Suppose the probability density function of a random variable X is
f(x)=[tex]\left \{ {{cx^{2}, 1\leq x\leq 2 } \atop {0, else}} \right.[/tex]

a. Find the value of constant c
b. Find the value of P(X>3/2)

The length of unknown side is,

⇒ 4.92

We have to given that;

A triangle is shown in image.

Let the length of unknown side = x

Now, From trigonometry formula we get;

⇒ tan 51° = x / 4

⇒ 1.23 = x / 4

⇒ x = 1.23 × 4

⇒ x = 4.92

Thus, the length of unknown side is,

⇒ x = 4.92

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Answer:

12.5 is the answer

Step-by-step explanation:

7 × 60 ( 7 being the minutes, 60 being the hour )

= 420

+ 30

= 450

450 ÷ 3600 × 100

45000 ÷ 3600

450 ÷ 30

= 12.5

12.5 being the answer

She earn fοr each year frοm 2014 thrοugh 2016 is 80628.40. The average οf her last five years οf wοrking $78,302.80. His annual retirement benefit was $28,423.92.

a) Salary οf 2014 : $76000

Salary in 2014 is the salary in 2015 increased by 3% οf the salary

= $76,000+3%

Salary οf 2015 = $78,280

Salary in 2015 is the salary in 2016 increased by 3% οf the salary

=$78,280+3%

Salary οf 2016 = $80,628.40

Hence, she earn fοr each year frοm 2014 thrοugh 2016 is 80628.40

b) The average οf the last three years is the sum οf the salaries divided by the number οf salaries.

= $76,000+$78,280+$80,628.40 / 3

= $78,302.80

Hence, the average οf her last five years οf wοrking $78,302.80

c ) The annual retirement benefit is the prοduct οf the rate and the average and the number οf years οf service.

= 1.65%×$78,302.80×22

=$28,423.92

Hence,  his annual retirement benefit was $28,423.92.

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Answer:

To find the product of matrices AB, we need to multiply the elements of the rows of matrix A with the corresponding elements of the columns of matrix B, and then sum these products.

Since matrix A is a 2x2 matrix and matrix B is a 2x3 matrix, we can perform the multiplication as follows:

AB = | 1 2 |   | 1 2 3 |   | (1*1)+(2*4)  (1*2)+(2*5)  (1*3)+(2*6) |

    | 3 4 | x | 4 5 6 | = | (3*1)+(4*4)  (3*2)+(4*5)  (3*3)+(4*6) |

                          |              |              |              |

    |  9 12 15 |          |  9  12  15 |

Therefore, the product of matrices AB is a 2x3 matrix, and the answer is C) 2x3.

The equation for line n that is perpendicular to the x-axis and passes through the point (–3,–7) is x = -3, and its slope is undefined.

Since line n is perpendicular to the x-axis, it is parallel to the y-axis. Therefore, its slope is undefined since the y-axis is a vertical line with no defined slope.

To write the equation for line n, we know that the y-coordinate of every point on the line will be constant since the line is parallel to the y-axis. We also know that the line passes through the point (-3,-7), so we can write the equation as:

x = -3

This means that for any value of y, the x-coordinate will always be -3. Graphically, this represents a vertical line passing through the point (-3,-7) and parallel to the y-axis.

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The resulting matrix formed by performing R2 -> 4R1 + R2 on M is given as follows:

[tex]M = \left[\begin{array}{ccc}-4&3&1\\-18&9&8\end{array}\right][/tex]

The matrix in the context of this problem is defined as follows:

[tex]M = \left[\begin{array}{ccc}-4&3&1\\-2&-3&r\end{array}\right][/tex]

The rows of the matrix are given as follows:

Hence the row 2 of the resulting matrix has the elements given as follows:

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Answer: To compute the t-test statistic for the paired sample data, we need to first calculate the sample mean difference and the sample standard deviation of the differences. Then we can use the formula:

t = (sample mean difference - hypothesized mean difference) / (standard error of the mean difference)

where the standard error of the mean difference is calculated as:

standard error = sample standard deviation / sqrt(sample size)

Let's first calculate the sample mean difference:

x 9 6 7 5 12

y 6 8 3 6 7

The differences between each pair of x and y values are:

(9-6), (6-8), (7-3), (5-6), (12-7) = 3, -2, 4, -1, 5

The sample mean difference is the average of these differences:

sample mean difference = (3 - 2 + 4 - 1 + 5) / 5 = 1.8

Next, we need to calculate the sample standard deviation of the differences. To do this, we first calculate the deviations of each difference from the sample mean difference:

(3 - 1.8), (-2 - 1.8), (4 - 1.8), (-1 - 1.8), (5 - 1.8) = 1.2, -3.8, 2.2, -2.8, 3.2

The sample standard deviation of the differences is the square root of the sum of the squared deviations divided by (n-1):

sample standard deviation = sqrt[(1.2^2 + (-3.8)^2 + 2.2^2 + (-2.8)^2 + 3.2^2) / 4] = 3.153

Finally, we can calculate the t-test statistic:

t = (sample mean difference - hypothesized mean difference) / (standard error of the mean difference)

where the hypothesized mean difference is 0, and the standard error of the mean difference is:

standard error = sample standard deviation / sqrt(sample size) = 3.153 / sqrt(5) = 1.410

Substituting the values, we get:

t = (1.8 - 0) / 1.410 = 1.277

Rounding the final answer to three decimal places, we get:

t = 1.277

Therefore, the correct option is B. t = 1.292.

Answer:

C

Step-by-step explanation:

the prime factorization of 84 is 2 x2 x 3 x 7

I can rewrite the problem

-3[tex]\sqrt{84x^{3} }[/tex]

-3[tex]\sqrt{(2)(2)(3)(7)xxx}[/tex]  pull out the pairs

-3(2)x[tex]\sqrt{(3)(7)x}[/tex]

-6x[tex]\sqrt{21x}[/tex]

Helping in the name of Jesus.

Suppose you are a small business owner who sells handmade crafts. You have a budget of $100 to purchase materials for your next batch of products. You know that each craft requires some amount of materials, which costs $8 per unit. Additionally, you will need to pay a fixed cost of $14 for other expenses related to production and shipping.

To make a profit, you must ensure that the cost of materials and fixed expenses does not exceed the $100 budget. Therefore, you can write an inequality to represent this situation:

8x + 14 < 100

Here, x represents the number of units of materials needed for each craft. The inequality states that the total cost of materials (8x) plus fixed expenses ($14) must be less than $100.

To solve this inequality, you can subtract 14 from both sides:

8x < 86

Finally, you can divide both sides by 8:

x < 10.75

This means that for each craft, you can use no more than 10.75 units of materials in order to stay within budget and make a profit.

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The factor of 2xy + 30x^2 - xy - 16y^4 - yx - 5x^2 is 25x2−16y4=(5x+4y2)(5x−4y2)

We are given that;

2xy + 30x^2 - xy - 16y^4 - yx - 5x^2

Now,

Combine like terms by adding or subtracting the coefficients of the same variables. For example, 2xy - xy - yx = 0xy, and 30x^2 - 5x^2 = 25x^2. The polynomial becomes:

25x2−16y4

Check if there is a common factor for all the terms. In this case, there is no common factor other than 1, so we cannot use the greatest common factor method.

Check if the polynomial is a difference of two squares, which means it has the form a2−b2. In this case, we can see that both terms are perfect squares: 25x2=(5x)2 and 16y4=(4y2)2. Therefore, we can use the difference of two squares formula:

a2−b2=(a+b)(a−b)

Substituting a=5x and b=4y2, we get:

25x2−16y4=(5x+4y2)(5x−4y2)

Check if each factor can be further factored using any of the methods. In this case, neither factor can be further factored, so we are done.

Therefore, by factorization the answer will be 25x2−16y4=(5x+4y2)(5x−4y2)

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After considering all the given options we conclude that the number is atleast 21 and the less than 25, which is Option C.

It is given to us that two events are independent if they take place then one event does not trigger the probability of the other event.

Now if the taking place of a certain event triggers the other event then it is referred as dependent

For the given case, we have four events A, B, Q and L.

A = the state when the given number is At least 21

B = is the sate when the given number is Between 12 and 25

Q = is the sate when the given number is Odd

L = is the state when the given number is a Less than 25

It is clearly visible that events A and L are independent due to the number being at least 21, it doesn't affect whether it's less than 25 or not. So, events B and Q are independent because if we know that a number is between 12 and 25, it doesn't affect whether it's odd or not.

Hence, option C) A and L is correct.

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The complete question is

S = [10, 11, 13, 19, 21, 23, 29, 30]

A = The number is At least 21

B= the number is Between 12 and 25

Q = number is Odd

L= number is a Less than 25

Which events are independent.

Question options:

A) A and B

B) A and O

C) A and L

D) Band O

E) Land B

F) Land O

G) None of the 2 events are independent

The number of acres pollinated by the bees in 18 beehives is 48 acres.

A proportional relationship is defined as follows:

y = kx.

In which k is the constant of proportionality.

The variables for this problem are given as follows:

x: number of beehives.

y: number of acres.

From the table, the constant is obtained as follows:

3k = 8

k = 8/3

Hence the equation is of:

y = 8x/3.

The number of acres that will be pollinated by 18 beehives is then given as follows:

y = 8(18)/3

y = 48 acres.

Therefore, the number of acres pollinated by the bees in 18 beehives is 48 acres.

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The percentage of percentage of the apples have diameters that are between 6.34cm and 8.5cm is given as follows:

99.7%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

The measures of 6.34 cm and 8.50 cm are the bounds exactly within three standard deviations of the mean, hence the percentage is given as follows:

99.7%.

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Answer:

do it by yourself what you do while teacher is teaching

Answer:

2x+9y=18

Step-by-step explanation:

Answer:

Step-by-step explanation:

C (Wait for another defendant, check with him and write this answer)

The area of the shaded region is 117.8 m².

Given is a semicircle and a triangle in it, we need to find the area that is shaded,

So, we know that a triangle in a semicircle is always a right triangle whose hypotenuse is the diameter of the semicircle,

So, let us find the hypotenuse of the triangle (diameter of the semicircle) using the Pythagorean theorem,

diameter / hypotenuse = √21.6²+9²

= 23.4m

So the radius = 23.4/2 = 11.7 m

To find the area of the shaded region we will subtract the area of triangle from the semicircle,

So,

Area of the shaded region = π×r²/2 - 1/2×base×height

= 3.14×11.×7²/2 - 1/2×21.6×9

= 215 - 97.2

= 117.8 m²

Hence the area of the shaded region is 117.8 m².

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The value of m and n are 2√3 and 4√3/3 respectively.

The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

Sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

There are special angles in trigonometry, examples are; 60° , 30° and 90°. This angles have exact values and can be calculated without using calculator.

Tan 60 = n/2

√3 = n/2

n = 2√3

cos 60 = 2/m

√3/2 = 2/m

m√3 = 4

m = 4/√3

= 4√3/3

Therefore the value of m and n are 2√3 and 4√3/3 respectively.

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Answer:

1. $990

2. Sn = Sn-1 + (r/100) * Sn-1

3. $27,037.44

Step-by-step explanation:

1. The interest earned each year can be calculated using the simple interest formula:

Simple Interest = (Principal * Rate * Time) / 100

Here, Principal = $22,000, Rate = 4.5%, and Time = 1 year

So, the interest earned each year would be:

= (22,000 * 4.5 * 1) / 100

= $990

Therefore, the interest earned each year would be $990.

2. The recurrence relation to model the value of investment from year to year is:

Sn = Sn-1 + (r/100) * Sn-1

where Sn represents the value of the investment after n years, Sn-1 represents the value after n-1 years, and r represents the annual interest rate.

Using this recurrence relation, we can calculate the value of the investment for different years:

- S1 = 22,000 + 990 = 22,990

- S2 = 22,990 + (4.5/100) * 22,990 = 24,026.55

- S3 = 24,026.55 + (4.5/100) * 24,026.55 = 25,103.46

And so on...

3. To determine the value of the investment after 5 years, we can simply substitute n = 5 in the recurrence relation:

S5 = S4 + (r/100) * S4

= S3 + (r/100) * S3 + (r/100) * S3

= S2 + (r/100) * S2 + (r/100) * S2 + (r/100) * S2

= S1 + (r/100) * S1 + (r/100) * S1 + (r/100) * S1 + (r/100) * S1

Substituting values from previous calculations:

S1 = 22,000 + 990 = 22,990

So,

S5 = 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990

= $27,037.44

Therefore, the value of the investment after 5 years would be $27,037.44.

Answer: To determine the correct answer, we need to calculate the t-test statistic using the given information.

The formula for the t-test statistic for a one-sample t-test is:

t = (x - μ) / (s / sqrt(n))

where x is the sample mean, μ is the hypothesized population mean (which is not given in this question), s is the sample standard deviation, and n is the sample size.

Here, x = 63.2 minutes, s = 7.7 minutes, and n = 24 cases. We are not given a hypothesized population mean, so we cannot calculate the exact t-test statistic. However, we can use the sample mean as an estimate of the population mean for the purposes of this question.

Plugging in the values, we get:

t = (63.2 - 60) / (7.7 / sqrt(24))

t = 2.04

Therefore, the correct answer is (a) t = 2.04.

To determine Duane's taxable income, we need to know his tax rate. Let's assume that his tax rate is a flat 20%.

We can use the formula:

taxable income = tax bill / tax rate

Plugging in the values given, we get:

taxable income = $1000 / 0.20

taxable income = $5000

Therefore, Duane's taxable income was $5000.

Answer:

Without more information, it is not possible to determine Duane's taxable income.

The Taylor polynomial T3(x) for the function f centered at the number a.

f(x) = e^x, a = 1 is

[tex]T3(x0 = e + e(x-1) + e(x-1)^ 2/2 + e(x-1)^3/6[/tex]

A polynomial is  described as an expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

The Taylor polynomial of degree n for a function f(x) centered at a is given by the formula:

[tex]pn(x)=f(c)+f′(c)(x−c)+f′′(c)2!....[/tex]

In this formula, f'(1), f''(1), f'''(1), and f^(n)(1) are the first, second, third, and nth derivatives of f(x), respectively, evaluated at a.

The symbol n! denotes the factorial of n, which is the product of all positive integers from 1 to n.

In the scenario above, we are given the function f(x) = e and a = 1.

The first derivative of e^x is e^x,

the second derivative is e^x,

and the third derivative is also eˣ.

We then evaluate these derivatives at a = 1, we get:

f(1) = e¹ = e

f'(1) = e¹ = e

f''(1) = e¹ = e

f'''(1) = e¹ = e

We substitute  these values into the formula for the Taylor polynomial, and simplify to get Taylor polynomial T3(x) for the function:

[tex]T3(x0 = e + e(x-1) + e(x-1)^ 2/2 + e(x-1)^3/6[/tex]

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Answer: 15 x 162 = 2430$

The station earned $1485 in revenue from selling the gas.

In this problem, we are given two vectors, v and w, representing the number of gallons of gas sold at a station and the corresponding prices per gallon, respectively. We are asked to find the dot product of these two vectors, which is a scalar quantity known as the "Vw". We will then interpret the meaning of this dot product in practical terms.

To find the dot product of v and w, we will use the formula:

Vw = v . w = (210)(2.8) + (300)(2.99)

Vw = 588 + 897 = 1485

Therefore, the value of Vw is 1485.

Practical terms: The dot product of two vectors is a scalar quantity that represents the "projection" of one vector onto the other. In this case, the dot product Vw represents the total revenue earned by selling regular and premium gasoline at the given station.

The components of v represent the number of gallons of regular and premium gas sold, while the components of w represent the respective prices per gallon for each kind of gas. Multiplying the number of gallons sold by the price per gallon gives the total revenue earned for each type of gas. Adding these two values together gives the total revenue earned for both types of gas.

Therefore, the dot product Vw represents the total revenue earned by selling all the gas at the given station. In practical terms, this means that the station earned $1485 in revenue from selling the gas.

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Answer:

6

Step-by-step explanation:

bc

1. The standard quantity of plastic allowed to make 3,500 helmets is 2,310 kilograms.

2. The standard materials cost allowed to make 3,500 helmets is $16,170.

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.

1. The standard quantity of kilograms of plastic (SQ) allowed to make 3,500 helmets can be calculated as:

SQ = Standard quantity per unit × Actual output

= 0.66 kg/helmet × 3,500 helmets

= 2,310 kg

Therefore, the standard quantity of plastic allowed to make 3,500 helmets is 2,310 kilograms.

2. The standard materials cost allowed (SQ × SP) to make 3,500 helmets can be calculated as:

Standard materials cost allowed = Standard price × Standard quantity allowed

= $7.00/kg × 2,310 kg

= $16,170

Therefore, the standard materials cost allowed to make 3,500 helmets is $16,170.

3. The materials spending variance can be calculated as the difference between the actual cost incurred and the standard cost allowed:

Materials spending variance = Actual materials cost - Standard materials cost allowed

= $16,401 - $16,170

= $231 (Favorable)

Therefore, the materials spending variance is $231 (Favorable).

4. The materials price variance and the materials quantity variance can be calculated as follows:

Materials price variance = (Actual price - Standard price) × Actual quantity

= ($16,401/2,485 kg - $7.00/kg) × 2,485 kg

= $9,141.77 (Unfavorable)

Materials quantity variance = (Actual quantity - Standard quantity allowed) × Standard price

= (2,485 kg - 2,310 kg) × $7.00/kg

= $1,225 (Unfavorable)

Therefore, the materials price variance is $9,141.77 (Unfavorable) and the materials quantity variance is $1,225 (Unfavorable).

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The formula for the linear function is f(p) = -50p + 1750. So, correct option is E.

To find the formula for the linear function from the given table, we need to determine the equation of a straight line that passes through the three given points. We can use the point-slope form of the equation of a straight line to solve for the slope and y-intercept.

Let's choose two of the given points, say (15, 1000) and (25, 500), to calculate the slope:

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

= (500 - 1000)/(25 - 15)

= -50

Now, we can use the slope and one of the given points, say (15, 1000), to solve for the y-intercept using the point-slope form:

y - y₁ = m(x - x₁)

y - 1000 = -50(x - 15)

y - 1000 = -50x + 750

y = -50x + 1750

Therefore, the formula for the linear function is f(p) = -50p + 1750, which means that for each additional dollar increase in price per shirt, the number of shirts sold decreases by 50. Option (e) is the correct answer.

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Answer:

9.38

Step-by-step explanation:

I did the math

The length of circumference of this circle is 21.9911 inches.

The area of this circle is, approximately, 38.4845 (in²).

The circumference of the circle can be easily found by utilizing the "π" (pi) number.This number is one of the most recognizible and emblematic numbers in math because it's the value of the ratio between any circle's length of circumference to it's diameter. Therefore, another way to express π is:

[tex]\sf \pi =\dfrac{s}{d}[/tex], where "s" is the length of the circumference of a circle, and "d" is the diameter of that same circle.

The value of π is not a variable, it is a constant for all circles, and it's, approximately, 3.141592653589793238... But don't worry, majority of calculator, if not all of them, have a button with the π so you can just click on it and have that value written automatically.

Fun fact: This number doesn't really have an end for its decimal figures because it's irrational.

So now, taking the equation of π presented previously, we can rewrite the equation by solving it for "s", which is our variable of interest for this first parth. This is the process of that rewritting:

[tex]\sf \pi(d) =\dfrac{s}{d}(d)\\ \\\\\pi(d) =s\\ \\ \\s=\pi(d)[/tex]

We're given the diameter of this circle, which is 7 inches. Now, substitute letter "d" on the formula by "7 inches" and calculate:

[tex]\sf s=\pi(7(in))=\boxed{\sf 21.9911(in)}.[/tex].

Remember that π is an irrational number, so any calculation involving it will result in an irrational answer aswell, unless the π is cancelled by another π.

The length of circumference of this circle is 21.9911 inches.

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These are the most commonly used formulas for the area of circles:

[tex]\sf1) A=\pi r^{2}[/tex]; where "r" is the radius of the circle.

[tex]\sf2) A=\dfrac{\pi d^{2}}{4}[/tex]; where "d" si the diameter of the circle.

Remember that the difference between the radius and diameter is just that the radius is half of the diamaterer. So, technically, everytime you have either the diameter of the radius, you can get both of the parameters.

Let's use the area formula that directly involves diameter to avoid any conversions.

Substitute letter "d" by the length of the diameter (7 inches).

[tex]\sf A=\dfrac{\pi d^{2}}{4}=\dfrac{\pi (7(in))^{2}}{4}=\pi \dfrac{49}{4} (in^{2} )=\boxed{\sf 38.4845(in^{2} )}.[/tex]

Therefore, the area of this circle is, approximately, 38.4845 (in²).

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