Find any relative extrema. f(x,y) = x³ – 12xy + 8y³ A. f(2,1)= - 8, relative minimum B. f(1,2)= 9, relative minimum c. f(1,2)= 9, relative maximum D. f(2,1)= - 8, relative maximum

(a) The general solution of the homogeneous equation is [tex]yh(t) = e^{(-2t)}(Acos(t) + Bsin(t)).[/tex]

(b) The particular solution of the forced equation is yp(t) = (-9/5)t + 47/5.

(c) The solution of the forced equation subject to the initial conditions is [tex]y(t) = e^{(-2t)}(sin(t))(9/5) - (9/5)t + 47/5.[/tex]

(d) The overall behavior of the solution y(t) approaches (-9/5)t as t goes to infinity.

(a) The characteristic equation of the homogeneous equation y" + 4y' + 5y = 0 is given by r² + 4r + 5 = 0. Solving for r, we get r = -2 ± i. Therefore, the general solution of the homogeneous equation is [tex]yh(t) = e^{(-2t)}(Acos(t) + Bsin(t)).[/tex]

(b) (1). To find a particular solution of the forced equation, we assume a solution of the form yp(t) = At + B.

Taking the derivatives of yp(t), we get yp'(t) = A and yp''(t) = 0. Substituting these into the original equation, we get:

0 + 4A + 5(At + B) = [tex]1 - e^{(-2t)}[/tex]

Solving for A and B, we get A = -9/5 and B = 47/5. Therefore, a particular solution of the forced equation is yp(t) = (-9/5)t + 47/5.

(c) The general solution of the forced equation is [tex]y(t) = yh(t) + yp(t) = e^{(-2t)}(Acos(t) + Bsin(t)) - (9/5)t + 47/5.[/tex] Using the initial conditions y(0) = 0 and y'(0) = 0, we get:

y(0) = A = 0, therefore A = 0

y'(0) = -2A + B - (9/5) = 0, therefore B = (9/5)

Thus, the solution of the forced equation subject to the initial conditions is [tex]y(t) = e^{(-2t)}(sin(t))(9/5) - (9/5)t + 47/5.[/tex]

(d) The forcing function [tex]F(t) = 1 - e^{(-2t)}[/tex] approaches 1 as t goes to infinity. As t approaches infinity, the exponential term [tex]e^{(-2t)}[/tex] approaches zero, and the particular solution yp(t) approaches (-9/5)t.

Therefore, the overall behavior of the solution y(t) approaches (-9/5)t as t goes to infinity.

The function [tex]g(t) = e^{(-2t)}(sin(t))(9/5)[/tex] approaches zero as t goes to infinity, so it does not have a significant impact on the long-term behavior of the solution.

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The value of x by the given data is x=−8 or x=2.

We are given that;

Height=4

Base=3

Now,

By Pythagoras theorem;

(3+x)^2 = 3^2 + 4^2

32+42=52=25

32+2×3×x+x2−25=0

x2+6x−16=0

Factor the quadratic equation: (x+8)(x−2)=0

Set each factor to zero and solve for x: x+8=0 or x−2=0

Therefore, by the Pythagoras theorem the answer will be x=−8 or x=2.

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We cannot consider the event of the cost exceeding $315 to be highly unlikely, as its probability is greater than 5%. Based on the given information, it is not highly unlikely that the cost of completing the tests will exceed $315.

Based on the given information, we know that 40% of the employees have positive indications of asbestos. Therefore, if we randomly select three employees, the probability that all three have positive indications is:
P(all three have positive indications) = (0.4)(0.4)(0.4) = 0.064
This means that the probability that at least one of the selected employees does not have a positive indication is:
P(at least one does not have a positive indication) = 1 - 0.064 = 0.936
Now, to estimate the cost of completing the tests, we need to consider the mean and variance of the cost of finding three employees with positive indications. We know that the mean cost is $150 and the variance is $4,500. Since the cost is a continuous variable, we can use the normal distribution to estimate the probability that the cost exceeds $315. We need to standardize the value of $315 using the mean and variance:
z = (315 - 150) / sqrt(4500) = 1.732
Looking at a standard normal distribution table, we find that the probability of a value being greater than 1.732 standard deviations above the mean is 0.042, which is slightly higher than 0.05.

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The volume of the triangular pyramid is 92.4 cubic inches having a height of 11 inches and a base that has a side of 7.2 inches with an altitude of 7 inches

To calculate the volume of a triangular pyramid, we can use the formula;

Volume = (1/3) × Base Area × Height

Given; Height (h) = 11 inches

Base side (a) = 7.2 inches

Altitude (b) = 7 inches

First, we need to calculate the area of the triangular base. Since we are given the base side (a) and the altitude (b) of the base, we can use the formula for the area of a triangle;

Base Area = (1/2) × Base side × Altitude

Plugging in the given values;

Base Area = (1/2) × 7.2 inches × 7 inches

Base Area = 25.2 square inches

Now, we can substitute the values for Base Area and Height into the formula for the volume of a triangular pyramid;

Volume = (1/3) × Base Area × Height

Plugging in the given values;

Volume = (1/3) × 25.2 square inches × 11 inches

Volume = 92.4 cubic inches

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Answer:

There is no integer or rational solution to the equation 4 = x^5 + x^4. This can be verified by trying integer and rational values of x and seeing that none of them satisfy the equation. Therefore, the solution is "no solution".

Answer:

64cm³

Step-by-step explanation:

Take length 8cm width 2cm and height 4cm,multiply to get the volume

The unit rate at which the turtle is crawling is 6/25 mile per hour

From the question, we have the following parameters that can be used in our computation:

Ths unit rate is then calculated as

Unit rate = distance/time

Substitute the known values in the above equation, so, we have the following representation

Unit rate = (2/25)/(1/3)

Evaluate

Unit rate = 6/25

Hence, the unit rate is 6/25 mile per hour

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Two collections that are not well defined will be a list of the most loved restaurants in a neighborhood and a list of the best five candy choices. A well-defined set will be the top 20 government-approved restaurants in a neighborhood or the five top-selling candies in a company.

A well-defined set is one in which people can clearly tell the content of the set. For instance, the first 50 numbers starting from 1 will be classified as a well-defined set because we can easily tell the content of this set.

However, a not-well-defined set will be a vague list like the most loved restaurants in a neighborhood. This is not finite. It is possible to change a not well-defined set to a defined one.

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The expression for w can be written as the product of matrix A and vector V w = a1 * x + a2 * y.

To answer your question, let's first assume that we have found the expression for w in the previous step as w = a1 * x + a2 * y. Now, we will rewrite this expression as the product of a matrix A and a vector V.

Matrix A will consist of the coefficients a1 and a2, while vector V will contain the variables x and y:

Matrix A: | a1  a2 |
Vector V: | x |
         | y |

To write the expression w as the product of matrix A and vector V, we will perform matrix multiplication:

w = A * V
w = | a1  a2 | * | x |
                 | y |

To multiply a 1x2 matrix by a 2x1 vector, we perform the following steps:

1. Multiply the first element of the matrix's row (a1) by the first element of the vector's column (x): a1 * x
2. Multiply the second element of the matrix's row (a2) by the second element of the vector's column (y): a2 * y
3. Add the results from steps 1 and 2: a1 * x + a2 * y

Thus, the expression for w can be written as the product of matrix A and vector V:

w = a1 * x + a2 * y

This representation allows us to efficiently work with the expression for w in linear algebra and perform calculations involving other matrices and vectors.

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To find the standard matrix of the given linear transformation T: R3 → R3, we can combine the individual transformations and determine their matrix representations. Let's break down the steps:

1. Rotation counterclockwise about the positive x-axis by an angle of 2π/3:

The standard matrix for this rotation is:

```

[1       0            0      ]

[0  cos(2π/3) -sin(2π/3)]

[0  sin(2π/3)  cos(2π/3)]

```

2. Reflection about the xz-plane:

The standard matrix for this reflection is:

```

[1  0  0]

[0 -1  0]

[0  0  1]

```

3. Dilation with a factor of 4:

The standard matrix for this dilation is:

```

[4  0  0]

[0  4  0]

[0  0  4]

```

To obtain the composite transformation, we multiply the matrices in the reverse order of their operations:

```

[A] = [Dilation] · [Reflection] · [Rotation]

   = [4  0  0] · [1  0  0] · [1       0            0      ]

     [0  4  0]   [0 -1  0]   [0  cos(2π/3) -sin(2π/3)]

     [0  0  4]   [0  0  1]   [0  sin(2π/3)  cos(2π/3)]

```

Multiplying the matrices gives us the standard matrix for the given linear transformation:

```

[A] = [4  0  0] · [1  0  0] · [1       0            0      ]

     [0  4  0]   [0 -1  0]   [0  cos(2π/3) -sin(2π/3)]

     [0  0  4]   [0  0  1]   [0  sin(2π/3)  cos(2π/3)]

   = [4  0        0         ]

     [0 -4        0         ]

     [0  0  cos(2π/3) -sin(2π/3)]

     [0  0  sin(2π/3)  cos(2π/3)]

```

Therefore, the standard matrix of the given linear transformation T: R3 → R3 is:

```

[4  0        0         ]

[0 -4        0         ]

[0  0  cos(2π/3) -sin(2π/3)]

[0  0  sin(2π/3)  cos(2π/3)]

```

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A better method to determine the more popular sport is by conducting a comprehensive, unbiased survey with a random sample

a. Concluding that baseball is more popular than soccer based on a poll at a championship event is not valid due to potential sample bias, self-selection bias, limited sample size, and question phrasing.

b. A better method to determine the more popular sport is by conducting a comprehensive, unbiased survey with a random sample of students in a neutral setting, using clear and unbiased questions that allow for all preferences

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General Computers Inc. bought a server for $53,500 and took a loan for the remaining amount at 5% compounded semi-annually. They paid $2,350.08 at the end of every quarter for 7 years to fully pay off the loan. The total interest paid was around $15,698.14.

General Computers Inc. purchased a computer server for $53,500, paying 35% of the value ($18,725) as a down payment and receiving a loan for the balance of $34,775. The loan was charged an interest rate of 5% compounded semi-annually.

To repay the loan, General Computers Inc. made payments of $2,350.08 at the end of every quarter. These payments were high enough to cover the interest as well as repay the principal, which allowed the loan to be fully paid off in 7 years (28 quarters).

The interest on the loan is calculated using the formula for compound interest:

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

where A is the amount after t years, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

Substituting the values for the loan, we get:

[tex]A = 34,775(1 + 0.05/2)^{(2 \times7)} \approx 50,473.14[/tex]

Therefore, the total interest paid on the loan was approximately $50,473.14 - $34,775 = $15,698.14.

In summary, General Computers Inc. purchased a computer server for $53,500 and received a loan for the balance at 5% compounded semi-annually. It made payments of $2,350.08 at the end of every quarter to settle the loan, which allowed the loan to be fully paid off in 7 years. The total interest paid on the loan was approximately $15,698.14.

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The probability that an unprepared student, who can eliminate one of the possible answers on the first four questions, will guess all six questions correctly on a multiple-choice test with 5 options per question.

Each question has 5 possible answers, and the student can eliminate one option, so she has a 1/4 chance of guessing the correct answer. On the first four questions, the student will have a 1/4 chance of guessing correctly since she can eliminate one option, and a 3/4 chance of choosing one of the incorrect answers.

On the last two questions, she has a 1/5 chance of guessing the correct answer. Therefore, the probability that she guesses all six questions correctly is:

(1/4)^4 x (1/5)^2 x 5^4 = 0.000015625 or 1/64,000.
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The solution to the recurrence relation an = 2(an/2) + d with an = 1 is simply an = 1.

To solve the given recurrence relations, we can use a technique called substitution. Let's solve each relation separately.

Recurrence relation: an = (an/2) + d

To solve this relation, we need to express the term an in terms of smaller terms until we reach the base case. Let's substitute an/2 in place of an:

an = (an/2) + d

= [(an/4) + d] + d

= (an/4) + 2d

Continuing this process, we can express an in terms of smaller terms:

an = (an/8) + 3d

= (an/16) + 4d

In general, we can write:

an = (an/2^k) + kd

Now, let's find the value of k when an = 1 (initial condition):

1 = [tex](1/2^{k})[/tex]+ kd

Rearranging the equation:

1 - kd = [tex]1/2^{k}[/tex]

Multiplying both sides by [tex]2^{k}[/tex]:

[tex]2^{k} - k2^{k} d = 1[/tex]

This equation cannot be solved analytically in general. However, we can approximate the value of k using numerical methods or by using software tools such as Wolfram Alpha or numerical solvers in programming languages.

Once we have the value of k, we can substitute it back into the general formula to find the nth term, an, for any given n.

Recurrence relation: an = 2(an/2) + d

Using the same substitution technique as above, we can express an in terms of smaller terms:

an = 2(an/2) + d

= 2[2(an/4) + d] + d

= 4(an/4) + 3d

Continuing this process, we have:

an = 2^k (an/2^k) + kd

Again, to find the value of k when an = 1:

[tex]1 = 2^{k} (1/2^{k}) + kd[/tex]

1 = 1 + kd

Since kd = 0 for k = 0 (initial condition), we have k = 0.

Therefore, the solution to the recurrence relation an = 2(an/2) + d with an = 1 is simply an = 1.

Please note that if the value of d is non-zero, the recurrence relation may have different solutions or properties.

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Answer: x=-150

Step-by-step explanation:

The probability that the male sample will have 4 to 6 more days of absences than the female sample is approximately 0.7887.

The difference in means between boys and girls is 14 - 9 = 5, and the difference in standard deviations is 4 - 3 = 1. We can use the Central Limit Theorem to approximate the distribution of the difference in sample means.

The mean of the difference in sample means is 5, and the standard deviation is √((4²/100) + (3²/64)) = 0.754.

To find the probability that the male sample will have 4 to 6 more days of absences than the female sample, we need to find the z-scores for the values x1 = 4 and x2 = 6:

z₁ = (4 - 5) / 0.754 = -1.325

z₂ = (6 - 5) / 0.754 = 1.325

Using a standard normal table or calculator, we find that the probability of a z-score falling between -1.325 and 1.325 is approximately 0.7887.

Therefore, the probability that the male sample will have 4 to 6 more days of absences than the female sample is approximately 0.7887.

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Rita's chances for becoming a president of the key club is 1/30

From the question, we have the following parameters that can be used in our computations

Number of people = 10 i.e. 9 and other people

P(Junior) = 1/3

This means that

P(Rita) = P(Junior) * P(Selected from Junior)

The value of P(Selected from Junior) is calculated as

P(Selected from Junior) = 1/10

So, we have

P(Rita) = P(Junior) * P(Selected from Junior)

Substitute the known values in the above equation, so, we have the following representation

P(Rita) = 1//3 * 1/10

Evaluate

P(Rita) = 1//30

Hence, the probability is 1/30

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The mean of a data represented by the stem and leaf is 108.75

The data represented by the stem and leaf plot given can be also written as,

Row 1 : 10, 10, 12, 18

Row 2: 31, 37, 39

Row 3 : 50, 55, 57, 57, 57

Row 4 : 113, 114, 116

Row 5: 223

Row 6: 235

Row 7: 310, 312, 319

The mean of a data represented by the stem and leaf can be calculated as,

Total number of observations = 20

Total value of the observations in the data set = ( 10 + 10 + 12 + 18 + 31 + 37 + 39+ 50 + 55 + 57 + 57+ 57 + 113 + 114 + 116 + 223 + 235 + 310 + 312 + 319)

= 2175

Mean = (Total value of the observations in the data set) / (Total number of observations) = 2175/ 20 = 108.75

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A. The probability that the first pie sold in August will be a cherry or apple pie is 0.98 or 98%.

B. Mama's Bakery should plan to order 308 blueberry pies for the month of August.

Part A:

We are given that Mama's Bakery sold 247 total pies during July, of which 74 were cherry and 55 were blueberry.

So, the number of apple pies sold during July is:

247 - 74 - 55 = 118

The total number of cherry and apple pies that Mama's Bakery sold during July:

74 + 118 = 192

Therefore, the probability that the first pie sold in August will be a cherry or apple pie is:

P(cherry or apple) = (74 + 118)/247 = 0.98

So the probability is 0.98 or 98%.

Part B:

We know that Mama's Bakery sold 247 pies during July, and we can express this as:

cherry + blueberry + apple = 247

We also know that Mama's Bakery wants to make a total of 500 pies, so we can express this as:

cherry + blueberry + apple = 500

Subtracting the first equation from the second equation gives:

blueberry + apple = 253

We know that 118 apple pies were sold during July, so Mama's Bakery should plan to make:

500 - 118 = 382

apple pies in total. Therefore, the number of blueberry pies Mama's Bakery should plan to order for is:

blueberry = 500 - cherry - apple = 500 - 74 - 118 = 308

Therefore, Mama's Bakery should plan to order 308 blueberry pies for the month of August.

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We reject the null hypothesis and conclude that adding bupropion to nicotine patches is associated with a higher quit rate than using regular nicotine patches alone.

However, note that this conclusion is based on a hypothetical quit rate for the bupropion patch group, and the actual quit rate may be different.

To determine whether adding antidepressants to nicotine patches helps smokers quit, we can compare the quit rates of the two groups using statistical analysis.

Let p1 be the quit rate for the group receiving the patch with bupropion and p2 be the quit rate for the group receiving regular nicotine patches.

The null hypothesis is that there is no difference in quit rates between the two groups: p1 - p2 = 0.

The alternative hypothesis is that the quit rate for the group receiving the patch with bupropion is higher:

p1 > p2.

We can use a one-tailed z-test to test this hypothesis, since we are interested in whether the quit rate for the bupropion patch group is higher than the regular patch group.

The test statistic is:

[tex]z = (p1 - p2) / \sqrt{(p*(1-p)*(1/n1 + 1/n2))}[/tex]

where p = (x1 + x2) / (n1 + n2) is the pooled proportion of smokers who quit, x1 and x2 are the number of smokers who quit in each group, and n1 and n2 are the sample sizes.

From the given information, we know that 98 out of 246 smokers who received regular nicotine patches quit after a year, so x2 = 98 and n2 = 246.

We don't have information about the number of smokers who quit in the bupropion patch group, so we will assume a hypothetical quit rate of 60%, or x1 = 0.6*230 = 138 and n1 = 230.

The pooled proportion is:

p = (x1 + x2) / (n1 + n2) = (138 + 98) / (230 + 246) = 0.415

The standard error of the difference in proportions is:

[tex]SE = \sqrt{(p*(1-p)(1/n1 + 1/n2)}[/tex]

[tex]= \sqrt{(0.415(1-0.415)*(1/230 + 1/246)}[/tex]

≈ 0.053.

The z-score is:

z = (p1 - p2) / SE

= (0.6 - 98/246) / 0.053

≈ -6.22

Using a standard normal distribution table or calculator, we can find that the p-value is very small, much smaller than the conventional alpha level of 0.05.

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The  correct choices are:

The vector w is not in Col(A) because Ax=w is an inconsistent system. One row of the reduced row echelon form of the augmented matrix [A|w] is in the form [0 0 b] where b ≠ 0.

The vector w is not in Nul(A) because Aw ≠ 0.

To determine whether the vector w = [3 1] is in the column space of the matrix A = [−15 45 −5 15], we can row reduce the augmented matrix [A|w] and check if the resulting system is consistent.

[A|w] = [−15 45 −5 15 | 3 1]

Performing row reduction on [A|w], we get:

[1 -3 1/3 -1/3 | -1/5 0]

So, the system Ax = w is inconsistent, and therefore, w is not in the column space of A.

To determine whether the vector w is in the null space of A, we need to check if Aw = 0.

Aw = [−15 45 −5 15] [3 1]ᵀ = [(-15)(3) + (45)(1) + (-5)(0) + (15)(0) , (-15)(0) + (45)(0) + (-5)(3) + (15)(1)]ᵀ = [30, -30]ᵀ

Since Aw ≠ 0, we can conclude that w is not in the null space of A.

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The correct answer is: The argument is valid because all truth table rows that have true premises have true conclusions. The argument form presented in the question is called a disjunctive syllogism, which states that if one of the two premises is true, then the conclusion is also true. In this case, the premises are P and ~9, and the conclusion is P v 9.

To determine the validity of the argument, we need to analyze the truth table provided. The premise column(s) are the ones that represent the initial propositions or assumptions that the argument is based on. In this case, the premise columns are P and ~9. The conclusion column, on the other hand, represents the final statement or inference that the argument is trying to make. In this case, the conclusion column is P v 9.

Looking at the truth table, we can see that all rows where the premises are true (TT, FT) also have a true conclusion. This means that the argument is valid because all truth table rows that have true premises have true conclusions. Therefore, the correct answer is: "The argument is valid because all truth table rows that have true premises have true conclusions."

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Answer:

Please help me answer my question

True. If the derivative of the function f(x) exists and is nonzero for all values of x, then the function must be continuously increasing or decreasing.

Therefore, the value of f(8) will not be equal to the value of f(0), unless the function is a constant function.

To determine whether the statement is true or false, we can analyze it step by step.

1. The condition given is that f'(x) exists and is nonzero for all x. This means that the function f(x) is differentiable and has a nonzero slope everywhere.

2. If a function is differentiable everywhere, it is also continuous everywhere. This is because differentiability implies continuity.

3. However, the condition that f'(x) is nonzero for all x does not guarantee that f(8) ≠ f(0). It is possible for a function to be differentiable and have a nonzero derivative everywhere, yet still have equal values at two distinct points.

Therefore, the statement is False.

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The concentration of the antibiotic will drop below 20.0 micrograms/milliliter at around 11.1 hours after injection.

To find when the concentration drops below a level of 20.0 micrograms/milliliter, we need to solve the equation:

f(t) < 20 for t, where f(t) = t^2 * e^(t-12) and t >= 0.

Set up the inequality
t^2 * e^(t-12) < 20

Solve the inequality
Unfortunately, there's no simple algebraic way to solve this inequality. We'll need to use numerical methods or graphical analysis to approximate the value of t.

One way to approach this is by using a graphing calculator or software to graph the function f(t) = t^2 * e^(t-12) and then finding the value of t where the graph is below the horizontal line y = 20.

Upon analyzing the graph, you'll find that the concentration drops below 20.0 micrograms/milliliter at approximately t ≈ 11.1 hours (keep in mind that the actual value might slightly vary depending on the accuracy of the method used).

So, the concentration of the antibiotic will drop below 20.0 micrograms/milliliter at around 11.1 hours after injection.

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T(t) = (7cos(t)sec(t))i + (7)j.To find the unit tangent vector T(t), we need to find the derivative of the position vector r(t) with respect to t and then normalize it. The position vector r(t) is given as r(t) = 7ln(sec(t))i + 7tj.

Taking the derivative, we have dr/dt = (7cos(t)sec(t))i + 7j.To normalize dr/dt, we divide it by its magnitude, which is √((7cos(t)sec(t))^2 + 7^2).

Simplifying this expression gives √(49cos^2(t)sec^2(t) + 49), which simplifies further to 7sec(t). Dividing dr/dt by its magnitude, we get T(t) = (7cos(t)sec(t))i + (7)j.

Therefore, T(t) = (7cos(t)sec(t))i + (7)j.

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This can be answered by the concept of Matrix. we have proved that tr(ab) = tr(a)tr(b).

To prove that tr(ab) = tr(ba), we have:

tr(ab) = ∑ᵢ(ab)ᵢᵢ = ∑ᵢ∑ⱼaᵢⱼbⱼᵢ (using matrix multiplication)

Interchanging the order of summation, we get:

tr(ab) = ∑ⱼ∑ᵢbⱼᵢaᵢⱼ = ∑ᵢ(ba)ᵢᵢ = tr(ba)

Therefore, we have proved that tr(ab) = tr(ba).

Now, to prove that tr(ab) = tr(a)tr(b), we have:

tr(ab) = ∑ᵢ(ab)ᵢᵢ = ∑ᵢ∑ⱼaᵢⱼbⱼᵢ (using matrix multiplication)

We can rewrite the terms aᵢⱼ and bⱼᵢ as follows:

aᵢⱼ = [a(i,1), a(i,2), ..., a(i,n)] * [0, 0, ..., 1, ..., 0]ᵀ, where the 1 is in the j-th position.

bⱼᵢ = [b(1,j), b(2,j), ..., b(n,j)] * [0, 0, ..., 1, ..., 0]ᵀ, where the 1 is in the i-th position.

Therefore, we have:

tr(ab) = ∑ᵢ∑ⱼ[a(i,1), a(i,2), ..., a(i,n)] * [0, 0, ..., b(1,j), ..., 0]ᵀ * [b(1,j), b(2,j), ..., b(n,j)] * [0, 0, ..., 1, ..., 0]ᵀ

Using the associative and distributive properties of matrix multiplication, we can rewrite this expression as:

tr(ab) = ∑ᵢ[a(i,1), a(i,2), ..., a(i,n)] * [b(1,i), b(2,i), ..., b(n,i)] * [0, 0, ..., 1, ..., 0]ᵀ * [0, 0, ..., 1, ..., 0] * [0, 0, ..., 1, ..., 0]ᵀ

Notice that the term [a(i,1), a(i,2), ..., a(i,n)] * [b(1,i), b(2,i), ..., b(n,i)] is just the dot product of the i-th row of a with the i-th column of b, which is equal to the (i,i)-th element of the matrix product ab.

Therefore, we have:

tr(ab) = ∑ᵢ(ab)ᵢᵢ = tr(ab)

Using the fact that tr(a) = ∑ᵢaᵢᵢ, we can rewrite the expression for tr(ab) as:

tr(ab) = ∑ᵢ∑ⱼaᵢⱼbⱼᵢ = ∑ᵢaᵢᵢ ∑ⱼbⱼⱼ = tr(a) tr(b)

Therefore, we have proved that tr(ab) = tr(a)tr(b).

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Answer: The distance between Bob's home and his school is 2.5 miles.

Step-by-step explanation:

What is speed?

Speed is the distance covered in unit time.

Speed of Bob = 3mph

Speed of his sister = 12mph

Distance between Bob's home and his school = distance covered by bob in 10 minutes + distance covered by his sister in 10 minutes.

Distance covered by Bob in 10 minutes = 3*10/60 = 0.5 mile.

Distance covered by His sister in 10 minutes = 12*10/60 = 2 miles.

So, the distance between Bob's home and his school =0.5+2 = 2.5miles.

Therefore, the distance between Bob's home and his school is 2.5 miles.

Answer:

600

Step-by-step explanation:

324 is 54% of what amount?

We Take

(324 ÷ 54) x 100 = 600

So, 324 is 54% of 600.

The probability that a randomly selected point within the square falls in the

the red-shaded triangle is 1/6.

We have,

Area of the square.

= Side²

Side = 6

So,

= 6 x 6

= 36

And,

Area of a triangle.

= 1/2 x base x height

Base = 3

Height = 4

So,

= 1/2 x 3 x 4

= 3 x 2

= 6

Now,

The probability that a randomly selected point within the square falls in the

red-shaded triangle.

= 6/36

= 1/6

Thus,

The probability that a randomly selected point within the square falls in the

the red-shaded triangle is 1/6.

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