A square has a perimeter of 20 cm
calculate the area of the square

t : r 2 → r 2 is a linear transformation, Formula for [tex]t^{-1}(y)[/tex] as:

[tex]t^{-1}(y) = [(7y_1 + 8y_2)/10, (2y_1 + 2y_2)/10][/tex]

We need to show that it is both one-to-one and onto.

First, let's check the one-to-one property. We can do this by checking whether the nullspace of the transformation only contains the zero vector.

To do so, we need to solve the homogeneous system of equations Ax = 0, where A is the matrix that represents the transformation t.

[tex]2x_1 - 8x_2 = y_1[/tex]

[tex]-2x_1 + 7x_2 = y_2[/tex]

The solution to this system is [tex]x_1 = 0[/tex] and [tex]x_2 = 0[/tex], which means that the nullspace only contains the zero vector. Therefore, t is one-to-one.

Next, let's check the onto property. We can do this by checking whether the range of the transformation covers all of[tex]R^2[/tex]. In other words, we need to show that for any vector y in [tex]R^2[/tex], there exists a vector x in R^2 such that t(x) = y.

Let y = (y1, y2) be an arbitrary vector in [tex]R^2[/tex]. We need to find [tex]x = (x_1, x_2)[/tex]such that t(x) = y.

[tex]2x_1 - 8x_2 = y_1[/tex]

[tex]-2x_1 + 7x_2 = y_2[/tex]

Solving this system of equations, we get:

[tex]x_1 = (7y_1 + 8y_2)/62[/tex]

[tex]x_2 = (2y_1 + 2y_2)/62[/tex]

Therefore, for any vector y in R^2, we can find a vector x in R^2 such that t(x) = y. Hence, t is onto.

Since t is both one-to-one and onto, it is invertible. To find the formula for t^-1, we can use the formula:

[tex]t^{-1}(y) = A^{-1}y[/tex]

where A is the matrix that represents the transformation t. The matrix A is:

[ 2 -8 ]

[-2 7 ]

To find [tex]A^{-1}[/tex], we can use the formula:

[tex]A^{-1} = (1/det(A)) * adj(A)[/tex]

where det(A) is the determinant of A and adj(A) is the adjugate of A (which is the transpose of the matrix of cofactors of A).

det(A) = (27) - (-2-8) = 10

adj(A) = [ 7 8 ]

[ 2 2 ]

Therefore,

[tex]A^{-1} = (1/10) * [ 7 8 ; 2 2 ][/tex]

Finally, we can write the formula for [tex]t^{-1}(y)[/tex] as:

[tex]t^{-1}(y) = (1/10) * [ 7 8 ; 2 2 ] * [ y_1 ; y_2 ][/tex]

Simplifying, we get:

[tex]t^{-1}(y) = [(7y_1 + 8y_2)/10, (2y_1 + 2y_2)/10][/tex]

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t : r 2 → r 2 is a linear transformation, Formula for [tex]t^{-1}(y)[/tex] as:

[tex]t^{-1}(y) = [(7y_1 + 8y_2)/10, (2y_1 + 2y_2)/10][/tex]

We need to show that it is both one-to-one and onto.

First, let's check the one-to-one property. We can do this by checking whether the nullspace of the transformation only contains the zero vector.

To do so, we need to solve the homogeneous system of equations Ax = 0, where A is the matrix that represents the transformation t.

[tex]2x_1 - 8x_2 = y_1[/tex]

[tex]-2x_1 + 7x_2 = y_2[/tex]

The solution to this system is [tex]x_1 = 0[/tex] and [tex]x_2 = 0[/tex], which means that the nullspace only contains the zero vector. Therefore, t is one-to-one.

Next, let's check the onto property. We can do this by checking whether the range of the transformation covers all of[tex]R^2[/tex]. In other words, we need to show that for any vector y in [tex]R^2[/tex], there exists a vector x in R^2 such that t(x) = y.

Let y = (y1, y2) be an arbitrary vector in [tex]R^2[/tex]. We need to find [tex]x = (x_1, x_2)[/tex]such that t(x) = y.

[tex]2x_1 - 8x_2 = y_1[/tex]

[tex]-2x_1 + 7x_2 = y_2[/tex]

Solving this system of equations, we get:

[tex]x_1 = (7y_1 + 8y_2)/62[/tex]

[tex]x_2 = (2y_1 + 2y_2)/62[/tex]

Therefore, for any vector y in R^2, we can find a vector x in R^2 such that t(x) = y. Hence, t is onto.

Since t is both one-to-one and onto, it is invertible. To find the formula for t^-1, we can use the formula:

[tex]t^{-1}(y) = A^{-1}y[/tex]

where A is the matrix that represents the transformation t. The matrix A is:

[ 2 -8 ]

[-2 7 ]

To find [tex]A^{-1}[/tex], we can use the formula:

[tex]A^{-1} = (1/det(A)) * adj(A)[/tex]

where det(A) is the determinant of A and adj(A) is the adjugate of A (which is the transpose of the matrix of cofactors of A).

det(A) = (27) - (-2-8) = 10

adj(A) = [ 7 8 ]

[ 2 2 ]

Therefore,

[tex]A^{-1} = (1/10) * [ 7 8 ; 2 2 ][/tex]

Finally, we can write the formula for [tex]t^{-1}(y)[/tex] as:

[tex]t^{-1}(y) = (1/10) * [ 7 8 ; 2 2 ] * [ y_1 ; y_2 ][/tex]

Simplifying, we get:

[tex]t^{-1}(y) = [(7y_1 + 8y_2)/10, (2y_1 + 2y_2)/10][/tex]

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The optimal strategy for the row player is to always play row 2, as it has the lowest expected value for the column player's choices.

To reduce the 4 x 4 game matrix and find the optimal strategy for the row player, we need to calculate the expected value for each row based on the column player's choices.

For the first row, the expected value is (4x57601 + 3x10 + 9x5/6 + 7x1/60)/60 = 42.72/60 = 0.712.

For the second row, the expected value is (-7x57601 + -5x10 + -3x5/6 + 5x1/60)/60 = -410.16/60 = -6.836.

For the third row, the expected value is (-1x57601 + 4x10 + 5x5/6 + 8x1/60)/60 = -38.58/60 = -0.643.

For the fourth row, the expected value is (3x57601 + -5x10 + -1x5/6 + 5x1/60)/60 = 214.42/60 = 3.574.

From these expected values, we can see that the optimal strategy for the row player is to always play row 2, as it has the lowest expected value for the column player's choices.

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The resulting one-hot-encoded matrix would be:

[[0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

To convert y into a one-hot-encoded matrix, we can use the following steps:

1. Create an empty matrix of size (m x 10), where m is the number of samples in y and 10 is the number of unique values that y can take on.

2. For each value in y, create a row vector of size (1 x 10) where all elements are 0, except for the element corresponding to the value, which is set to 1.

3. Replace the corresponding row in the empty matrix with the row vector created in step 2.

For example, if y is a vector of length m = 5 with values [3, 5, 2, 5, 1], and assuming y can take on 10 unique values, the resulting one-hot-encoded matrix would be:

[[0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

Each row in the matrix corresponds to a sample in y, and the value 1 in each row indicates the position of the value in y. For example, the first row indicates that the first sample in y has the value 3.

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The resulting one-hot-encoded matrix would be:

[[0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

To convert y into a one-hot-encoded matrix, we can use the following steps:

1. Create an empty matrix of size (m x 10), where m is the number of samples in y and 10 is the number of unique values that y can take on.

2. For each value in y, create a row vector of size (1 x 10) where all elements are 0, except for the element corresponding to the value, which is set to 1.

3. Replace the corresponding row in the empty matrix with the row vector created in step 2.

For example, if y is a vector of length m = 5 with values [3, 5, 2, 5, 1], and assuming y can take on 10 unique values, the resulting one-hot-encoded matrix would be:

[[0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

Each row in the matrix corresponds to a sample in y, and the value 1 in each row indicates the position of the value in y. For example, the first row indicates that the first sample in y has the value 3.

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Answer:

some parts are missing.where is the taxable income

⇒ The population increases the fastest when it is at half of the carrying capacity, which is 1750.

⇒ The solution to the differential equation is,

y = 200exp(0.11t + ln(3/197)) / (1 + 19exp(0.11t + ln(3/197)))

⇒ It will take about 3.04 years for the fish population to increase to 2250.

To determine when the population increases the fastest,

we have to find the maximum value of the derivative P'(t).

We can start by setting the derivative equal to zero and solving for p,

⇒ P'(t) = 0.9(1 - p/3500) = 0

⇒ 1 - p/3500 = 0

⇒ p/3500 = 1

⇒ p = 3500

So, the population will increase the fastest when p = 3500.

To confirm that this is a maximum,

Take the second derivative of P(t),

⇒ P''(t) = -0.9/3500

Since P''(t) is negative, P(t) has a maximum at p = 3500.

Therefore, the population increases the fastest when it is at half of the carrying capacity, which is 1750.

To solve the given differential equation ,

First, separate the variables by dividing both sides by (y(1 - y/200)),

⇒ (1 / (y(1 - y/200))) dy = 0.11 dt

Integrate both sides. Let's first integrate the left side,

⇒ ∫ (1 / (y(1 - y/200))) dy = ∫ (1 / y) + (1 / (200 - y)) dy

                                       = ln(y) - ln(200 - y) + C1

where C1 is the constant of integration.

Now we can integrate the right side,

⇒ 0.11t + C2

Where C2 is another constant of integration.

Putting it all together, we have,

⇒ ln(y) - ln(200 - y) = 0.11t + C

where C = C2 - C1.

To solve for y, we can exponentiate both sides,

⇒y / (200 - y) = exp(0.11t + C)

Multiplying both sides by (200 - y), we get,

⇒ y = 200exp(0.11t + C) / (1 + 19exp(0.11t + C))

Using the initial condition y(0) = 2,

Solve for C and get:

⇒ C = ln(3/197)

Therefore, the solution to the differential equation is:

⇒ y = 200exp(0.11t + ln(3/197)) / (1 + 19exp(0.11t + ln(3/197)))

a) To find the equation for the fish population,

we can use the logistic growth model,

⇒ P(t) = K / (1 + Aexp(-r*t))

where P(t) is the population at time t,

K is the carrying capacity,

A is the initial population,

r is the growth rate, and

e is the base of natural logarithms.

We know that

A = 500,

K = 4500, and

P(1) = 710.

Use these values to solve for r,

⇒ r = ln((P(1)/A - 1)/(K/A - P(1)/A))

⇒r = ln((710/500 - 1)/(4500/500 - 710/500))

⇒r = 0.4542

Now we can plug in all the values to get the equation,

⇒P(t) = 4500 / (1 + 4exp(-0.4542t))

b) We want to find t when P(t) = 2250.

Use the equation we found in part a) and solve for t,

⇒ 2250 = 4500 / (1 + 4exp(-0.4542t))

⇒ 1 + 4exp(-0.4542t) = 2

⇒        exp(-0.4542t) = 0.25

⇒                -0.4542t = ln(0.25)

⇒                              t = ln(0.25) / (-0.4542)

⇒                              t ≈ 3.04 years.

So it will take about 3.04 years for the fish population to increase to 2250.

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The answer is: x-intercept = (0.34, 0), (1.66, 0) : y-intercept = (0, -3)

To find the x-intercept(s), we set y to 0 and solve for x. For the given equation, 0 = 5x^2 - 6x - 3. To find the y-intercept, we set x to 0 and solve for y.x-intercept:0 = 5x^2 - 6x - 3We can use the quadratic formula to find the solutions for x:x = (-b ± √(b^2 - 4ac)) / 2ax = (6 ± √((-6)^2 - 4(5)(-3))) / 2(5)x ≈ 1.08, -0.55y-intercept:y = 5(0)^2 - 6(0) - 3y = -3So, the x-intercepts are (1.08, 0) and (-0.55, 0), and the y-intercept is (0, -3).Your answer: x-intercept =(1.08, 0),(-0.55, 0): y-intercept =(0, -3)

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If students' opinions change during their time at school when comparing a group of college freshmen and a group of sophomores regarding the quality of the university cafeteria. The correct answer is C. This scenario should not be analyzed using paired data because the groups are independent and do not have a natural pairing.

To explain, paired data is used when each observation in one group has a unique match in the other group, and these pairs are related in some way. In this scenario, college freshmen and sophomores are two separate groups with no direct relationship between individual students in each group.

Therefore, the data is not naturally paired, and we cannot track individual changes in opinions over time as the students progress from freshmen to sophomores. Additionally, the groups are independent, meaning the opinions of one group do not influence the opinions of the other group.

College freshmen and sophomores have different experiences and are at different stages of their college life, so their opinions about the university cafeteria are not dependent on each other.

Thus, to analyze the difference in opinions between these two independent groups, an appropriate statistical method would be to use unpaired data analysis techniques such as an independent samples t-test or a chi-square test for independence, depending on the nature of the data collected.

In conclusion, when comparing the opinions of college freshmen and sophomores about the quality of the university cafeteria, we should not use paired data analysis because the groups are independent and do not have a natural pairing.

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If f(x) = 7x and g(x) = 3x+1, the value of given function (fog)(x) is 21x+7. Therefore, the correct option is option D among all the given options.

In mathematics, a function is a statement, rule, or law that establishes the relationship between an independent variable and a dependent variable. In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. It is equivalent to linear forms in linear algebra, which are linear mappings from a vector space into their scalar field.

f(x) = 7x

g(x) = 3x+1

(fog)(x) = f(g(x))

(fog)(x) = f(3x+1)

(fog)(x) =7(3x+1)

⇒(fog)(x) = 21x+7

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The 4th partial sum of the given series is approximately 0.1164.

The given series is:

[tex]s = \sum_{n=1}^\infty 10/(6n^5)[/tex]

To compute the 4th partial sum, we add up the terms from n=1 to n=4:

[tex]s_4 = \sum_{n=1}^4 10/(6n^5) = (10/6) (1/1^5 + 1/2^5 + 1/3^5 + 1/4^5)[/tex]

We can simplify this expression using a calculator:

[tex]s_4[/tex]= (10/6) (1 + 1/32 + 1/243 + 1/1024) ≈ 0.1164

Therefore, the 4th partial sum of the given series is approximately 0.1164.

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The 4th partial sum of the given series is approximately 0.1164.

The given series is:

[tex]s = \sum_{n=1}^\infty 10/(6n^5)[/tex]

To compute the 4th partial sum, we add up the terms from n=1 to n=4:

[tex]s_4 = \sum_{n=1}^4 10/(6n^5) = (10/6) (1/1^5 + 1/2^5 + 1/3^5 + 1/4^5)[/tex]

We can simplify this expression using a calculator:

[tex]s_4[/tex]= (10/6) (1 + 1/32 + 1/243 + 1/1024) ≈ 0.1164

Therefore, the 4th partial sum of the given series is approximately 0.1164.

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Answer:

The percentage tax charged is $1,500 / $12,000 = 12.5%.

Step-by-step explanation:

0_0

Answer:

12.5%

Step-by-step explanation:

12000 - 10500 = 1500

1500 / 12000 = 0.125

0.125 * 100 = 12.5%

The distribution property Elizabeth used in her solution

What is distribution property?

The distribution property is a fundamental property of arithmetic and algebra that states that multiplication can be distributed over addition or subtraction, and vice versa. It is a property that is used extensively in mathematics, science, engineering, and other fields that involve mathematical calculations.

The distribution property can be expressed in various ways, but the most common form is:

a × (b + c) = (a × b) + (a × c)

This means that if you have a number "a" and you want to multiply it by the sum of two other numbers "b" and "c", you can do so by multiplying "a" by each of the two numbers "b" and "c" separately, and then adding the results together.

For example, if a = 3, b = 4, and c = 5, then:

3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27

The distribution property can also be used in reverse, which means that you can factor out a common factor from an expression. For example:

3x + 6x = (3 + 6)x = 9x

In this example, the distribution property was used to factor out the common factor of "3x" from the expression "3x + 6x".

The distribution property is a very powerful tool in mathematics, and it can be used to simplify and solve many different types of problems. It is especially useful in algebra, where it is used to expand and simplify expressions, factor polynomials, and solve equations.

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Correct question is ''Which property did Elizabeth use in her solution? Explain the property."

the mode of the distribution is x = 2.

To find the mode of the distribution, we need to find the value of x that corresponds to the peak of the distribution function. In other words, we need to find the value of x at which the probability density function (pdf) is maximized.

To do this, we first need to find the pdf. We can do this by taking the derivative of the cumulative distribution function (cdf):

[tex]f[x] = \frac{d}{dx} F[x][/tex]

For 0 ≤ x < 3, we have:

[tex]f[x] = \frac{d}{dx} {[(1/9) (2 x^2 - x^{3/3}]}\\f[x] = 1/9 {(4x - x^2)}[/tex]

For x ≥ 3, we have:

f[x] = d/dx (1)
f[x] = 0

Therefore, the pdf is:

[tex]f[x] = (1/9) (4x - x^2)[/tex]for 0 ≤ x < 3
f[x] = 0 for x ≥ 3

To find the mode, we need to find the value of x that maximizes the pdf. We can do this by setting the derivative of the pdf equal to zero and solving for x:

[tex]\frac{df}{dx} = (4/9) - (2/9) x = 0[/tex]
x = 2

Therefore, the mode of the distribution is x = 2.

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The calculatd value of the median number of spots rolled is 4

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

Spots = 6 5 5 3 4 3 4

Start by sorting the number of spots in ascending order

So, we have

6  5 5 4 4 3 3

As a general rule.

The median is the middle number

Using the above as a guide, we have the following:

Median = middle number = 4

Hence, the value of the median is 4

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A- The least-squares equation for the given (x, y) data pairs is ŷ = 4.759 - 0.115x, rounded to three digits after the decimal.

B- The least-squares equation for the given (x, y) data pairs is ŷ = 1.505 + 0.461x, rounded to three digits after the decimal.

(a) To find the least-squares equation for the given (x, y) data pairs, we first calculate the means of x and y:

Mean of x = (2 + 3 + 5) / 3 = 3.333

Mean of y = (4 + 3 + 6) / 3 = 4.333

Next, we calculate the sample covariance of x and y and the sample variance of x:

Sample covariance of x and y = [(2 - 3.333)(4 - 4.333) + (3 - 3.333)(3 - 4.333) + (5 - 3.333)(6 - 4.333)] / 2

= -0.333

Sample variance of x = [(2 - 3.333)^2 + (3 - 3.333)^2 + (5 - 3.333)^2] / 2

= 2.888

Finally, we can use these values to calculate the slope and intercept of the least-squares line:

Slope = sample covariance of x and y / sample variance of x = -0.333 / 2.888 = -0.115

Intercept = mean of y - (slope * mean of x) = 4.333 - (-0.115 * 3.333) = 4.759

Therefore, the least-squares equation for the given (x, y) data pairs is ŷ = 4.759 - 0.115x, rounded to three digits after the decimal.

(b) Following the same steps as in part (a), we find:

Mean of x = (4 + 3 + 6) / 3 = 4.333

Mean of y = (2 + 3 + 5) / 3 = 3.333

Sample covariance of x and y = [(4 - 4.333)(2 - 3.333) + (3 - 4.333)(3 - 3.333) + (6 - 4.333)(5 - 3.333)] / 2

= 1.333

Sample variance of x = [(4 - 4.333)^2 + (3 - 4.333)^2 + (6 - 4.333)^2] / 2

= 2.888

Slope = sample covariance of x and y / sample variance of x = 1.333 / 2.888 = 0.461

Intercept = mean of y - (slope * mean of x) = 3.333 - (0.461 * 4.333) = 1.505

Therefore, the least-squares equation for the given (x, y) data pairs is ŷ = 1.505 + 0.461x, rounded to three digits after the decimal.

(d) To solve the least-squares equation from part (a) for x, we can rearrange the equation as follows:

x = (y - 4.759) / (-0.115)

Therefore, x = (-8.130y + 37.069), rounded to three digits after the decimal.

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If the functions f(x) = x²+3, g(x) = 5−x, then the values of,

(a) (f g)(x) = 28 - 10x + x²

(b) (f - g)(x) = x² + x - 2

(c) (fg)(x) = -x³ + 2x² + 15x - 15

(d) (f/g)(x) = (5x² + 8x + 15) / (x² - 25), where x ≠ 5.

(a) (f g)(x) represents the composition of two functions f(x) and g(x), where the output of g(x) is the input to f(x).

So, (f g)(x) = f(g(x)) = f(5-x) = (5-x)² + 3 = 28 - 10x + x².

Therefore, (f g)(x) = 28 - 10x + x².

(b) (f - g)(x) represents the subtraction of one function from another.

So, (f - g)(x) = f(x) - g(x) = (x² + 3) - (5 - x) = x² + x - 2.

Therefore, (f - g)(x) = x² + x - 2.

(c) (fg)(x) represents the multiplication of two functions.

So, (fg)(x) = f(x) × g(x) = (x² + 3) × (5 - x) = -x³ + 2x² + 15x - 15.

Therefore, (fg)(x) = -x³ + 2x² + 15x - 15.

(d) (f/g)(x) represents the division of one function by another.

So, (f/g)(x) = f(x) / g(x) = (x² + 3) / (5 - x).

Note that (5 - x) cannot equal 0, otherwise the denominator would be undefined. Therefore, the domain of (f/g)(x) is all real numbers except x = 5.

Simplifying (f/g)(x) by multiplying the numerator and denominator by the conjugate of the denominator (5 + x), we get

(f/g)(x) = (x² + 3) / (5 - x) × (5 + x) / (5 + x)

= (x² + 3) (5 + x) / (25 - x²)

= (5x² + 8x + 15) / (x² - 25)

Therefore, (f/g)(x) = (5x² + 8x + 15) / (x² - 25), where x ≠ 5.

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The given question is incomplete, the complete question is:

If f(x) = x²+3, g(x) = 5−x  find the values of (a) (f g)(x)

(b) (f − g)(x)  

(c) (fg)(x)

(d) (f/g)(x)

Answer:

[tex]a_{11} = 45[/tex]

Step-by-step explanation:

An arithmetic sequence can be defined by an explicit formula in which , [tex]a_n = a_1 + d(n-1)[/tex], where d is the common difference between consecutive terms.

Plugging in [tex]a_1\\[/tex] as 5, [tex]d[/tex] as 4, and [tex]n[/tex] as 11, we get the equation [tex]a_n = 5 + 4(11-1)[/tex]. [tex]11-1=10[/tex], and [tex]4[/tex] × [tex]10\\[/tex] [tex]=40\\[/tex], and finally [tex]40 + 5 = 45[/tex].

Thus, [tex]a_{11} = 45[/tex].

Answer:

35

Step-by-step explanation:

we know that,

formula of arithmatic sequence is a+(n-1)d.

a11=a+(n-1)d

a11=-5+(11-1)4

a11=-5+10*4

a11=-5+40

a11=35.

The arithmatic sequence of a11=35.

Thank you

To find the function r(t) given its derivative r′(t) and an initial condition, we need to integrate r′(t) and apply the initial condition.

Step 1: Integrate r′(t) component-wise:
For the i-component: ∫(te^(-t^2)) dt
For the j-component: ∫(-e^(-t)) dt
For the k-component: ∫(1) dt

Step 2: Find the antiderivatives for each component:
For the i-component: -(1/2)e^(-t^2) + C1
For the j-component: e^(-t) + C2
For the k-component: t + C3

Step 3: Combine the antiderivatives to obtain the general solution for r(t):
r(t) = [-(1/2)e^(-t^2) + C1]i + [e^(-t) + C2]j + [t + C3]k

Step 4: Apply the initial condition r(0):
r(0) = [-(1/2)e^(0) + C1]i + [e^(0) + C2]j + [0 + C3]k
Given r(0), we can determine the constants C1, C2, and C3.

Without the provided value for r(0), I can't find the specific constants, but you can use the general solution r(t) and plug in r(0) to find the exact function for r(t).

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The values of C and a that satisfy the given conditions are C = 9 and a = 3/2, respectively. Thus, the function f(x) = Ca is given by:

f(x) = 9(3/2)x = 27/2x

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

Since we are given that f(x) = Ca, we have to determine the values of C and a such that the given conditions f(1) = 9 and f(2) = 27 are satisfied.

First, we have f(1) = Ca(1) = C. Therefore, we have:

C = 9

Next, we have f(2) = Ca(2) = 2aC. Since we know that C = 9, we can substitute it into the expression for f(2) to obtain:

f(2) = 2aC = 2a(9) = 18a

We are also given that f(2) = 27, so we can substitute this value to get:

18a = 27

Solving for a, we obtain:

a = 3/2

Therefore, the values of C and a that satisfy the given conditions are C = 9 and a = 3/2, respectively. Thus, the function f(x) = Ca is given by:

f(x) = 9(3/2)x = 27/2x

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Answer: Complementary: x= 5

Step-by-step explanation:

First we know that the angles are complementary because they add to 90 degrees.

Next to find 5x we can subtract 65 from 90: 90-65=25

Solve: 5x=25

x=5

The value of x is √42

Similar triangles are triangles that have the same shape, but their sizes may vary. The corresponding ratio of similar triangles are equal.

Therefore,

represent the hypotenuse of the small triangle by y

y/13 = 6/y

y² = 13×6

y² = 78m

Using Pythagoras theorem,

y² = 6²+x²

78 = 36+x²

x² = 78-36

x² = 42

x = √42

therefore the value of x is √42

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Answer:

Step-by-step explanation:

To find the area of the circle, we need to use the formula:

A = πr^2

where D is the diameter of the circle and r is the radius, which is half of the diameter.

Given that D = 22ft, we can find the radius by dividing the diameter by 2:

r = D/2 = 22ft/2 = 11ft

Now we can substitute the value of r into the formula for the area:

A = πr^2 = π(11ft)^2

Using 3.14 as an approximation for π, we get:

A ≈ 3.14 × 121ft^2 ≈ 380.13ft^2

Therefore, the area of the circle is approximately 380.13 square feet.

➢ Radius of Circle:-

➢ Area of Circle:-

The chain rule to find du/dt: du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt) + (∂u/∂z)(dz/dt)
du/dt = (y/z)(4p3r4) + ((x - u)/z)(4p-3r4) + [tex](-xy/z^2)(-4p3r)[/tex]Now, you can substitute the given expressions for x, y, and z to compute du/dt in terms of p, r, and t.

To use the chain rule, we need to find the partial derivatives of u with respect to x, y, and z, and then multiply them together.

∂u/∂x = y/y z = 1/z

∂u/∂y = x/z

∂u/∂z = -xy/y^2 z

Now we can apply the chain rule:

∂u/∂p = (∂u/∂x)(∂x/∂p) + (∂u/∂y)(∂y/∂p) + (∂u/∂z)(∂z/∂p)

= (1/z)(3r) + (p-3r)/(p-3r+4t)(-3) + (-xy/y^2 z)(3r)

Simplifying, we get:

∂u/∂p = (3r/z) - (3xyr)/(y^2 z(p-3r+4t))

Note: The simplification assumes that y is not equal to zero. If y=0, the function u is undefined.
To find the derivative of the function u(x, y, z) with respect to t using the chain rule, you need to find the partial derivatives of u with respect to x, y, and z, and then multiply them by the corresponding derivatives of x, y, and z with respect to t.

Given u = xy/yz and x = p3r4t, y = p-3r4t, z = p3r-4t.

First, find the partial derivatives of u with respect to x, y, and z:

∂u/∂x = y/z
∂u/∂y = (x - u)/z
∂u/∂z = -xy/z^2

Next, find the derivatives of x, y, and z with respect to t:

dx/dt = 4p3r4
dy/dt = 4p-3r4
dz/dt = -4p3r

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The error interval for the area "a" of the triangle is:  13.17 cm² ≤ a < 16.065 cm²

Given data ,

Let's write "b" for the triangle's base and "h" for the height of the perpendicular.

The alternative values for "b" would be 7 cm or 6 cm, depending on whether the actual value of the base is closer to 7.5 cm or 6.5 cm, respectively.

The range of potential values for "h" is 4.45 cm to 4.55 cm, depending on whether the actual height value is more closely related to 4.45 cm or 4.55 cm, respectively

Now , area of the triangle = ( 1/2 ) x Length x Base

When base "b" is 7 cm and height "h" is 4.45 cm:

Minimum possible area = (1/2) * 7 * 4.45 = 15.615 cm²

When base "b" is 7 cm and height "h" is 4.55 cm:

Maximum possible area = (1/2) * 7 * 4.55 = 16.065 cm²

When base "b" is 6 cm and height "h" is 4.45 cm:

Minimum possible area = (1/2) * 6 * 4.45 = 13.17 cm²

When base "b" is 6 cm and height "h" is 4.55 cm:

Maximum possible area = (1/2) * 6 * 4.55 = 13.63 cm²

Hence , the error interval for the area "a" of the triangle is:

13.17 cm² ≤ a < 16.065 cm²

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The distance between the line and the point is D = 10/3 units

Given data ,

Let the two points be P ( -4 , 0 ) and Q ( 0 , -2 )

To find the slope (m)

m = (y2 - y1) / (x2 - x1)

m = (-2 - 0) / (0 - (-4))

m = -2 / 4

m = -1/2

So, the equation of the line is:

y = (-1/2)x + b

To find the y-intercept (b), we can plug in the coordinates of one of the points.

-2 = (-1/2)(0) + b

b = -2

So, the equation of the line is

y = (-1/2)x - 2

Now , Distance of a point to line D = | Ax₀ + By₀ + C | / √ ( A² + B² )

On simplifying , we get

( 1/2 )x + y + 2 = 0

A = 1/2 , B = 1 and C = 2

D = | ( 1/2 )4 + 1 + 2 | / √(9/4)

D = 5 / 3/2

D = 10/3 units

Hence , the distance is D = 10/3 units

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We can make the conclusion that the series ∑ 3cos(n)/n is convergent.

The series ∑ 3cos(n)/n satisfies the conditions of the integral test if we consider the function f(x) = 3cos(x)/x.

Using integration by parts, we can find that the integral of f(x) from 1 to infinity is equal to 3sin(1) + 3/2 ∫1^∞ sin(x)/x^2 dx.

Since the integral ∫1^∞ sin(x)/x^2 dx converges (as it is a known convergent integral), we can conclude that the series ∑ 3cos(n)/n also converges by the integral test.

Using the Integral Test, we can determine the convergence or divergence of the series ∑ (3cos(n)/n) from n=1 to infinity. The Integral Test states that if a function f(n) is continuous, positive, and decreasing for all n≥1, then the series ∑ f(n) converges if the integral ∫ f(x)dx from 1 to infinity converges, and diverges if the integral diverges.

In this case, f(n) = 3cos(n)/n. Unfortunately, this function is not always positive, as the cosine function oscillates between -1 and 1. Due to this property, the Integral Test is not applicable to the given series, and we cannot draw any conclusions about its convergence or divergence using this test.

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Thus, the given set in spherical coordinates can be described as: ρ = 8, 0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π.

The given set can be described in spherical coordinates as follows: ρ² = 64 and z ≥ 0, where ρ (rho) is the radial distance, θ (theta) is the polar angle, and φ (phi) is the azimuthal angle.
In spherical coordinates, the relationship between Cartesian and spherical coordinates is:
x = ρ × sin(θ) × cos(φ)
y = ρ × sin(θ) × sin(φ)
z = ρ × cos(θ)
For x² + y² + z² = 64, we can substitute the spherical coordinates:
(ρ * sin(θ) × cos(φ))² + (ρ × sin(θ) × sin(φ))² + (ρ × cos(θ))² = 64
ρ² * (sin²(θ) × cos²(φ) + sin²(θ) × sin²(φ) + cos²(θ)) = 64
Since sin²(θ) + cos^2(θ) = 1, the equation simplifies to:
ρ² = 64
So, ρ = 8, as the radial distance must be non-negative.
For z ≥ 0, we use the relationship z = ρ × cos(θ):
8 × cos(θ) ≥ 0
This inequality is satisfied when 0 ≤ θ ≤ π/2, as the cosine function is non-negative in this range.
Since the azimuthal angle φ covers the entire range of possible angles in the xy-plane, we have 0 ≤ φ ≤ 2π.

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Answer: 54.98 sq. ft.

Step-by-step explanation:

The derivative of the function is y'(x) = 24x(x² + 8)^2.

Given: y = 4(x² + 8)^3, and f(u) = 4eu.

First, we need to find the function g(x) such that y = f(g(x)). Comparing y and f(u), we get:

4(x² + 8)^3 = 4e^(g(x))

We can deduce that g(x) must be of the form:

g(x) = ln((x² + 8)^3)

Now, let's find the derivatives f'(u) and g'(x).

f'(u) = d(4eu)/du = 4eu

g'(x) = d[ln((x² + 8)^3)]/dx = 3(x² + 8)^2 * (2x) / (x² + 8)^3 = 6x / (x² + 8)

Lastly, we'll find the derivative of the function y(x) using the chain rule:

y'(x) = f'(g(x)) * g'(x)

y'(x) = [4e^(ln((x² + 8)^3))] * [6x / (x² + 8)]

y'(x) = [4(x² + 8)^3] * [6x / (x² + 8)]

y'(x) = 24x(x² + 8)^2

So the derivative of the function y(x) is:

y'(x) = 24x(x² + 8)^2

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