [\frac{1}{2}+ \frac{1}{2}] - [\frac{-1}{2} - \frac{1}{2}] + [\frac{1}{2} + \frac{1}{2}]- [ \frac{-1}{2} - \frac{1}{2}])\\ [/tex]

[tex]= 3\sqrt{2}(6)[/tex] =[tex]= 18\sqrt{2}[/tex]

Hence, the required distance is [tex] 18\sqrt{2}[/tex].

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The probability that exactly one of the computers will be defective is 0.141

To calculate the probability of exactly one defective computer being selected out of four computers, we can use the binomial distribution formula

P(X = k) = (n choose k) × p^k × (1 - p)^(n - k)

where P(X = k) is the probability of exactly k successes (in this case, one defective computer), n is the total number of trials (in this case, four computers being selected), p is the probability of success (in this case, the probability of selecting a defective computer), and (n choose k) is the binomial coefficient, which represents the number of ways to choose k items out of n.

The probability of selecting a defective computer is 4/15, since there are 4 defective computers out of 15 in total. Therefore, the probability of selecting exactly one defective computer out of four is:

P(X = 1) = (4 choose 1) × (4/15)^1 × (11/15)^3

= 4 × 4/15 × 1331/3375

= 0.141

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The given question is incomplete, the complete question is:

A computer retail store has 15 personal computers in stock. A buyer wants to purchase 4 of them. Unknown to either the retail store or the buyer, 4 of the computers in stock have defective hard drives. Assume that the computers are selected at random. What is the probability that exactly one of the computers will be defective?

So, the circle graph entitled New York City visitor's transportation with five sections labelled walk 40%, bicycle 8%, car service 15%, bus 10%, and subway 27% is the proper option that displays the data in the circular graphs.

A circle represents 360 degrees. A circle can be split up into smaller sections. An arc is a segment of a circle, and arcs are named based on their angles.

To illustrate information and data, use a circle graph or pie chart. Often, a circle graph is used to quickly and proportionately display the findings of an investigation.

Given data:

Type of Transportation      Number of Visitors  Percentage

Walk                                    120                            120*100/ 300 = 40%

Bicycle                                 24                            24*100/ 300 = 8%        

Car Service                         45                            45*100/ 300 = 15%

Bus                                      30                            30*100/ 300 = 10%

Subway                               81                             81*100/ 300 = 27%

Total                                   300

So, the circle graph entitled New York City visitor's transportation with five sections labelled walk 40%, bicycle 8%, car service 15%, bus 10%, and subway 27% is the proper option that displays the data in the circular graphs.

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After Derek drinks 1/4 ounce of water, he is left with 95/3 ounces of water in his glass.

Derek initially had 10 2/3 ounces of water in his glass. To find out how much water is left after he drinks 1/4 ounce of water, we need to subtract 1/4 from 10 2/3. To do that, we need to convert the mixed number (10 2/3) into an improper fraction.

To convert a mixed number into an improper fraction, we multiply the whole number by the denominator of the fraction and then add the numerator. In this case, we have:

10 x 3 + 2 = 32/3

Therefore, Derek initially had 32/3 ounces of water in his glass.

To subtract 1/4 from 32/3, we need to make sure the denominators are the same. We can convert 1/4 into twelfths by multiplying both the numerator and denominator by 3. This gives us:

1/4 x 3/3 = 3/12

Now that we have a common denominator of 12, we can subtract:

32/3 - 3/12 = 383/12 - 3/12 = 380/12

To simplify, we can divide both the numerator and denominator by 4:

380/12 ÷ 4/4 = 95/3

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Complete question is:

Derek had 10 2/3 ounce of water in a glass He drank 1/4 of an ounce of the water from his glass how much water is left in Derek's glass?

We can say with 82% confidence that the true difference in mean times for left-handed and right-handed people is between -0.978 s and -0.222 s and t-distribution is used.

To calculate the certainty interim for the contrast between left-handed and right-handed individuals, we are able to utilize the taking-after equation:

CI = (x1 - x2) ± t*SE 

where:

x1 = the meantime for left-handed people

x2 = the meantime for right-handed people

t = the critical value for the t-distribution with n1+n2-2 degrees of freedom and the desired confidence level (in this case, 82%)

SE = the standard error of the difference in means, which is calculated as:

[tex]SE = sqrt((s1^2/n1) + (s2^2/n2))[/tex]

where:

s1 = the standard deviation for left-handed people

s2 = the standard deviation for right-handed people

n1 = the sample size for left-handed people

n2 = the sample size for right-handed people

Plugging in the given values, we get:

x1 - x2 = 1.1 - 1.7 = -0.6

s1 = 0.4

s2 = 1.4

n1 = 47

n2 = 42

[tex]SE = sqrt((0.4^2/47) + (1.4^2/42)) = 0.261[/tex]

To find the critical value for the t-distribution, we can use a t-table or a calculator. With n1+n2-2 = 87 degrees of freedom and an 82% confidence level, the critical value is approximately 1.360.

Thus, the 82% confidence interval for the difference in left-handed or right-handed people is:

CI = (-0.6) ± (1.360*0.261) = (-0.978, -0.222)

hence, we can say with 82% confidence that the true difference in mean times for left-handed and right-handed people is between -0.978 s and -0.222 s.

Note that we used the t-distribution because the sample sizes for both groups are relatively small (n1 = 47, n2 = 42).

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The number of hours it will take to complete the resurfacing project is:

51 hours

We are told that the department of  transportation is resurfacing 12. 75 miles of roadways in Collin County.

Now, we are also given that:

Number of miles completed each hour = 1/4 mile in section = 0.25 mile per hour

Thus, the number of hours it will take to complete the resurfacing project is:

That would be 12.75 divided by 0.25 hours

= 12.75/0.25

= 51 hours

Thus, we conclude that is the total time it will take to complete the project

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Answer:10g-3

Step-by-step explanation:

Answer:

g=0.5

Step-by-step explanation:

2(5—1)=3

10g—2=3

10g—2+2=3+2

10g=5

10g/10=5/10

g=0.5

Answer:

Step-by-step explanation:

The concept of absolute value can be used to determine which temperature is warmer between -4.5°C and 3.5°C. Absolute value is a mathematical function that returns the magnitude, or distance, of a number from zero, regardless of whether the number is positive or negative. In other words, the absolute value of a number is always a positive number.

In this case, we can calculate the absolute value of each temperature as follows:

| -4.5°C | = 4.5°C

| 3.5°C | = 3.5°C

The absolute value of -4.5°C is 4.5°C, which is greater than the absolute value of 3.5°C, which is 3.5°C. This means that -4.5°C is further from zero than 3.5°C, and therefore it is a colder temperature. So, the temperature of 3.5°C is warmer than the temperature of -4.5°C.

Therefore, we can use the concept of absolute value to determine which temperature is warmer by comparing the absolute values of the temperatures. The temperature with the greater absolute value is farther from zero, and therefore colder, while the temperature with the smaller absolute value is closer to zero, and therefore warmer.

Answer:

x=4

Step-by-step explanation:

If the equation is:

[tex]x^{2} =16[/tex]

then the answer is 4, because to solve this, you would have to take the square root of both sides to cancel out the square on the left side of the equation:

[tex]\sqrt{x^{2} } =\sqrt{16}\\ x=4[/tex]

Hope this helps :)

Answer: If there are 60 questions and the student has to answer 75% of questions then the number of questions the student must answer is 45

Step-by-step explanation:

No of questions= 60,

Answers to be given= 75%

then no of questions to pass the exam= No of questions*Answers to be given in percentage

=60*75/100\

=45

Answer: Width: 12 Height: 24 Depth: 18

Step-by-step explanation: Width= 12, 12x2=24 and 12x1.5=18. 12x24x18=5184in3

The cylinder's volume is 245 cubic units as a result.

It is possible to determine how much material a cylinder can carry by using its volume, which is known as its capacity in geometry. Cubic measurements for a cylinder's volume include cm³, m³, in³, etc.

The formula V = r²h is used to determine the volume of a cylinder, where r is the radius of its circular base and h is the height between the bases' perpendicular centers.

As a result, if r = 7 units and x = 5 units, we can use the formulas below to get the cylinder's volume:

V = πr²h

V = π(7)²(5)

= 245 cubic units per volt

Consequently, the size of the cylinder 245 cubic units as a result.

Complete question given below:

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If r = 7 units and x = 5 units, then what is the volume of the cylinder(height=x)?

The volume of the cylinder is 245π cubic units.

The cylinder is a three-dimensional shape having a circular base. A cylinder can be seen as a set of circular disks that are stacked on one another.

The formula for the volume of a cylinder is:

V = πr²h

where r is the radius of the base, h is the height, and π is the mathematical constant pi (approximately equal to 3.14).

Substituting r = 7 units and h = 5 units into the formula, we get:

V = π(7)²(5)

V = π(49)(5)

V = 245π

Therefore, the volume of the cylinder is 245π cubic units. Note that the volume is an exact value in terms of π, and cannot be simplified further without a numerical approximation of π.

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Using percentages we know that Terrel has the best free throw statistics which is 70.83%.

In mathematics, a quantity or ratio known as A% stands for a percentage of 100.

There are several different ways to depict a dimensionless connection between two numbers, including ratios, fractions, and decimals.

To represent percentages, the symbol "%" is typically written after the number.

In mathematics, a percentage is a number or ratio that is expressed as a fraction of 100.

So, in the given situation where we need to get whose statistics are the best.

We need to calculate the percentages in all cases as follows:

Jack: 18/39*100 = 46.15%

Henry: 54%

Terrel: 17/24*100 = 70.83%

Riley: 9/23*100 = 39.13%

Since Terrel has the highest %, so he has the best free throw statistics.

Therefore, using percentages we know that Terrel has the best free throw statistics which is 70.83%.

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The value of the standard deviation of the weekly earnings of the bus drivers is $14.57.

To solve this problem, we can use the standard normal distribution table or a calculator that provides the inverse of the standard normal distribution. Let X be the weekly earnings of the bus drivers, then we have:

P(X > $429.35) = 0.011

Using the standard normal distribution table or calculator, we can find the z-score that corresponds to a probability of 0.011, which is approximately -2.33. We can then use the formula for standardizing a normal variable:

z = (X - μ) / σ

where μ is the mean and σ is the standard deviation of the distribution. Substituting the given values, we have:

-2.33 = ($429.35 - $395) / σ

Solving for σ, we get:

σ = ($429.35 - $395) / (-2.33) = $14.57

Therefore, the value of the standard deviation of the weekly earnings of the bus drivers is $14.57.

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The Law of Large Numbers is a fundamental concept in probability theory and statistics stating the number of trials of a random event increases, the average of the outcomes approaches the expected value of the event. It's better to use 500 trials.

In other words, as we repeat an experiment a large number of times, the proportion of times that a certain outcome occurs will converge to its probability of occurrence. This is an important concept in statistical inference, as it helps to ensure that the results of a sample are representative of the population from which it was drawn.

Now, if you were using the relative frequency of an event to estimate the probability of the event, it would be better to use 500 trials rather than 100 trials. The reason is that the larger the number of trials, the more accurate the estimate of the probability is likely to be.

Using a larger sample size reduces the impact of random variation and provides a more precise estimate of the true probability. The Law of Large Numbers tells us that as the sample size increases, the relative frequency of the event should converge to the true probability, so a larger sample size will generally give a more reliable estimate

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The area of sector DEF is 15.09 square units

The formula for area of a sector is:

A = (θ/360) * πr²

Where θ is the angle subtended at the center and r is the radius of the circle.

In this case, DE is the radius.

Thus r = 6 cm and θ = 48°

Substituting:

A = (48/360) * π * 6²

A = (48/360) * (22/7) * 36

A = 15.09 square units

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