an experiment of study times versus test scores found a correlation coefficient of r = 0.49. how would you describe this relationship?

The distributional derivatives of the given function f(x) are:

f'(x) = 3x2 + 4x for x<1, 4x3 + 1 for x>1, and f'(1-) = 5, f'(1+) = 7, and F"(1) = 2.

To evaluate the distributional derivatives of the given function f(x), we need to consider two cases: x<1 and x>1.

Case 1: x<1

For x<1, f(x) = x3 + 2x2 - 1, which is a smooth function. Therefore, f'(x) = 3x2 + 4x and F"(x) = 6x + 4.

Case 2: x>1

For x>1, f(x) = x4 + x + 1, which is a smooth function. Therefore, f'(x) = 4x3 + 1 and F"(x) = 12x2.

At x=1, the function f(x) is discontinuous. We can evaluate the distributional derivatives at x=1 using the following formula:

f'(1-) = lim(x→1-) [f(x) - f(1)]/(x-1) = lim(x→1-) [x3 + 2x2 - 1 - 2]/(x-1) = 5

f'(1+) = lim(x→1+) [f(x) - f(1)]/(x-1) = lim(x→1+) [x4 + x + 1 - 6]/(x-1) = 7

F"(1) = f'(1+) - f'(1-) = 7 - 5 = 2

Therefore, the distributional derivatives of the given function f(x) are:

f'(x) = 3x2 + 4x for x<1, 4x3 + 1 for x>1, and f'(1-) = 5, f'(1+) = 7, and F"(1) = 2.

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Answer:

A. The answer is actually correct .

Step-by-step explanation:

Eliminate B. because the answer is correct

Eliminate C. & D. since step 2 & 3 are correct.

Hope this helped

To calculate the flux of the vector field \vec{F}(x,y,z) = 4 \vec{i} - 7 \vec{j} + 9 \vec{k} through the square of side length 5 lying in the plane 4x + 4y + 2z = 1, we need to use the flux integral:

\iint_S \vec{F} \cdot d\vec{S}

where S is the square and d\vec{S} is the outward-pointing unit normal vector to the surface.

To parametrize the square, we can use the variables x and y as parameters, and solve for z in terms of x and y using the equation of the plane:

z = (1 - 4x - 4y) / 2

The bounds for x and y are 0 to 5, since the side length of the square is 5. So we have:

0 <= x <= 5

0 <= y <= 5

The outward-pointing unit normal vector to the surface can be found by taking the gradient of the equation of the plane and normalizing it:

\nabla(4x + 4y + 2z) = 4\vec{i} + 4\vec{j} + 2\vec{k}

|\nabla(4x + 4y + 2z)| = \sqrt{4^2 + 4^2 + 2^2} = 6

\vec{n} = \frac{1}{6}(4\vec{i} + 4\vec{j} + 2\vec{k})

Now we can evaluate the flux integral:

\iint_S \vec{F} \cdot d\vec{S} = \iint_S (4\vec{i} - 7\vec{j} + 9\vec{k}) \cdot \vec{n} dS

Substituting in the parametrization of the square and the unit normal vector, we get:

\iint_S (4\vec{i} - 7\vec{j} + 9\vec{k}) \cdot \frac{1}{6}(4\vec{i} + 4\vec{j} + 2\vec{k}) dxdy

= \iint_S \frac{2}{3}(2x + 2y + 1) dxdy

Now we can evaluate the double integral over the square:

\int_0^5 \int_0^5 \frac{2}{3}(2x + 2y + 1) dxdy

= \frac{2}{3} \int_0^5 \left[\int_0^5 (4x + 4y + 2) dy\right] dx

= \frac{2}{3} \int_0^5 (20x + 10) dx

= \frac{2}{3} \left[\frac{1}{2}(20x^2 + 10x)\right]_0^5

= \frac{2}{3} (525)

= \boxed{350}

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Let x be the cost of one rose bush and y be the cost of one carnation.

Then we have the system of equations:

x * 9 + y * 12 = 210

x * 3 + y * 1 = 40

To determine the method that is most efficient to solve this system, we can use the fact that one of the equations has already solved for one of the variables, y.

We can use substitution to solve for the other variable, x.

Substitute y = (40 - 3x) into the first equation:

x * 9 + (40 - 3x) * 12 = 210

Simplify:

x * 9 + 480 - 36x = 210

Solve for x:

9x - 36x = 210 - 480

-27x = -270

x = 10

Substitute x = 10 into the second equation:

10 * 3 + y = 40

Solve for y:

y = 10

Therefore, one rose bush costs $10 and one carnation costs $10.

The most efficient method to solve the system was substitution, since one of the equations already solved for one of the variables.

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Answer:

x=2; x=-3

Step-by-step explanation:

3[tex]x^{2}[/tex]+3x-18=0

We can use the method of completing the square to solve (you can also use the quadratic formula):

3([tex]x^{2}[/tex]+x)-18=0

We can add [tex]\frac{1}{4}[/tex] inside the parentheses because this completes the square, as you will see soon. By adding [tex]\frac{1}{4}[/tex] in the parentheses, we are actually adding [tex]\frac{3}{4}[/tex] to the equation because everything in the parentheses is multiplied by 3. Therefore, we have to add [tex]\frac{3}{4}[/tex] to the other side of the equation to keep both sides equal.

3([tex]x^{2}[/tex]+x+[tex]\frac{1}{4}[/tex])-18=[tex]\frac{3}{4}[/tex]

Add 18 to both sides.

3([tex]x^{2}[/tex]+x+[tex]\frac{1}{4}[/tex])=[tex]\frac{75}{4}[/tex]

Divide by 3 on both sides.

([tex]x^{2}[/tex]+x+[tex]\frac{1}{4}[/tex])=[tex]\frac{25}{4}[/tex]

[tex](x+\frac{1}{2}) ^{2}[/tex]=[tex]\frac{25}{4}[/tex]

Now, take the square root of both sides. Note that there will be a plus minus because squaring the negative of a number will get the same answer as squaring the positive.

x+[tex]\frac{1}{2}[/tex] = ±[tex]\sqrt{\frac{25}{4}}[/tex]

x+[tex]\frac{1}{2}[/tex]=±[tex]\frac{5}{2}[/tex]

We now have two equations and can solve both.

x+[tex]\frac{1}{2}[/tex]=[tex]\frac{5}{2}[/tex]

Subtract 1/2 on both sides to get

x=2

and

x+[tex]\frac{1}{2}[/tex]=-[tex]\frac{5}{2}[/tex]

Subtract 1/2 on both sides to get

x=-3

This confidence interval indicates that we are 99% confident that the true population mean depression score of juveniles living in a group home is between 72.96 and 77.04. Hence, we can say that the sample of juveniles living in a group home with a mean score of 75 on a measure of depression is unlikely to be a chance effect.

Confidence Interval:The confidence interval provides a range of values within which the true population mean is likely to lie with a given probability (level of confidence).Calculating Confidence Interval:To calculate the confidence interval, the formula used is:CI = X ± Zc (SEM)WhereX is the sample meanZc is the critical value of the standard normal distribution corresponding to a given level of confidence (Zc = 2.58 for a 99% confidence level)SEM is the standard error of the meanSEM = SD / √nWhereSD is the sample standard deviationn is the sample sizeCalculation of Confidence Interval:Given,Sample mean, X = 75SD = 2.5n = sample sizeFor a 99% confidence level, Zc = 2.58 (from standard normal distribution table)SEM = 2.5 / √n99% confidence interval is calculated as follows:CI = X ± Zc (SEM)CI = 75 ± 2.58(2.5/√n)CI = 75 ± 2.04CI = (75 - 2.04, 75 + 2.04)CI = (72.96, 77.04)Therefore, the 99% confidence interval is (72.96, 77.04).Results:This confidence interval indicates that we are 99% confident that the true population mean depression score of juveniles living in a group home is between 72.96 and 77.04. Hence, we can say that the sample of juveniles living in a group home with a mean score of 75 on a measure of depression is unlikely to be a chance effect.

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Is a juvenile justice system necessary? Why or why not?Explain in 5-6 complete sentences.

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The estimate for the number of times a person's heart beats in their lifetime is approximately [tex]6.2 x 10^8.[/tex]

To estimate the number of times a person's heart beats in their lifetime, we need to calculate the total number of heartbeats per day and then multiply it by the number of days in a person's lifetime.

Given that a person's heart beats approximately [tex]10^5[/tex] times each day, we can multiply this value by the number of days in 81 years. To convert years to days, we multiply 81 by 365 (assuming there are 365 days in a year).

Calculating the total number of heartbeats in a lifetime:

Number of heartbeats per day = [tex]10^5[/tex][tex]6.2 x 10^8.[/tex]

Number of days in 81 years = 81 * 365

Total number of heartbeats in a lifetime = [tex](10^5) * (81 * 365)[/tex]

Simplifying the calculation:

Total number of heartbeats in a lifetime = [tex]8.1 x 10^4 * 2.96 x 10^4[/tex]

Multiplying the values:

Total number of heartbeats in a lifetime = 2.3976 x 10^9

Rounding to two significant figures:

Total number of heartbeats in a lifetime ≈[tex]6.2 x 10^8[/tex]

Therefore, the estimate for the number of times a person's heart beats in their lifetime is approximately[tex]6.2 x 10^8.[/tex]

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The exponential function that models the value of the car is given as follows:

[tex]f(t) = 18000(0.84)^t[/tex]

The monthly rate of change is given as follows:

Decay of 1.44%.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter values for this problem are given as follows:

Hence the function is:

[tex]f(t) = 18000(0.84)^t[/tex]

After one month, the value of the car is given as follows:

[tex]f\left(\frac{1}{12}\right) = 18000(0.84)^{\frac{1}{12}}[/tex]

[tex]f\left(\frac{1}{12}\right) = 17740.3607[/tex]

The percentage is:

17740.3607/18000 = 98.56%.

Hence it is a decay of 100 - 98.56 = 1.44%.

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1. The amplitude of the function is 2

2. The period of the function is 1

To find the parameters, we examine the equation to identify the functions present and compare with a general formula

The cos function is written considering the general formula in the form

sine function, y = A sin (bx + c) + d

where

A = amplitude

b = 2π / period

c  = phase shift

d = vertical shift

In the problem the values equation is G(x) = 2 cos [2π(x+2π/3)] - 5

rewriting the equation results to

G(x) = 2 cos (2πx + 4π²/3) - 5

A = 2

b = 2π / period = 2π

period = 1

Phase shift, c

c = 4π²/3

vertical translation of the function, d  = -5

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The formula for the area of a circle A = πr^2, where A is the area, π is the mathematical constant pi, and r is the radius.

In this case, the radius So, the area of the tabletop is:

A = πr^2

A = (22/7) x (5/2)^2

A = (22/7) x (25/4)

A = 550/28 radius is given as 2 1/2 feet, or 5/2 feet in fractional form.

A = 19 11/28 square feet

Therefore, the area of Melba's tabletop is approximately 19 11/28 square feet

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The answers is explained below.

Events in Probability:

In Statistics, an event is any target outcome or set of outcomes. Perhaps we want to obtain outcome A or B in one single trial, or perhaps we want outcome A followed immediately by outcome B in two subsequent trials. The probability of an event can be calculated from the probabilities of the outcomes in the event.

The definition of mutually exclusive and independent events. Then, let's see how they relate to one another.

First, two events are mutually exclusive if they cannot happen at the same time.

For example, the outcomes of head and tails on a coin flip are mutually exclusive, because a coin can't be showing heads and tails at the same time.

Second, two events are independent if they do not influence each other. For example, obtaining a result of heads on the first coin toss is independent of obtaining a result of heads on the second coin toss. This is because the coin doesn't remember the first outcome it generated, and so the outcome of the second coin toss is not related in any way to the result of the first coin toss.

Using the examples that we gave as part of our definitions, we can see that this statement is true.

This is because when we have mutually exclusive events, they are dependent on one another. This is because in order for one outcome to occur, the other outcome cannot occur, and vice versa.

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The total number of ways to choose 12 ponds from 40 ponds is given by the combination:

C(40, 12) = 40! / (12! * 28!) = 95,171,280

Out of these 12 ponds, we want to choose 8 ponds with crayfish and 4 ponds without crayfish. The number of ways to do this is given by the product of two combinations:

C(25, 8) * C(15, 4) = (25! / (8! * 17!)) * (15! / (4! * 11!)) = 4,989,600

The probability of choosing 8 ponds with crayfish and 4 ponds without crayfish is the number of favorable outcomes divided by the total number of outcomes:

P = 4,989,600 / 95,171,280 = 0.0524

Therefore, the probability of choosing 8 ponds with crayfish and 4 ponds without crayfish is approximately 0.0524 or 5.24%.

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Its components by their greatest common factor, which is 2:

To find a normal vector to the plane with the equation 8x - 14y - 6z = 0, we can simply read off the coefficients of x, y, and z and use them as the components of the normal vector. So, the normal vector is:

⟨8, -14, -6⟩

Note that this vector can be simplified by dividing all its components by their greatest common factor, which is 2:

⟨4, -7, -3⟩

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The experimental probability of Sarah arriving after 7:30 A.M. on her next day of work is 0.5167.

Experimental probability is the likelihood of an event occurring based on actual outcomes of an experiment or trial.

It is calculated by dividing the number of times an event occurs by the total number of trials or experiments performed.

Let's calculate the experimental probability of Sarah arriving after 7:30 A.M on her next day of work:

Total number of days Sarah has worked = 60

Number of days she arrived before 7:30 A.M. - 29

Number of days she arrived after 7:30 A.M.

= 60 - 29

= 31

Experimental probability of Sarah arriving after 7:30 A.M.

on her next day of work = Number of times she arrived after 7:30 A.M. / Total number of days she has worked= 31/60

= 0.5167 (rounded to four decimal places)

Therefore, the experimental probability of Sarah arriving after 7:30 A.M. on her next day of work is 0.5167.

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Answer: By using double-angle formula  sin(2) = 120/169.

Step-by-step explanation:

We can use the following double-angle formula for the sine function: sin(2θ) = 2sin(θ)cos(θ).

First, we need to get the value of cos().

We can use the fact that sec() is negative, which means that cos() is also negative.

We know that:

sec() = 1/cos()Since sec() is negative, we can conclude that cos() is also negative.

Now, we can use the Pythagorean identity to get cos():

cos() = -sqrt(1 - sin()^2) = -sqrt(1 - (-5/13)^2) = -12/13

Next, we can use the double-angle formula to get sin(2):

sin(2) = 2sin()cos() = 2(-5/13)(-12/13) = 120/169

Therefore, sin(2) = 120/169.

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This verifies that y = cos(4x) also satisfies the differential equation.

The given solutions satisfy the differential equation.

The given differential equation is y'' + 16y = 0, and the proposed solutions are y = sin(4x) and y = cos(4x). To verify, we need to find the second derivative (y'') of each solution and plug it into the equation.

For y = sin(4x), the first derivative (y') is 4cos(4x) and the second derivative (y'') is -16sin(4x). Now, substitute y and y'' into the equation: (-16sin(4x)) + 16(sin(4x)) = 0, which simplifies to 0 = 0. This verifies that y = sin(4x) satisfies the differential equation.

For y = cos(4x), the first derivative (y') is -4sin(4x) and the second derivative (y'') is -16cos(4x). Substitute y and y'' into the equation: (-16cos(4x)) + 16(cos(4x)) = 0, which simplifies to 0 = 0.

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a. the Temp variable has a roughly normal distribution with a peak around 80°F. b. a cluster of points with higher ozone concentrations at lower temperatures.

a. The histogram of Ozone and Temp shows that Ozone has a skewed distribution with a long right tail, while the Temp variable has a roughly normal distribution with a peak around 80°F.

b. The scatterplot of temperature and ozone indicates a negative correlation between the two variables. As temperature increases, ozone concentration tends to decrease. There are a few interesting features, such as a cluster of points with higher ozone concentrations at lower temperatures.

c. It is not clear whether the linear regression model would be a good choice for these data without further investigation. The error terms for different days are likely to be correlated with one another, as air quality is affected by many factors that persist over time, such as weather patterns and seasonal changes.

d. The linear regression model estimates a slope of -0.052 and an intercept of 3.472. The slope suggests that for each one-degree increase in temperature, the ozone concentration decreases by 0.052 parts per billion, on average. The intercept represents the estimated ozone concentration when the temperature is 0°F. However, the interpretation of the intercept may not be meaningful given that the range of temperatures in the data is much higher than 0°F.

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The area of the triangle is 15.5 square units.

To find the area of the triangle with the given vertices, we can use the formula:

A = 1/2 * |(x1y2 + x2y3 + x3y1) - (x2y1 + x3y2 + x1y3)|

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices.

Substituting the given values, we get:

A = 1/2 * |(03 + 58 + 30) - (50 + 33 + 08)|

A = 1/2 * |(0 + 40 + 0) - (0 + 9 + 0)|

A = 1/2 * |31|

A = 15.5

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(2) of Theorem 7.8 in additive notation states that if G is a finite additive group of order n, then for all elements a in G, a^n = 0.

Statements (1)-(3) of Theorem 7.9 in additive notation are:

(1) For all a,b in G, ab = ba (commutativity property)

(2) There exists an element 0 in G such that a + 0 = a for all a in G (identity property)

(3) For all a in G, there exists an element -a in G such that a + (-a) = 0 (inverse property)

Explanation:

Theorems 7.8 and 7.9 are important results in abstract algebra that pertain to additive groups. The additive notation used in the theorems allows us to better understand the properties and behavior of these groups.

Theorem 7.8 tells us that in a finite additive group of order n, every element raised to the power of n equals 0. This is a powerful result that can be used to prove many other theorems in group theory.

Theorem 7.9 outlines the properties that must hold true in any additive group. These properties include commutativity (property 1), identity (property 2), and inverse (property 3). These properties are essential for understanding the behavior of additive groups and how they interact with each other.


Theorems 7.8 and 7.9 provide important insights into the behavior and properties of additive groups. The additive notation used in the theorems allows us to more easily understand and analyze the behavior of these groups. By understanding these theorems and the properties of additive groups, we can better understand many other important results in abstract algebra.

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The equivalent equation for a two-input XOR gate is y = ab’ + a’b.

A two-input XOR gate is a logic gate that outputs a high or 1 signal only when the two inputs are different. In other words, the output of an XOR gate is 1 when one input is 0 and the other input is 1, and vice versa.

To represent an XOR gate with an equation, we can use Boolean algebra. The Boolean expression for an XOR gate is y = ab’ + a’b, where y is the output, a and b are the two inputs, and a' and b' represent the complement (or NOT) of a and b, respectively.

This equation can be derived using the laws of Boolean algebra. For example, we know that the product of a variable and its complement is always 0, i.e., a a' = 0. Using this property, we can simplify the equation y = ab’ + a’b as follows:

y = ab’ + a’b
 = ab’ + ab’’ + a’b     (adding a' and b')
 = ab’ + a’b + ab’’    (rearranging terms)
 = ab’(1) + a’b(1)     (using a a' = 0 and b b' = 0)
 = ab’ + a’b            (simplifying)

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One dose of tolbutamide for this order is one half (1/2) of a 0.5 g scored tablet or one full 250 mg tablet.

To calculate the dose of tolbutamide for one administration, we first need to know how many tablets are needed. The supply of tolbutamide is in 0.5 g scored tablets, which is the same as 500 mg.
For the order of tolbutamide 250 mg p.o. b.i.d. (twice a day), we need to divide the total daily dose (500 mg) by the number of doses per day (2). This gives us 250 mg per dose.
Therefore, one dose of tolbutamide for this order is one half (1/2) of a 0.5 g scored tablet or one full 250 mg tablet.

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Answer: We can evaluate the iterated integral in the following order:

First, we integrate with respect to z from 0 to xy:

∫ xy 0 y^(-1)z dz = 1/2y^(-1)z^2 |_0^xy = 1/2x^2

Next, we integrate the result with respect to y from 1 to x:

∫ 1+x 1 1/2x^2 dy = 1/2x^2[y]_1^(1+x) = 1/2x^2(1+x-1) = x^3/4

Finally, we integrate the previous result with respect to x from 0 to 1:

∫ 1 0 x^3/4 dx = 1/4 * x^4/4 |_0^1 = 1/16

Therefore, the value of the iterated triple integral is 1/16.

Answer:

(a) 1, 2, 3, 4, 6, 8, 12, 24

(b) 1, 2, 3, 4, 6, 9, 12, 18, 36

(c) 1, 5, 7, 35

(d) 1, 2, 4, 8, 16, 32

It will take approximately 4.67 years (rounded to two decimal places) for the fish population to reach 9000 fish.

To find the time it takes for the fish population to reach 9000 fish, we need to solve for t in the equation P(t) = 9000. Substituting the given values, we get:

9000 = 2000 + (8000 - 2000)/(1 + 3e^(-0.2t))

Simplifying this equation, we get:

(1 + 3e^(-0.2t))(9000 - 2000) = 8000

3e^(-0.2t) = 1

Taking the natural logarithm of both sides, we get:

ln(3) - 0.2t = 0

0.2t = ln(3)

t = ln(3)/0.2 ≈ 4.67

Therefore, it will take approximately 4.67 years (rounded to two decimal places) for the fish population to reach 9000 fish.

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A simple random sample (SRS) is described by option d: every possible sample of a given size has the same chance to be selected.

A simple random sample is a sampling method where each possible sample of a given size has an equal chance of being selected from the population.

Among the given options, option d is the one that accurately describes a simple random sample. It states that every possible sample of a given size has the same probability of being selected.

In a simple random sample, each member of the population has an equal and independent chance of being included in the sample. This ensures that the sample is representative of the population and minimizes bias. By selecting samples randomly, we eliminate the potential for systematic or intentional selection, ensuring that all individuals in the population have an equal opportunity to be included.

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Answer: No solution

Step-by-step explanation:

Hi there, to set up this problem you are first going to draw a triangle and label the angles A, B, and C. The sides opposite from the vertexes are going to be labeled a, b and c. Fill in the information as provided to you in the problem.

You are given angle m<A=123 , the side across is a=45, and c=24. You know to use law of sines for this problem because you are given pieces of information that correspond with the same letter (A and a).

Start by setting up a proportion with that looks like

(45/sin(123)) = (24/sin(C))  

You are looking to solve for the remaining angles and sides, but when you cross multiply and divide, you end up with arcsin(1.573), which does not provide a solution for m<C and also means that there are no solutions to this triangle.

Hope this helps.

The only possible triangle that satisfies the given conditions has sides of length a = 45, b = 57.58, and c = 24, and angle measures of A = 123º, B = 31.7º, and C = 25.3º.

According to the Law of Sines, in a triangle ABC:

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

Where a, b, and c are the lengths of the sides, and A, B, and C are the opposite angles, respectively.

Using the given information:

a = 45

c = 24

∠a = 123º

We can solve for the remaining parts of the triangle as follows:

sin(A) = a/csc(∠a) = 0.298

Since sin(A) < 1, there is only one possible triangle that can satisfy the given conditions.

Using the Law of Sines:

b/sin(B) = c/sin(C)

b/sin(B) = 24/sin(∠B)

b = 24(sin(A))/sin(∠B) = 57.58

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The probability that student chosen at random is 16 years is 0.28.

The table shows the distribution of student by age in a high school with 1500 students. Therefore, the probability that randomly chosen student is  16 years can be found as follows:

Therefore,

probability = number of favourable outcome to age / total number of possible outcome

Hence,

probability that student chosen at random is 16 years =  420 / 150

probability that student chosen at random is 16 years = 0.28

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A solution x to the homogeneous linear system of equations Ax = 0 is orthogonal to the row vectors of A because the dot product of x and each row vector in A is equal to 0.

Let's consider a solution, x, to the homogeneous linear system of equations Ax = 0, and discuss why x is orthogonal to the row vectors of A.
The homogeneous linear system of equations can be represented as Ax = 0,

where A is the matrix of coefficients, x is the solution vector, and 0 is the zero vector.

When we say that x is orthogonal to the row vectors of A, we mean that the dot product of x and each row vector is equal to 0.

Let's consider the i-th row vector of A, represented as [tex]a_i.[/tex]

To find the dot product of x and a_i, we multiply the corresponding elements of the two vectors and then sum up the results: [tex]a_i . x = a_i1 \times  x1 + a_i2 \times  x2 + ... + a_in \times  xn.[/tex].
Now, let's recall the matrix-vector multiplication in Ax = 0.

Each element in the result vector 0 is obtained by taking the dot product of a row vector from A and the solution vector x.

So, for the i-th element in the zero vector, we have:[tex]0 = a_i . x.[/tex]
Since the dot product of each row vector [tex]a_i[/tex] and the solution vector x is equal to 0, we can conclude that x is orthogonal to the row vectors of A.

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Question:  Let x be a solution to the m×n homogeneous linear system of equations Ax=0. Explain why x is orthogonal to the row ve…

Writing Let x be a solution to the m×n homogeneous linear system of equations Ax=0. Explain why x is orthogonal to the row vectors of A

We are given i.i.d. Bernoulli trials with success probability p, and we need to find the conditional probability mass function of Sm, given Sn-k. The distribution that arises in this problem is the binomial distribution.

The binomial distribution is the probability distribution of the number of successes in a sequence of n independent Bernoulli trials, with a constant success probability p. In this problem, we are considering a subsequence of n-k trials, and we need to find the conditional probability mass function of the number of successes in a subsequence of m trials, given the number of successes in the remaining n-k trials. Since the Bernoulli trials are independent and identically distributed, the probability of having k successes in the remaining n-k trials is given by the binomial distribution with parameters n-k and p.

Using the definition of conditional probability, we can write:

P(Sm = s | Sn-k = k) = P(Sm = s and Sn-k = k) / P(Sn-k = k)

=[tex]P(Sm = s)P(Sn-k = k-s) / P(Sn-k = k)[/tex]

=[tex](n-k choose s)(p^s)(1-p)^(m-s) / (n choose k)(p^k)(1-p)^(n-k)[/tex]

where (n choose k) =n! / (k!(n-k)!)  is the binomial coefficient.

We can use this conditional probability mass function to find E[Sm | Sn-k]. By the law of total expectation, we have:

[tex]E[Sm] = E[E[Sm | Sn-k]][/tex]

=c[tex]sum{k=0 to n} E[Sm | Sn-k] P(Sn-k = k)\\= sum{k=0 to n} (m(k/n)) P(Sn-k = k)[/tex]

where we have used the fact that E[Sm | Sn-k] = mp in the binomial distribution.

Thus, the conditional probability mass function of Sm, given Sn-k, leads to an expression for the expected value of Sm in terms of the probabilities of Sn-k.

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To determine the size of carpets that will maximize the company's revenue, we need to find the dimensions that will generate the highest total sales. Let's analyze the situation step by step.

We know that the company can sell 200 carpets per week when the size is 3ft by 3ft. Beyond this size, for each additional foot of length and width, the number sold decreases by 4.

Let's denote the additional length and width beyond 3ft as x. Therefore, the dimensions of the carpets will be (3 + x) ft by (3 + x) ft.

Now, we need to determine the relationship between the number of carpets sold and the dimensions. We can observe that for each additional foot of length and width, the number sold decreases by 4. So, the number of carpets sold can be expressed as:

Number of Carpets Sold = 200 - 4x

Next, we need to calculate the revenue generated from selling these carpets. The price per square foot is $5, and the area of the carpet is (3 + x) ft by (3 + x) ft, which gives us:

Revenue = Price per Square Foot * Area

= $5 * (3 + x) * (3 + x)

= $5 * (9 + 6x + [tex]x^2)[/tex]

= $45 + $30x + $5[tex]x^2[/tex]

Now, we can determine the dimensions that will maximize the revenue by finding the vertex of the quadratic function. The x-coordinate of the vertex gives us the optimal value of x.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = $5 and b = $30.

x = -30 / (2 * 5)

x = -30 / 10

x = -3

Since we are dealing with dimensions, we take the absolute value of x, which gives us x = 3.

Therefore, the additional length and width beyond 3ft that will maximize the revenue is 3ft.

The dimensions of the carpets that the company should sell to maximize its revenue are 6ft by 6ft.

To calculate the maximum weekly revenue, we substitute x = 3 into the revenue function:

Revenue = $45 + $30x + $[tex]5x^2[/tex]

= $45 + $30(3) + $5([tex]3^2)[/tex]

= $45 + $90 + $45

= $180

Hence, the maximum weekly revenue for the company is $180.

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