Use induction to prove that if A1, A2,...,An and B are sets,
then (A1 −B) ∩(A2 −B) ∩...∩(An−B) = (A1 ∩A2 ∩...∩An) −B.

Given the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1. We have to determine the number of zeros of the function in the disk D[0,2].

According to the Fundamental Theorem of Algebra, a polynomial function of degree n has n complex zeros, counting multiplicity. Here, the degree of the given polynomial function is 4. Therefore, it has exactly 4 zeros.Let the zeros of the function f(z) be a, b, c, and d. The function can be written as the product of its factors:$$f(z) = (z-a)(z-b)(z-c)(z-d)$$$$\Rightarrow f(z) = z^4 - (a+b+c+d)z^3 + (ab+ac+ad+bc+bd+cd)z^2 - (abc+abd+acd+bcd)z + abcd$$

According to the Cauchy's Bound, if a polynomial f(z) of degree n is such that the coefficients satisfy a_0, a_1, ..., a_n are real numbers, and M is a real number such that |a_n|≥M>|a_n-1|+...+|a_0|, then any complex zero z of the polynomial satisfies |z|≤1+M/|a_n|.

We can write the polynomial function as $$f(z) = z^4 - 2z^3 + 9z^2 + z - 1 = (z-1)^2(z+1)(z-1+i)(z-1-i)$$The zeros of the function are 1 (multiplicity 2), -1, 1 + i, and 1 - i. We have to count the zeros that are in the disk D[0,2].Zeros in the disk D[0,2] are 1 and -1.Therefore, the number of zeros of the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1 in the disk D[0,2] is 2.

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Answer:

6

Step-by-step explanation:

2,400 divided by 4 = 6

For values of n other than 0 or 1 in a Bernoulli differential equation, the substitution [tex]u = y^{(1-n)[/tex] is used to transform it into a linear equation.

A Bernoulli differential equation is given by the form:

dy + P(x)y dx = Q(x)[tex]y^n[/tex] (*)

If we consider the case when n = 0 or n = 1, the Bernoulli equation becomes linear. Let's examine each case:

When n = 0:

Substituting[tex]u = y^{(-n) }= y^{(-0)} = 1[/tex], the differential equation becomes:

[tex]dy + P(x)y dx = Q(x)y^0[/tex]

dy + P(x)y dx = Q(x)

This is a linear differential equation of the first order.

When n = 1:

Substituting [tex]u = y^{(-n) }= y^{(-1)},[/tex] we have:

[tex]u = y^{(-1)[/tex]

Taking the derivative of both sides with respect to x:

[tex]du/dx = -y^{(-2)} \times dy/dx[/tex]

Rearranging the equation:

[tex]dy/dx = -y^2\times du/dx[/tex]

Now substituting the expression for dy/dx in the original Bernoulli equation:

[tex]dy + P(x)y dx = Q(x)y^1\\-y^2 \times du/dx + P(x)y dx = Q(x)y\\-y \times du + P(x)y^3 dx = Q(x)y[/tex]

This equation is also a linear differential equation of the first order, but with the variable u instead of y.

In summary, when n is equal to 0 or 1, the Bernoulli equation becomes linear. For other values of n, a substitution u = y^(-n) is typically used to transform the Bernoulli equation into a linear differential equation, allowing for easier analysis and solution.

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Answer:

6

0

2

Step-by-step explanation:

I assume we are taking the y-value and dividing it by x, as indicated by the y/x

-6/-1

6

0/1

0

6/3

2

Answer:

2ax(3b+4)

Step-by-step explanation:

there you go your answer

Answer:

(4,2)

Step-by-step explanation:

Answer:

(4, 2)

Step-by-step explanation:

Answer: Mean = 2.36 Median = 4 Range = 0

Step-by-step explanation:

Mean - the sum of the data values divided by the number of data values

Median - the middle number in an ordered set of data

Range - the difference between the greatest and least numbers in a data set

This coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.

The binomial formula is used to expand binomials of the form (a + b)ⁿ, where a, b, and n are integer.

In general, the formula is given by:

[tex]$(a+b)^n=\sum_{k=0}^{n}{n \choose k}a^{n-k}b^k$[/tex]

The coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² can be found by using the binomial formula.

To find this coefficient, we need to determine the value of k for which the term [tex]y^{22-k} (3x)^k[/tex] has y¹²⁰x²  as a product.

Let's write out the first few terms of the expansion of (y + 3x)²²:

[tex]$(y + 3x)^{22} = {22 \choose 0}y^{22}(3x)^0 + {22 \choose 1}y^{21}(3x)^1 + {22 \choose 2}y^{20}(3x)^2 + \cdots$[/tex]

Notice that each term in the expansion has the form {22 choose k}[tex]y^{22-k} (3x)^k[/tex]

Thus, the coefficient of the y¹²⁰ x²  term is given by the binomial coefficient {22 choose k}, where k is the value that makes 22 - k equal to the exponent of y in y¹²⁰  (i.e., 120). Therefore, we have:

22 - k = 120k = 22 - 120k = -98

Thus, the coefficient of the y¹²⁰ x² term is given by the binomial coefficient {22 choose -98}.

However, this coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²²  is 0.

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Answer:

[tex]\frac{4}{9}[/tex]

Step-by-step explanation:

[tex](\frac{2}{3} )^{2} =\frac{2^2}{3^2} =4/9[/tex]

Hope that helps :)

Answer:

87 ft

Step-by-step explanation:

SohCahToa is your best friend here.

You have two values you need to pay attention to:

The length that is adjacent to the 74°C, 25 ft. And the length opposite of the 74°C, the height of how high the rocket traveled.

So adjacent and opposite, O & A. "Toa", find the tangent of 74°C.

tan(74) = [tex]\frac{x}{25}[/tex]

x = (tan(74))(25)

x = 87 ft

Answer:

1

Step-by-step explanation:

The y-intercept in the equation is 1 because the equation uses the format y=mx+b. The b in y=mx+b represents the y-intercept So, in this equation the y-intercept is 1 because b=1.

The subspace topology for Z, which is a subset of R with its usual (standard) topology, is the set of open sets in Z.

In other words, the subspace topology on Z is obtained by considering the intersection of Z with open sets in R.

To find the subspace topology for Z, we need to determine which subsets of Z are open. In the usual topology on R, an open set is a set that can be represented as a union of open intervals. Since Z is a subset of R, its open sets will be the intersection of Z with open intervals in R.

For example, let's consider the open interval (a, b) in R. The intersection of (a, b) with Z will be the set of integers between a and b (inclusive) that belong to Z. This intersection is an open set in Z.

By considering all possible open intervals in R and their intersections with Z, we can generate the collection of open sets that form the subspace topology for Z. This collection of open sets will satisfy the axioms of a topology, including the properties of openness, closure under unions, and closure under finite intersections.

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jesse has never used the sliding board at his daycare because he is afraid. His teacher has encouraged him, but he refuses to slide down. one day his mother stands at the bottom And say's Jesse let's go and slides into his mother's arms Laughing The situation is a example of Social referencing. Option (d) is correct.

Situation refers to a group of conditions or a current state of events.

Social reference is the method through which newborns control their behavior toward surrounding items, people, and circumstances by observing the emotive displays of an adult.

For adaptive social functioning to occur, one must recognize and make use of the emotional communication of others. The ability to negotiate complicated and frequently ambiguous settings is known as social referencing in the developmental literature and social appraisal in adult studies.

Therefore, Option (d) is correct. The situation is a example of Social referencing.

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Answer:

x = 120°

Step-by-step explanation:

this is a 7-sided polygon and the sum of the interior angles is (7-2)×180° = 900°

add all 7 angles together and set equal to 900

x + 150 + x - 20 + 140 + 120 + x + 20 + 130 = 900

combine 'like terms'

3x + 540 = 900

3x = 360

x = 120

The standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients is 1.6487.

Given that a healthcare research agency reported that 41% of people who had coronary bypass surgery in 2008 were over the age of 65 and twelve coronary bypass patients are sampled.

To determine the mean number of people over the age of 65 in a sample of 12 coronary bypass patients, we use the formula below:

Mean = np

Where n = 12 and p = 0.41.

Mean = 12(0.41)

Mean = 4.92

Therefore, the mean number of people over the age of 65 in a sample of 12 coronary bypass patients is 4.92.

To determine the standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients, we use the formula below:

Standard deviation, σ = √(n p q)

Where n = 12, p = 0.41, and q = 1 - p.

Standard deviation, σ = √(12 × 0.41 × 0.59)

Standard deviation, σ = √2.71948

Standard deviation, σ = 1.6487 (rounded to four decimal places).

Therefore, the standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients is 1.6487.

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The equation for the time of the trip, t, as a function of the speed, s, is t = (50/s) + (75/(s-10)).

To find the equation for the time of the trip as a function of the speed of the car before slowing down, we need to consider two parts of the journey. The first part is driving 50 miles at the original speed, which takes (50/s) hours, where s is the speed. The second part is driving 75 miles at a slower speed of (s-10) mph, which takes (75/(s-10)) hours.

To calculate the total time, we add the times for both parts: t = (50/s) + (75/(s-10)). This equation allows us to determine the time of the trip for any given speed before slowing down.

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The population of interest in the given scenario is all named storms that have made landfall in the United States since 2000. "All named storms that have made landfall in the United States since 2000".

The given scenario is focusing on determining the mean sustained wind speed of all named storms that have made landfall in the United States since 2000. Therefore, the population of interest in this scenario is all named storms that have made landfall in the United States since 2000. The population of interest is the entire group of individuals, objects, events, or processes that researchers want to investigate to answer their research questions.

The researchers want to determine the mean sustained wind speed of all named storms that have made landfall in the United States since 2000. Hence, they will collect data on the wind speed of all named storms that have made landfall in the United States since 2000, and calculate the mean sustained wind speed for the entire population.

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Answer:

5 drinks will be americanos

Step-by-step explanation:

2:8  (2/8)

simplify

1:4  (1/4)

divide 20 by 4

5:20

The number of next 20 drinks which may be Americanos is 5.

Probability is simply the possibility of getting an event. Or in other words, we are predicting the chance of getting an event.

The value of probability will be always in the range from 0 to 1.

Given that,

Total drinks sold = 8

Number of drinks that is Americanos = 2

Probability of finding Americano = 2/8 = 1/4

If the total number of drinks next is 20,

Number of Americanos expected = Probability of Americanos × Number of drinks

= 1/4 × 20

= 5

Hence the number of Americanos expected is 5.

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Answer:

$400

Step-by-step explanation:

all you do is add 320+540+230+450+280+580/6 and the asnwe comes out to 400

Answer:

-1.4  ; -0.7  ; 0.003   ; 3%    ;  0.3   ;  2/3  ;    7/8  ;   100/50

Step-by-step explanation:

0.003 ;  -1.4 ;  100/50 = 2 ;  0.85 ; 2/3 = 0.67  ; 3% = 3/100 = 0.03 ; 7/8 = 0.875 ;

0.3 , -0.7

0.003 ;  -1.4 ;   2 ;  0.85 ;  0.67  ;   0.03 ;   0.875 ; 0.3    ; -0.7

Least to greatest:

-1.4 ; - 0.7   ; 0.003  ;   0.03  ;   0.3   ;  0.67  ; 0.85    ;2

-1.4  ; -0.7  ; 0.003   ; 3%    ;  0.3   ;  2/3  ;    7/8  ;   100/50

Negative numbers have least value.Then  in decimal numbers, the number having the least value in tenth is the least

Answer:

[tex] 2 \sqrt{3} [/tex]

Step-by-step explanation:

Geometric mean of 4 and 3

[tex] = \sqrt{4 \times 3} \\ = \sqrt{ {2}^{2} \times 3 } \\ = 2 \sqrt{3} [/tex]

The value of the lawn ornament set in the year 2009 was $51.375. The value of the lawn ornament set in the year 2021 was $558.181. The instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986. The instantaneous rate of change of the value of the lawn ornament set in the year 2021 was $351.076.  The estimated value of the lawn ornament set in 2022 was $909.257.

a)

To find the value of the lawn ornament set in the year 2009, we have to plug in t = 3, as t = 0 corresponds to the year 2006.

V(3) = 0.06(3)4 – 1.05(3)3 + 3.47(3) – 8.896 + 269.95

V(3) = 51.375

So, the value of the lawn ornament set in the year 2009 was $51.375.

b)

To find the value of the lawn ornament set in the year 2021, we have to plug in t = 15, as t = 0 corresponds to the year 2006.

V(15) = 0.06(15)4 – 1.05(15)3 + 3.47(15) – 8.896 + 269.95

V(15) = $558.181

So, the value of the lawn ornament set in the year 2021 is $558.181.

c)

To find the instantaneous rate of change of the value of the lawn ornament set in the year 2013, we have to find V'(7), where V(t) is the given function.

V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts 15

V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95

V'(7) = 0.24(7)3 – 3.15(7)2 + 10.41(7) + 269.95

V'(7) = $230.986

So, the instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986.

d) To find the instantaneous rate of change of the value of the lawn ornament set in the year 2021, we have to find V'(15), where V(t) is the given function.

V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts

15V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95

V'(15) = 0.24(15)3 – 3.15(15)2 + 10.41(15) + 269.95

V'(15) = $351.076

So, the instantaneous rate of change of the value of the lawn ornament set in the year 2021 is $351.076.

e)

To ESTIMATE the value of the lawn ornament set in 2022, we can use the formula

V(t) ≈ V(a) + V'(a)(t – a),

where a is the year 2021.

V(a) = V(15) = $558.181

V'(a) = V'(15) = $351.076t = 16 (as we need to estimate the value of the lawn ornament set in 2022)

V(t) ≈ V(a) + V'(a)(t – a)

V(t) ≈ 558.181 + 351.076(16 – 15)

V(t) ≈ $909.257

So, the estimated value of the lawn ornament set in 2022 is $909.257.

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The values of  base-ten numerals to the indicated bases are:

a. 861 in base six is 3553.

b. 2157 in base nine is 2856.

c. 131 in base three is 11221.

To convert the base-ten numerals to the indicated bases:

a. 861 in base six:

To convert 861 to base six, we divide the number by six repeatedly and note down the remainder until the quotient becomes zero.

861 ÷ 6 = 143 remainder 3

143 ÷ 6 = 23 remainder 5

23 ÷ 6 = 3 remainder 5

3 ÷ 6 = 0 remainder 3

Reading the remainders in reverse order, the base-six representation of 861 is 3553.

b. 2157 in base nine:

To convert 2157 to base nine, we follow a similar process.

2157 ÷ 9 = 239 remainder 6

239 ÷ 9 = 26 remainder 5

26 ÷ 9 = 2 remainder 8

2 ÷ 9 = 0 remainder 2

Reading the remainders in reverse order, the base-nine representation of 2157 is 2856.

c. 131 in base three:

To convert 131 to base three, we apply the same procedure.

131 ÷ 3 = 43 remainder 2

43 ÷ 3 = 14 remainder 1

14 ÷ 3 = 4 remainder 2

4 ÷ 3 = 1 remainder 1

1 ÷ 3 = 0 remainder 1

Reading the remainders in reverse order, the base-three representation of 131 is 11221.

Therefore:

a. 861 in base six is 3553.

b. 2157 in base nine is 2856.

c. 131 in base three is 11221.

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Answer:

13388

Step-by-step explanation:

12600 will be 100% so we want to get at what price its sold when there is a 15%dicount

So will minus 15% from the 100% of the Mp

100%-15%=85%

so if 100%=12600

what about 85%=?

we crossmultiply

85%×12600/100%=10710

so10710 is what the radio will be sold if a 15% dicount is given but we want to get wat price the shopkeeper got in that he made a profit of25%

so if 100%=10710

what about 125%

125%×10710/100%=13387.5 which is 13388/=

Using the matrix exponential method, the solution to the non-homogeneous IVP y'(t) = -x(t), with initial conditions x(0) = 1 and y(0) = 0, is given by X(t) = [1 - t; -t 1]. Alternatively, solving the system of equations x'(t) = 3y(t) and y'(t) = 3x(t) yields [tex]\[x(t) = \frac{3yt^2}{2} + t\][/tex] and [tex]\[y(t) = \frac{3xt^2}{2}\][/tex] as the solution.

Here is the explanation :

(a) Using the matrix exponential method:

The given system of equations can be written in matrix form as:

X' = A*X + B, where X = [y; x], A = [0 -1; 0 0], and B = [0; -1].

To solve this system using the matrix exponential method, we first need to find the matrix exponential of A*t. The matrix exponential is given by:

[tex]\[e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \dotsb\][/tex]

To find the matrix exponential, we calculate the powers of A:

A² = [0 -1; 0 0] * [0 -1; 0 0] = [0 0; 0 0]

A³ = A² * A = [0 0; 0 0] * [0 -1; 0 0] = [0 0; 0 0]

...

Since A² = A³ = ..., we can see that Aⁿ = 0 for n ≥ 2. Therefore, the matrix exponential becomes:

[tex]\[e^{At} = I + At\][/tex]

Substituting the values of A and t into the matrix exponential, we get:

[tex][e^{At} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & -t \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -t \\ 0 & 1 \end{bmatrix}][/tex]

Now we can find the solution to the non-homogeneous system using the matrix exponential:

[tex]\[X(t) = e^{At} X(0) + \int_0^t e^{A\tau} B d\tau\][/tex]

Substituting the given initial conditions X(0) = [1; 0] and B = [0; -1], we have:

X(t) = [1 -t; 0 1] * [1; 0] + ∫[0, t] [1 -τ; 0 1] * [0; -1] dτ

Simplifying the integral and matrix multiplication, we get:

X(t) = [1 -t; 0 1] * [1; 0] + ∫[0, t] [0; -1] dτ

    = [1 -t; 0 1] * [1; 0] + [-t 1]

Finally, we obtain the solution:

X(t) = [1 -t; -t 1]

(b) Using another method:

Given the system of equations:

x' = 3y

y' = 3x

We can solve this system by taking the derivatives of both equations:

x'' = 3y'

y'' = 3x'

Substituting the initial conditions x(0) = 1 and y(0) = 0, we have:

x''(0) = 3y'(0) = 0

y''(0) = 3x'(0) = 3

Integrating the second-order equations, we find:

x'(t) = 3yt + C₁

y'(t) = 3xt + C₂

Applying the initial conditions x'(0) = 0 and y'(0) = 3, we get:

C₁ = 0

C₂ = 3

Integrating once again, we obtain:

[tex]\[\begin{aligned}x(t) &= \frac{3yt^2}{2} + C_1t + C_3 \\y(t) &= \frac{3xt^2}{2} + C_2t + C_4\end{aligned}\][/tex]

Substituting the initial conditions x(0) = 1 and y

(0) = 0, we have:

C₃ = 1

C₄ = 0

Therefore, the solution to the system is:

[tex]\[\begin{aligned}x(t) &= \frac{3yt^2}{2} + t \\y(t) &= \frac{3xt^2}{2}\end{aligned}\][/tex]

Thus, we have obtained the solutions for x(t) and y(t) using an alternative method.

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x'(t)= y(t)-1 1. Solve the non-homogeneous IVP: y'(t)=-X(t) (x(0)= 1,7(0) = 0 a. using the matrix exponential method, b. using any other method of your choice. . Find a Fundamental Matrix 0(t) and solve the IVP: x'= 3y 1 y' = 3* (x(0) = 1, y(0)=0 , for x(t) and y(t).

Answer: 2 1/2

Step-by-step explanation:

the answer is D i took the test here is proof

Answer:

the full answer is 215.859885inches cubed

Step-by-step explanation:

times the length, width and height together

The correct answer is option (C): 0.487

Given that the average time to receive an order at a restaurant is 15 minutes, we can use the exponential distribution to calculate the probability of receiving the order in the first 10 minutes.

The exponential distribution is defined by the probability density function (PDF): f(x) = (1/µ) * e^(-x/µ), where µ is the average value or mean.

In this case, the mean (µ) is 15 minutes. We want to find P(a ≤ X ≤ b), where a is 0 (the lower bound) and b is 10 (the upper bound).

To calculate this probability, we need to integrate the PDF from a to b:

P(0 ≤ X ≤ 10) = ∫[0 to 10] (1/15) * e^(-x/15) dx

Integrating this expression gives us:

P(0 ≤ X ≤ 10) = [-e^(-x/15)] from 0 to 10

Plugging in the values, we get:

P(0 ≤ X ≤ 10) = [-e^(-10/15)] - [-e^(0/15)]

Simplifying further:

P(0 ≤ X ≤ 10) = -e^(-2/3) + 1

Using a calculator, we can evaluate this expression:

P(0 ≤ X ≤ 10) ≈ 0.487

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To determine the value of b in the expression x^2b(x - 7)^2, we can compare it with the given equivalent expression x^2b49. By equating the two expressions, we can solve for b.

In the given expression x^2b(x - 7)^2, we can simplify it by multiplying the exponents:

x^2 * b * (x - 7)^2 = x^2b(x^2 - 14x + 49)

Comparing this with the equivalent expression x^2b49, we can equate the coefficients of the like terms:

x^2b(x^2 - 14x + 49) = x^2b49

From this equation, we can see that the coefficient of the x term is -14b. For it to be equivalent to 49, we have:

-14b = 49

Solving for b, we divide both sides by -14:

b = -49/14 = -7/2

Therefore, the value of b is -7/2.

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a. To take a sample of the student population that would not represent the population well, Susan could use a biased sampling method.

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