Find a tangent vector of unit length at the point with the given value of the parameter t.r(t) = 2 sin(t)i + 3cos(t)jt= pi/6Please express answer in terms of i + j

 The probability of [tex]3X1-1X2 + 3X3[/tex] being greater than 20 is given by[tex]P(3X1-1X2 + 3X3 > 20) = 1- Φ((20-3μ1+μ2-3μ3)/(√3σ11+σ22+3σ33))[/tex].

In this case, [tex]μ1=10, μ2=10, μ3=10, σ11=0.3, σ22=0.3, σ33=0.3,[/tex] so the probability of [tex]3X1-1X2 + 3X3[/tex] being greater than 20 is 1-Φ(-1.0).

1. To answer this question, we can use the formula for a multivariate normal distribution.

The probability of the profit for selling chocolate being greater than 6 million is given by P(X1 > 6) = 1- Φ(6-μ1)/(√σ11). In this case, μ1=10, σ11=0.3, so the probability of the profit being greater than 6 million is 1-Φ(2.667).

2. To answer this question, we need to use the formula for the conditional probability of a multivariate normal distribution.

The probability of the profit for selling chocolate being greater than 6 million, given the sales of marshmallow is 5 million and the sales of graham cracker is 5 million, is given by

[tex]P(X1>6 | X2=5, X3=5) = 1- Φ((6-μ1-Σ12*5-Σ13*5)/(√σ11-Σ12²-Σ13²))[/tex]. In this case,

[tex]μ1=10, σ11=0.3, Σ12=0.3, Σ13=0.3,[/tex]so the probability of the profit being greater than 6 million is 1-Φ(-0.1).

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The solution to the equation (3√x) - √(9x-17) = 1 is x = 9.

Given the equation in the question (3√x) - √(9x-17) = 1.

To solve for x in the given equation:

(3√x) - √(9x-17) = 1

We can start by isolating the square root term on one side of the equation. Adding √(9x - 17) to both sides, we get:

(3√x) = √(9x - 17) + 1

Squaring both sides of the equation, we get:

(3√x)² = (√(9x - 17) + 1)²

9x = -16 + 2√(9x - 17) + 9x

Solve for 2√(9x - 17)

2√(9x - 17) = 16

36x - 68 = 256

Add 68 to both sides

36x - 68 + 68 = 256 + 68

36x = 324

x = 324/36

x = 9

Therefore, the solution is x = 9.

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Answer:

mn(4m - 7n)

Step-by-step explanation:

➠ 4m^2n-7mn^2

➠ 2^2 x m^2n - 7mn^2

➠ mn([tex](\frac{2^2*m^2n}{mn} -\frac{7mn^2}{mn} )[/tex]

➠ mn([tex]2^2*m^{2-1}-(7n^{2-1}))[/tex]

➠ mn(4m - 7n)

We can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles

To answer this question, we need to use the following formula for a confidence interval for the mean: CI = (μ - z*(σ/√n), μ + z*(σ/√n)), Where μ is the population mean, z is the z-score for the given confidence level, σ is the population standard deviation, and n is the sample size. Using the given information, we can calculate the confidence interval for the mean:CI = (23500 - 2.575*(3900/√100), 23500 + 2.575*(3900/√100)), CI = (21342.6, 24637.4)

To summarize, we used the formula for a confidence interval for the mean and the given information to calculate the confidence interval for the average number of miles an automobile is driven annually in Alabama. This confidence interval is (21342.6, 24637.4), which means we can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles.

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The equatiοn is [tex]t = (-20 \± \sqrt{(400 - 4(-5)(-18))}) / 2(-5)[/tex]  tο sοlve fοr the time it takes fοr the οbject tο be at a height οf 18 feet.

Trigοnοmetric equatiοns are equatiοns that invοlve trigοnοmetric functiοns such as sine, cοsine, tangent, etc. These equatiοns usually invοlve finding values οf the unknοwn angle(s) that satisfy the given equatiοn. They can be sοlved using algebraic techniques οr by using the prοperties οf trigοnοmetric functiοns.

Accοrding tο the given infοrmatiοn:

The given functiοn is [tex]f(t) = -5t^2 + 20t[/tex], which mοdels the height οf an οbject in feet as a functiοn οf time in secοnds.

Tο find the number οf secοnds it will take fοr the οbject tο be at a height οf 18 feet after launch, we need tο sοlve the equatiοn [tex]-5t^2 + 20t = 18[/tex].

Tο sοlve this quadratic equatiοn using the quadratic fοrmula, we first identify the values οf a, b, and c frοm the general fοrm οf a quadratic equatiοn, [tex]ax^2 + bx + c = 0[/tex].

In this case, a = -5, b = 20, and c = -18. Substituting these values intο the quadratic fοrmula, we get:

[tex]t = (-b\± \sqrt{(b^2 - 4ac)}) / 2a[/tex]

Plugging in the values οf a, b, and c, we get:

[tex]t = (-20 \± \sqrt{+(20^2 - 4(-5)(-18)})) / 2(-5)[/tex]

Simplifying this expressiοn, we get:

[tex]t = (-20 \± \sqrt{(400 - 360))} / (-10)[/tex]

[tex]t = (-20\± \sqrt{(40)}) / (-10)[/tex]

[tex]t = (-20 \± 2\sqrt{(10)}) / (-10)[/tex]

[tex]t = 2 \± 0.632[/tex]

Therefοre, the twο pοssible values οf t are:

t = 2 + 0.632 = 2.632 secοnds

t = 2 - 0.632 = 1.368 secοnds

Therefοre, the equatiοn that cοrrectly shοws the quadratic fοrmula being used tο determine the number οf secοnds it will take fοr the οbject tο be at a height οf 18 feet after launch is:

[tex]t = (-b\± \sqrt{(b^2 - 4ac)}) / 2a[/tex]

[tex]t = (-20 \± \sqrt{(20^2 - 4(-5)(-18))}) / 2(-5)[/tex]

[tex]t = (-20\± \sqrt{(40)}) / (-10)[/tex]

[tex]t = (-20 \± 2\sqrt{(10)}) / (-10)[/tex]

t = 2 ± 0.632

Therefοre, the equatiοn is [tex]t = (-20 \± \sqrt{(400 - 4(-5)(-18))}) / 2(-5)[/tex] tο sοlve fοr the time it takes fοr the οbject tο be at a height οf 18 feet.

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Answer:

668.376

Step-by-step explanation:

Please hit brainliest if this was helpful!

To simplify 650 – 0.394 + 18.77, we can first add 650 and 18.77 since they're both whole numbers:

650 + 18.77 = 668.77

Then, we can subtract 0.394 from 668.77:668.77 - 0.394 = 668.376

Therefore, 650 – 0.394 + 18.77 simplifies to 668.376.

Step-by-step explanation:

x = +1

x = -2

x = -3

x = -4

Use number line to find the value and fit equation

The z-score for the 25th percentile of a standard normal distribution is approximately -0.625. Here option A is the correct answer.

To find the z-score for the 25th percentile of a standard normal distribution, we need to use a standard normal distribution table or calculator. The 25th percentile corresponds to a cumulative area under the standard normal curve of 0.25.

Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative area of 0.25 is about -0.68. This means that approximately 25% of the area under the standard normal curve lies to the left of -0.625.

So, among the given options, the correct answer is Option A, -0.625, Option D, -0.50, which is also incorrect. Option E, 0.00, is definitely incorrect because the 25th percentile is to the left of the mean.

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Answer:

I gave a couple solutions as I wasn't sure if you were asking for graphing purposes or substituting y=-x+6 into the second equation 2x-y=-9. So I gave both solutions just in case.

for the first equation y=-x+6, y intercept is (0,6)

for equation two 2x-y=-9, y intercept is (0,9)

In both of the equations the x value is 1.

Solving for y without graphing. Y=9+2x

and x=-1

Step-by-step explanation:substitute i

HOWEVER, if you are saying that the top equation is the value of y, then you substitute it into the bottom equation. 2x--x+6=-9 which would be x=-5

It really depends on what is expected of the question. I wasn't sure which one, so I gave a couple different approaches. If you could give more information, such as, are you graphing, that would be great. I'll keep an eye out for any comments.

The 90% confidence interval for the proportion of district residents in favor of starting the school day 15 minutes later is (0.392, 0.548). The true proportion is estimated to lie within this interval with 90% confidence.

To calculate the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later, we can use the following formula:

CI = p ± z*(√(p*(1-p)/n))

where:

CI: confidence interval

p: proportion of residents in favor of starting the day later

z: z- score based on the confidence level (90% in this case)

n: sample size

First, we need to calculate the sample proportion:

p = 94/200 = 0.47

Next, we need to find the z- score corresponding to the 90% confidence level. Since we want a two-tailed test, we need to find the z- score that cuts off 5% of the area in each tail of the standard normal distribution. Using a z-table, we find that the z- score is 1.645.

Substituting the values into the formula, we get:

CI = 0.47 ± 1.645*(√(0.47*(1-0.47)/200))

Simplifying this expression gives:

CI = 0.47 ± 0.078

Therefore, the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later is (0.392, 0.548). We can be 90% confident that the true proportion lies within this interval.

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Answer:

18+m=24, 6

Step-by-step explanation:

You will get the first part by understanding that 24 is the whole and 18 is the part. Part + the other part, m, is the whole. You will then solve this by isolating the variable m, and subtracting 18 on both sides of the equation. Since 24-18=6, that is the final answer.

Answer:

To solve the system of equations:

n + d = 21 ---(1)

0.05n + 0.10d = 1.70 ---(2)

We can use the substitution method by solving for one variable in terms of the other from equation (1) and substituting it into equation (2).

Solving equation (1) for n:

n = 21 - d

Substituting this expression for n into equation (2):

0.05(21 - d) + 0.10d = 1.70

Distributing the 0.05:

1.05 - 0.05d + 0.10d = 1.70

Combining like terms:

0.05d = 0.65

Dividing both sides by 0.05:

d = 13

Substituting this value of d into equation (1):

n + 13 = 21

Solving for n:

n = 8

Therefore, the solution to the system of equations is n = 8 and d = 13.

Answer:30

Step-by-step explanation: the formula is base x height over 2, so (6x10)/2 is 30.

The amount of interest earned on $4,000.00 for 4 years, compounding daily at 4.5% APR, is $1,064.08.

To calculate the amount of interest on $4,000.00 for 4 years, compounding daily at 4.5% APR, we can use the formula for compound interest:

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

where A is the final amount, P is the principal, 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 in years.

In this case, we have P = $4,000.00, r = 0.045, n = 365 (since interest is compounded daily), and t = 4. Plugging these values into the formula, we get:

A = $4,000.00(1 + 0.045/365)^(365*4)

A = $4,000.00(1.0001234)^1460

A = $4,889.68

The final amount is $4,889.68, which means that the interest earned is:

Interest = $4,889.68 - $4,000.00 = $889.68

We are given that the monthly interest table shows that $1.197204 in interest is earned for each $1.00 invested. Therefore, to find the interest earned on $4,000.00, we can multiply the interest earned by the factor:

$1.197204 / $1.00 = 1.197204

Interest earned = $889.68 x 1.197204 = $1,064.08

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Answer:

2x - 13

Step-by-step explanation:

If x represents the greater number, then the other number is x - 13. The sum of these two numbers is:

x + (x - 13) = 2x - 13

If the set of expressions represents measures of the sides of a triangle x, 4, 6 , the range of possible measures of x is 2 < x < 10.

To determine the range of possible measures of X if the set of expressions represents measures of the sides of a triangle x, 4, 6, we need to use the triangle inequality theorem. According to this theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Mathematically, this can be expressed as:

x + 4 > 6

x + 6 > 4

4 + 6 > x

Simplifying these inequalities, we get:

x > 2

x > -2

x < 10

The first two inequalities indicate that x must be greater than 2, since the sum of any two sides of a triangle must be greater than the third side. The third inequality indicates that x must be less than 10, since the longest side of a triangle cannot be greater than the sum of the other two sides.

This means that x can take any value between 2 and 10, but not including 2 or 10, in order for the set of expressions to represent the measures of the sides of a triangle.

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The average power output of a bike rider at a speed of 20 miles per hour is 556 W

Given, Riding a bike burns 100 Calories per mile. Speed of riding a bike is 20 miles per hour. Power is a measure of the rate at which work is done over time.

Therefore, the formula for average power is given by,

P = W/t

Where, P is power, W is work and t is time.

Now, We know that riding a bike burns 100 Calories per mile, but we have to find out the work done.

So, work done = 100 Calories × distance traveled= 100 × 1.609 × 1000 = 160900 J

Time taken = distance/speed= 1.609/20 = 0.08045 hours

Putting values in the formula of power, we get,

P = W/t= 160900/0.08045= 1999999.9 J/hour

Converting hour to seconds,1 hour = 3600 seconds1 J/s = 1 watt1 J/hour = 1 / 3600 watt

Therefore, power = 1999999.9 / 3600= 555.5556 watts (approx)

Therefore, the average power output of a bike rider at a speed of 20 miles per hour is 556 W (approx). Answer: 556

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Therefore, the net result of the transaction is a loss of $21.36. The answer is A) $21.36.

Selling price refers to the price at which a product or service is sold to customers or clients. It is the amount of money that a buyer pays to the seller in exchange for the product or service. The selling price is usually higher than the cost of producing or acquiring the product or service, and the difference between the selling price and the cost is the profit earned by the seller. In some cases, the selling price may also include additional charges such as taxes, shipping fees, or handling fees.

by the question.

To calculate the net result of the transaction, we need to determine how many shares were purchased with the $3,200 investment.

$3,200 divided by $11.95/sh = approximately 267.36 shares (rounded to the nearest hundredth)

Therefore, the total cost of purchasing 267 shares at $11.95/sh is:

267 shares x $11.95/sh = $3,195.65

The total revenue from selling 267 shares at $11.87/sh is:

267 shares x $11.87/sh = $3,174.29

To determine the net result of the transaction, we subtract the total revenue from the total cost:

$3,174.29 - $3,195.65 = -$21.36

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We cannot construct an interval using the normal approximation of survival rate between control and treatment groups because the samples must be random, independent, and their sample sizes must be sufficiently large.

The normal approximation is valid when the sample sizes are large enough to ensure that the sampling distribution of the mean of the variable is approximately normal.

The central limit theorem applies to the distribution of the sample mean when the sample size is large enough, according to the normal approximation.

As a result, the mean difference between the two groups must have a normal distribution. The normal distribution may not be an accurate representation of the underlying distribution of the difference between the two population means in the absence of this requirement, causing the confidence interval to be inaccurate. It will lead to incorrect inferences about the difference in the survival rates of the two groups.

The confidence interval constructed despite this problem will lead to incorrect inferences about the difference in the survival rates of the two groups. This would make it difficult to draw any conclusions based on the findings of this experiment.

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The value of the additional data point is  [tex]$19.17$[/tex].

Let us first find the mean of the given data:

[tex]Mean = \frac{\sum_{i=1}^{n} x_i}{n}=\frac{39 + 45 + 43 + 42 + 44}{5}= 42.6[/tex]

Now let's find the value of the additional data point. Let the value of the additional data point be x. Therefore, the new sum of data is

[tex]$(39+45+43+42+44+x)$[/tex].

Total numbers of data are 6 (five given in the set and one additional data point).So, the mean of the resulting data set is given by:

[tex]32.083 = \frac{(39+45+43+42+44+x)}{6}[/tex]

Multiplying both sides of the equation by 6 we get:

[tex]6 \times 32.083 = (39+45+43+42+44+x)[/tex]

We have the value of [tex]$39+45+43+42+44$[/tex] which is [tex]$213$[/tex].

Therefore, substituting all the values, we get:

[tex]193.83 + x = 213[/tex]

On subtracting [tex]$193.83$[/tex] from both sides, we get the value of

[tex]x. x = 213 - 193.83 = 19.17[/tex]

Therefore, the value of the additional data point is [tex]$19.17$[/tex]

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Hence, 28 toy soldiers are the correct answer.

A group in mathematics is created by combining a set with a binary operation. For instance, a group is formed by a set of integers with an arithmetic operation and a group is also formed by a set of real numbers with a differential operator.

Let's refer to the quantity of toy soldiers as "x".

We are aware that x is within the range of 27 and 54 thanks to the problem.

x can be divided by 4 without any remainders.

The residual is 6 when x is divided by 7.

The leftover after dividing x by five is three.

These criteria allow us to construct an equation system and find x.

Firstly, we are aware that x can be divided by 4 without any residual. As a result, x needs to have a multiple of 4. We can phrase this as:

x = 4k, where k is some integer.

Secondly, we understand that the remaining is 6 when x is divided by 7. This can be stated as follows:

x ≡ 6 (mod 7)

This indicates that x is a multiple of 7 that is 6 more than. We can solve this problem by substituting x = 4k:

4k ≡ 6 (mod 7)

We can attempt several values of k until we discover one that makes sense for this equation in order to solve for k. We can enter k in to equation starting using k = 1, as follows:

4(1) ≡ 6 (mod 7)

4 ≡ 6 (mod 7)

It is not true; thus we need to attempt a next value for k. This procedure can be carried out repeatedly until the equation is satisfied for all values of k.

k = 2:

4(2) ≡ 6 (mod 7)

1 ≡ 6 (mod 7)

k = 3:

4(3) ≡ 6 (mod 7)

5 ≡ 6 (mod 7)

k = 4:

4(4) ≡ 6 (mod 7)

2 ≡ 6 (mod 7)

k = 5:

4(5) ≡ 6 (mod 7)

6 ≡ 6 (mod 7)

k = 6:

4(6) ≡ 6 (mod 7)

3 ≡ 6 (mod 7)

k = 7:

4(7) ≡ 6 (mod 7)

0 ≡ 6 (mod 7)

We have discovered that the equation 4k 6 (mod 7) is fulfilled when k = 7. Thus, we can change k = 7 to x = 4k to determine that:

x = 4(7) = 28

This indicates that there are 28 toy troops. Yet we also understand that the leftover is 3 when x is divided by 5. We don't need to take into account any other values of x because x = 28 satisfies this requirement.

28 toy soldiers are the correct response.

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Answer: 0.04

Step-by-step explanation:

In order to get 4% as a decimal, you must divide 4 by 100.

4/100 = 0.04

Thus, the answer to your question is 0.04

Answer: 9, 10, and 11

Step-by-step explanation:

If the sum of three consecutive natural numbers is 30, then we can represent the smallest of the three numbers as x. The next two consecutive natural numbers would be x+1 and x+2. Therefore, we can form the equation:

x + (x+1) + (x+2) = 30

Simplifying the left-hand side, we get:

3x + 3 = 30

Subtracting 3 from both sides, we get:

3x = 27

Dividing by 3, we get:

x = 9

Therefore, the three consecutive natural numbers are 9, 10, and 11.

Answer:The prices of the adult movie tickets and the child movie tickets are $15 and $5 respectively.

Given that, the Jones family spent $45 on movie tickets for 2 adults and 3 children.

Step-by-step explanation:What is a linear system of equations?

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

Let cost of adult tickets be x and the cost of children tickets be c.

The Jones family spent $45 on movie tickets for 2 adults and 3 children.

2a+3c=45 ------(I)

The Smith family spent $40 for 2 adults and 2 children.

2a+2c=40

a+c=20 ------(II)

From equation (II), we have a=20-c

Substitute a=20-c in equation (I), we get

2(20-c)+3c=45

⇒ 40-2c+3c=45

⇒ c=$5

Put c=5 in equation (II), we get

a+5=20

⇒ a=$15

The rate at which the value of the computer system is changing 4 years after it was purchased is "Declining at the rate of $8,803.21 per year". The correct option is D.

We can use the exponential decay formula [tex]V(t) = V0 * e^{-kt}[/tex], where V(t) is the value of the computer system after t years, V0 is the initial value, and k is the decay rate.

We know that V(1) = $15,700 and V(0) = $28,000, so we can solve for k:

[tex]$15,700 = $28,000 * e^{-k*1}[/tex]

[tex]e^{-k} = 0.5607[/tex]

-k = ln(0.5607) ≈ -0.5797

k ≈ 0.5797

Therefore, the decay rate is approximately 0.5797 per year.

To find the rate of change of the value of the computer system 4 years after it was purchased, we can take the derivative of V(t) with respect to t:

dV/dt = -k * V0 * [tex]e^{-kt}[/tex]

Substituting t = 4, V0 = $28,000, and k ≈ 0.5797, we get:

dV/dt = -0.5797 * $28,000 * [tex]e^{-0.5797*4}[/tex] ≈ -$8,803.21 per year

Therefore, the value of the computer system is declining at the rate of approximately $8,803.21 per year 4 years after it was purchased.

The correct answer is (D) Declining at the rate of $8,803.21 per year.

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The solution for the other variable, according to the stated statement, is[tex]X = 6^y[/tex] and [tex]y = 21 * 2^X[/tex]

Exponential numbers are represented by an, where an is multiplied by itself n times. An easy example is 8=2³=222. In exponential notation, an is known as the base, whereas n is known as the power, exponent, or index. Scientific notation is an example of an exponential number, with 10 usually typically serving as the base number.

Exponential functions are often employed in the biological sciences to describe the amount of a certain quantity over time, such as population size. Experiment data graphs are often created with time on the x-axis and amount on the y-axis.

a. y = log6(X)

[tex]6^y = X[/tex]

[tex]X = 6^y[/tex]

b. [tex]X = log2(y/21)[/tex]

[tex]2^X = y/21[/tex]

[tex]21 * 2^X = y[/tex]

[tex]y = 21 * 2^X[/tex]

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First, re-read the problem until you understand it and can put it into your own words.  I re-wrote it like this:  "Find the area of a rectangle by first finding the length (L) and the width (W)."   [note that I added "find L and W," but that is how I'm going to solve the problem;  I could also have said that we will need the formulas, P=2L+2W and A=LW, but you knew that already, right?).

Translate the problem:

 "The length of a rectangle is 3 ft longer than its width"      means

         L                             =    3             +         W                 (eq1)

 "the perimeter of the rectangle is 30 ft"     means

            P                                = 50                      (eq2)

So, now the math is easy, just find L and W so we can compute the area:

     P = 50 = 2L + 2W                         (eq3; from eq2 and the formula for P)

          50 = 2(3+W)  +  2W              (use eq1 to substitute for L)

          50 = 6 + 2W   + 2W              (distribute)

           50 = 6 + 4W                       (collect terms)

            44 = 4W                        (subtract 6 from both sides)

            11 ft = W                        (divide both sides by 4)

Use the easiest equation (either eq1 or else eq3) to find L:

          L = 3 + W        (eq1)

          L = 3 + 11

         L = 14 ft

What is the area (A)?

          A = L*W

          A = (14 ft) x (11 ft)

          A = 154 sq ft

The nearest hundredth, we get:

C ≈ 12.56 feet.

Circle circumference (or perimeter) = 2R

where R denotes the circle's radius. 3.14 is the approximate (up to two decimal points) value of the mathematical constant. Again, Pi () is a special mathematical constant that represents the circumference to diameter ratio of any circle.

The circumference of a circle is calculated as follows:

C = 2πr

where C is the circumference, (pi) is a constant close to 3.14, and r is the radius of the circle.

When the given values are substituted, the following results are obtained:

C = 2(3.14)(2) \s= 12.56

We get the following when we round to the nearest hundredth:

C ≈ 12.56 feet.

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Answer:

x = -2 ,  y = 2

Step-by-step explanation:

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a) Graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380

b) The maximum profit of $5200 achieved.

Selling profit is the profit that a business makes on the sale of its products or services. It is the difference between the selling price and the cost of product.

a) Let x be the number of boards of mahogany sold and y be the number of boards of black walnut sold. Then, the constraints of the problem can be represented by the following system of inequalities:

x ≥ 0 (non-negative constraint)

y ≥ 0 (non-negative constraint)

x ≤ 260 (maximum number of mahogany boards available)

y ≤ 320 (maximum number of black walnut boards available)

x + y ≤ 380 (maximum number of boards that can be sold)

To graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380 on a coordinate plane and shade the feasible region that satisfies all of the constraints. The feasible region is the area that is bounded by these lines and includes the origin (0, 0).

b) The profit function P(x, y) can be defined as follows:

P(x, y) = 20x + 6y

To maximize the profit, we need to find the values of x and y that satisfy all of the constraints and maximize the profit function P(x, y).

One way to do this is to use the corner-point method. We can evaluate the profit function at each of the corners of the feasible region and find the corner that gives the maximum profit.

The corners of the feasible region are (0, 0), (0, 320), (260, 0), and (120, 260).

P(0, 0) = 0

P(0, 320) = 6(320) = 1920

P(260, 0) = 20(260) = 5200

P(120, 260) = 20(120) + 6(260) = 4720

Therefore, the maximum profit of $5200 can be achieved by selling 260 boards of mahogany and 0 boards of black walnut.

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