If α is chosen by the analyst to be .025 and X2o= 14.15 with 4 degrees of freedom, what is our conclusion for the hypothesis test if H1: σ > σ0?
Reject H0.
Fail to Reject H0.
Accept H1.
Reject H1

The correct statement is: the student is incorrect because 32 is less than 12.

This is because the student spent a total of 32 hours on the project (14 on Wednesday + 18 on Thursday), but claimed it took only 12 hours to complete. Therefore, the student's statement is not true.

The word "more" is used when one number is greater than another. Use more even when comparing two weights. For example, Joe went to the ice cream parlor. She likes the chocolate and vanilla flavor of the ice cream but wants to buy a cheaper ice cream cone. He asked the price of the chocolate and vanilla cones.

The seller shows the price of two types of cones: $10 for a dough cone and $5 for a vanilla cone. Then to compare the value of the two cones, Joe should use the concept of "more". As we can see, the chocolate cone is more expensive than the vanilla cone. The price of the cookie ($10) is more than the price of the vanilla cone ($5), so 10 > 5. So he compares the price of two ice creams and decides to buy the vanilla ice cream.

Just like two weights, distance, volume etc. Use the greater than sign as compare.

The student is incorrect because 38 is greater than 12. This is because the student spent 14 hours on Wednesday and 18 hours on Thursday, which totals 14 + 18 = 38 hours, and 38 hours is greater than the reported 12 hours.

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The stress (σ) for n = 100 is approximately 333.33 MPa.

The stress (σ) for n = 103 is also approximately 333.33 MPa.

To calculate the stress (σ) for n = 100 and 103 using the given Sut (ultimate tensile strength) and Fsut (factor of safety for ultimate tensile strength), we can use the formula:

σ = Sut / Fsut

Let's assume the given values of Sut and Fsut are as follows:

Sut = 500 MPa (megapascals)

Fsut = 1.5 (dimensionless)

For n = 100:

Plugging in the values into the formula, we get:

σ = Sut / Fsut

= 500 MPa / 1.5

≈ 333.33 MPa

So, the stress (σ) for n = 100 is approximately 333.33 MPa.

Similarly, for n = 103:

Using the same formula with the given values of Sut and Fsut:

σ = Sut / Fsut

= 500 MPa / 1.5

≈ 333.33 MPa

So, the stress (σ) for n = 103 is also approximately 333.33 MPa.

Please note that these calculations are based on the given values of Sut and Fsut, and the units are assumed to be in megapascals (MPa) as per the given formula.

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Using the central angle theorem, we can find the arm length of BD to be 118units.

Option C is correct.

The angle that an arc occupies at the centre of a circle is twice as large as the angle it occupies anywhere else around the circle's circumference, according to the central angle measure theorem.

Here in the question,

Length of the arc AC = 59.

As we can see that there is a quadrilateral inscribed inside the circle.

Arc AC = 1/2 arc BD

⇒ arc BD = 2 × arc length AC

⇒ arc BD = 2 × 59

⇒ arc BD = 118.

Therefore, the length of the arc BD = 118 units.

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Using the central angle theorem, we can find the arm length of BD to be 118units.

Option C is correct.

The angle that an arc occupies at the centre of a circle is twice as large as the angle it occupies anywhere else around the circle's circumference, according to the central angle measure theorem.

Here in the question,

Length of the arc AC = 59.

As we can see that there is a quadrilateral inscribed inside the circle.

Arc AC = 1/2 arc BD

⇒ arc BD = 2 × arc length AC

⇒ arc BD = 2 × 59

⇒ arc BD = 118.

Therefore, the length of the arc BD = 118 units.

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95% confidence interval for the population mean is (67.35, 82.65).

We can use the formula:

CI = x⁻ ± z*(σ/√n)

where x⁻ is the sample mean, σ is the population standard deviation, n is the sample size, and z is the critical value from the standard normal distribution corresponding to the desired confidence level (95% in this case).

First, we need to find the value of z. The area under the standard normal distribution curve between -z and z is 0.95. Using a table or calculator, we find that the critical value for a 95% confidence level is 1.96.

Now we can plug in the values we have:

CI = 75 ± 1.96*(20/√36)

= 75 ± 7.65

Therefore, the 95% confidence interval for the population mean is (67.35, 82.65). We can be 95% confident that the true population mean lies within this interval

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The value of x is 8.37

What is the area of the shaded region?

The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.

Here, we have

Given: The area of the shaded region is 20cm².

we have to find the value of x.

x in this case is the radius. In fact, both the height and base are the radius.

To find the radius, we need to form an equation. The only info given is with the area of the shaded area which is 20cm².

The area of the sector - an area of the triangle = the shaded area.

Area of the sector = πr²/4

Area of triangle = (1/2)bh

Area of the triangle = x²/2

The area of the triangle is in that way, as the height and base are x ( and x is the radius here!)

=  πr²/4 - x²/2

Multiply 4 with the whole equation as it is the LCM.

= 4(πr²/4 - x²/2) = 80

= πx² - 2x² = 80

= 1.142x² = 80

x² = 70.1

x = 8.37

Hence, the value of x is 8.37

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All the inequalities that represent the graph include the following:

B. y > -5/4(x) + 5

E. y + 5 > -1.25(x - 8)

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (5 - 0)/(0 - 4)

Slope (m) = 5/-4

Slope (m) = -5/4

At data point (0, 5) and a slope of -5/4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = -5/4(x - 0)  

y - 5 = -1.25(x - 0)

y = -5x/4 + 5

y > -5x/4 + 5 (shaded above the dashed line).

At data point (8, -5) and a slope of -5/4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-5) = -5/4(x - 8)  

y + 5 = -1.25(x - 8)

y + 5 > -1.25(x - 8)

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f ◦ g is onto, it does not guarantee that g must be onto.

If f ◦ g is onto, must g be onto? The answer is no, g does not necessarily have to be on. Let's explain this with the following steps:

1. Definition of ontological (surjective) function: A function g: A → B is onto if for every element b in the codomain B, there exists at least one element a in the domain A such that g(a) = b.

2. Definition of function composition (fg): Given two functions f: B → C and g: A → B, the composition f ◦ g: A → C is a function such that (f ◦ g)(a) = f(g(a)) for all an in A.

3. Given that f ◦ g is on, for every element c in the codomain C, there exists at least one element a in the domain A such that (f ◦ g)(a) = c.

4. However, the surjectivity of f ◦ g does not imply the surjectivity of g. This is because f may "compensate" for any lack of subjectivity in g. In other words, even if g does not map to every element in its codomain B, f might still map the outputs of g to every element in its codomain C.

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g has a relative minimum and absolute minimum at (x,y) = (2, -36).

To find the exact location of all the relative and absolute extrema of the function g(x) = 9x^2 - 36x, follow these steps,

1. Find the derivative of g(x) with respect to x:
g'(x) = d(9x^2 - 36x)/dx = 18x - 36

2. Set g'(x) equal to 0 to find critical points:
18x - 36 = 0
18x = 36
x = 2

3. Determine the type of extrema (minimum or maximum) by analyzing the second derivative:
g''(x) = d^2(9x^2 - 36x)/dx^2 = 18

Since g''(x) is positive at x = 2, there is a relative minimum at this point.

4. Calculate the value of g(x) at the critical point:
g(2) = 9(2)^2 - 36(2) = 9(4) - 72 = 36 - 72 = -36

5. Summarize the findings:
g has a relative minimum at (x,y) = (2, -36), and since this is the only extrema of the function, it is also an absolute minimum.

g has a relative minimum and absolute minimum at (x,y) = (2, -36).

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When measured in straight line, the distance of the cars apart would be = 10 miles.

To calculate the distance of the cars apart in a straight line, the Pythagorean formula should be used. That is;

C² = a²+b²

c² = 8²+6²

= 64+36

c² = 100

c = √100

= 10 miles

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The following parts can be answered by the concept of combination of functions.

20) fg: Since f is even and g is odd, the product (fg) will be an odd function.

21) fg: The answer is the same as #20. The product (fg) will be an odd function.

22) g∘f: For a composition of functions, the even/odd properties depend on the functions themselves. Since g is odd and f is even, the composition g∘f will also be an odd function.

23) f∘f: Since both functions are even, the composition of two even functions, f∘f, will result in an even function.

24) g∘g: Similarly, since both functions are odd, the composition of two odd functions, g∘g, will result in an even function.
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The following combinations nCr are:

The formula for nCr, where n is the total number of items and r is the number of items being chosen, is:

Using this formula, we get:

So, the combinations are 11, 330, 1, and 330 for (a), (b), (c), and (d) respectively.

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Null hypothesis: The type of airline ticket purchased is independent of the type of flight.

Alternative hypothesis: The type of airline ticket purchased is dependent on the type of flight.

To determine whether the type of airline ticket purchased is independent of the type of flight, we can use the chi-square test for independence. The invalid speculation is that the two factors are free, while the elective speculation is that they are reliant.

a. Speculations:

Negative hypothesis: The sort of aircraft ticket bought is free of the kind of flight.

Other possibilities: The kind of flight determines which kind of airline ticket is purchased.

To summarize the data, we can make a contingency table as follows:

                                   Domestic International Total

Business class           85         78           163

Economy class          157              130           287

First class                    20       22             42

Total                        262              230            492

We calculate a p-value of 0.150 and a test statistic of 3.794 using the chi-square test for independence. The critical value for a chi-square distribution with two degrees of freedom is 5.991 at a significance level of 0.05.

We are unable to reject the null hypothesis because the calculated test statistic (3.794) is lower than the critical value (5.991). We don't have enough evidence to say that the kind of airline ticket you buy depends on the type of flight.

b. According to the contingency table, the majority of tickets purchased were for domestic and international flights in economy class. However, compared to domestic flights, business class tickets are purchased slightly more frequently on international flights. On the other hand, compared to international flights, tickets in economy class are purchased slightly more frequently for domestic flights. These distinctions are not genuinely critical, however they really do recommend a few reliance between the kind of ticket and the sort of flight. This could be because of things like how long the flight is or what the trip is for.

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To obtain a feasible solution, you would need to revise the assignment limits or add additional constraints that do not violate the given constraint.

Based on the constraint you provided, x21 + x22 + x23 + x24 ≤ 3, it means that the sum of variables x21, x22, x23, and x24, representing the number of tasks assigned to agents 1, 2, 3, and 4 respectively, cannot exceed 3.

This constraint implies that agent 3 can be assigned to a maximum of 2 tasks (since x23 ≤ 2), and agent 2 can be assigned to a maximum of 3 tasks (since x22 ≤ 3).

However, there seems to be a contradiction with the statement that "agent 3 can be assigned to 2 tasks" and "agent 2 can be assigned to 3 tasks" because the sum of these maximum assignments would already exceed 3, which is not feasible according to the constraint.

Therefore, To obtain a feasible solution, you would need to revise the assignment limits or add additional constraints that do not violate the given constraint, such as reducing the maximum number of tasks that can be assigned to agent 2 or agent 3, or adjusting the total number of tasks available for assignment.

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(a) To find the price the company should charge for each unit to sell 6,500 units, we need to substitute q with 6.5 (since q is measured in thousands of units) in the demand function p = 26e^(-0.6q): p = 26e^(-0.6 * 6.5)

After calculating, we get: p ≈ $2.98

So, the company should charge approximately $2.98 per unit to sell 6,500 units.

(b) To find how many units will sell if the company prices the products at $6.50 each, we need to solve for q in the demand function p = 26e^(-0.6q) with p = $6.50: 6.50 = 26e^(-0.6q)

Now, we need to solve for q: q = ln(6.50/26) / -0.6 ≈ 1.884

Since q is measured in thousands of units, the company will sell approximately 1,884 units when the price is $6.50 each.

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The correct answer is option C i.e. 0.15 - 0.02 to 0.15 + 0.02

The correct calculation for a 95% confidence interval for the true proportion of all US adults who feel a gym membership is a necessity is:

Margin of error = z√(p(1-p)/n)

where z is the z-score corresponding to the desired level of confidence (95% in this case), p is the sample proportion (0.15), and n is the sample size (1522).

From a standard normal distribution table, the z-score for a 95% confidence level is approximately 1.96.

Substituting these values into the formula, we get:

Margin of error = 1.96 * √(0.15*(1-0.15)/1522) ≈ 0.02

Therefore, the 95% confidence interval is:

0.15 - 0.02 to 0.15 + 0.02

which simplifies to: [0.13, 0.17]

So, the correct answer is option C.

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The probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 can be calculated using the central limit theorem.

According to the central limit theorem, the sampling distribution of the sample mean becomes approximately normal, regardless of the distribution of the population, if the sample size is large enough (n > 30).

Therefore, we can assume that the sample mean of the 45 combo meals follows a normal distribution with a mean of $6.75 and a standard deviation of $1.08/sqrt(45) = $0.161.

To find the probability that the sample mean is less than $6.50, we need to standardize the distribution using the z-score formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (6.50 - 6.75) / (1.08 / sqrt(45)) = -1.73

Looking up the z-score in the standard normal distribution table, we find that the probability of a z-score less than -1.73 is approximately 0.04.

Therefore, the probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 is 0.04 or 4%.

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The absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.

To find the absolute maximum and absolute minimum values of f on the given interval [−1, 4], we first need to find the critical points of the function f(t) = t − 3√t.
Taking the derivative of f(t) with respect to t, we get:
f'(t) = 1 - (3/2)t^(-1/2)
Setting f'(t) = 0 to find critical points, we get:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 2.25
So the only critical point of f(t) on the interval [−1, 4] is t = 2.25.
Now we need to evaluate f(t) at the endpoints of the interval and at the critical point to determine the absolute maximum and minimum values of f on the interval:
f(-1) = -1 - 3√(-1) = -1 - 3i
f(4) = 4 - 3√4 = 4 - 6 = -2
f(2.25) = 2.25 - 3√2.25 = 2.25 - 3(1.5) = -2.25
Therefore, the absolute maximum value of f on the interval [−1, 4] is f(-1) = -1 - 3i, and the absolute minimum value of f on the interval is f(4) = -2.
To find the absolute maximum and minimum values of f(t) = t - 3√t on the interval [-1, 4], we need to evaluate the function at its critical points and endpoints.
First, we find the critical points by taking the derivative of the function and setting it to zero:
f'(t) = 1 - (3/2)t^(-1/2)
To solve for critical points, set f'(t) = 0:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 9/4
Since 9/4 is within the interval [-1, 4], it is a valid critical point.
Now, evaluate the function at the critical point and the endpoints:
f(-1) = -1 - 3√(-1)

(Note: This value is complex, and we're looking for absolute max/min in the real domain, so we'll ignore this endpoint)
f(9/4) = (9/4) - 3√(9/4) ≈ -0.1213
f(4) = 4 - 3√4 = -2
So, the absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.

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The solution of the differential equation is  U=x+2y. (A)

To solve the differential equation (x + 2y)dx + ydy = 0 using substitution, you can use the substitution U = x + 2y.


1. Substitute U for x+2y: dU = (dx + 2dy)

2. Replace (x + 2y)dx + ydy with dU - 2ydy + ydy: dU - ydy = 0

3. Factor out dy: dU - ydy = dy(U - y) = 0

4. Separate variables: (1/dU) dU = dy/y

5. Integrate both sides: ∫(1/dU) dU = ∫(dy/y)

6. Obtain the solution: ln|U| = ln|y| + C

7. Replace U with x+2y: ln|x+2y| = ln|y| + C

8. Exponentiate both sides: x+2y = k*y, where k = e^C

Thus, the differential equation (x + 2y)dx + ydy = 0 can be solved using the substitution U = x + 2y.(A)

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Answer: A

Step-by-step explanation:

y int = b= 4

slope=rise/run= rise of 4/ run of 2 = 2

so

y=2x+4

The mean is approximately 75.0, the variance is approximately 28.9, and the standard deviation is approximately 5.4.

To find the mean, variance, and standard deviation of a binomial distribution with n = 121 and p = 0.62, you can use the following formulas:

1. Mean (μ) = n * p
2. Variance (σ²) = n * p * (1 - p)
3. Standard Deviation (σ) = √(variance)

Step 1: Calculate the mean.
Mean (μ) = n * p = 121 * 0.62 ≈ 75.02

Step 2: Calculate the variance.
Variance (σ²) = n * p * (1 - p) = 121 * 0.62 * (1 - 0.62) ≈ 28.91

Step 3: Calculate the standard deviation.
Standard Deviation (σ) = √(variance) = √(28.91) ≈ 5.38

So, the mean is approximately 75.0, the variance is approximately 28.9, and the standard deviation is approximately 5.4.

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The average value of P=f(t)=2.04(1.03)^t for 0≤t≤30 is approximately 3.236. The average value of P=f(t)=2.04(1.03) for 0≤t≤30 is 2.10.

To find the average value of the function P=f(t)=2.04(1.03)^t for 0≤t≤30, you'll need to evaluate the appropriate integral and use the formula for the average value of a function.
The formula for the average value of a function is:
Average value = (1/(b-a)) * ∫[f(t) dt] from a to b
In this case, a = 0, b = 30, and f(t) = 2.04(1.03)^t.
Step 1: Evaluate the integral.
∫[2.04(1.03)^t dt] from 0 to 30
Step 2: Use a calculator to find the definite integral value.
We should find that the integral value is approximately 97.091.
Step 3: Substitute the integral value, a, and b into the average value formula.
Average value = (1/(30-0)) * 97.091
Step 4: Calculate the average value.
Average value ≈ (1/30) * 97.091 ≈ 3.236
So, the average value of P=f(t)=2.04(1.03)^t for 0≤t≤30 is approximately 3.236.

To find the average value of P=f(t)=2.04(1.03) for 0≤t≤30, we need to first evaluate the integral of the function over the given interval.
∫(0 to 30) 2.04(1.03) dt
Using a calculator, we can simplify and solve this integral as follows:
2.04(1.03)∫(0 to 30) dt
= 2.10t |(0 to 30)
= 2.10(30) - 2.10(0)
= 63.00
So, the integral of P=f(t) over the interval 0≤t≤30 is 63.00.
To find the average value of P over this interval, we divide this integral by the length of the interval:
Average value of P = (1/30-0) * 63.00
= 2.10
Therefore, the average value of P=f(t)=2.04(1.03) for 0≤t≤30 is 2.10.

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3.) The area of the shape would be = 77mm².

To determine the area of the given shape, the area of the trapezium and the rectangule is both calculated and summed up

The area of a rectangle = length×width

width = 5 mm

length = 10mm

area = 10×5 = 50 mm²

Area of trapezium = ½(a+b)h

where;

a = 10mm

b = 8mm

h = 8-5 = 3mm

area = 1/2(10+8)×3

= 54/2 = 27mm²

Therefore area of the shape = 50+27 = 77mm².

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The nine fundamental solutions to the differential equation are:

[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]

We have,

The characteristic equation of the given differential equation is:

[tex](r^2 + 6r + 10)^2 \times r^2 (r - 1)^3 = 0[/tex]

We can find the fundamental solutions by looking at the roots of the characteristic equation.

The roots can be categorized as follows:

Roots of multiplicity 2 = -3 + i and -3 - i

Roots of multiplicity 2 =  1

Root of multiplicity 1 =  0

Root of multiplicity 2 = -5 + i and -5 - i

For each of these roots, we need to find the corresponding fundamental solution.

For the roots (-3 + i) and (-3 - i), the corresponding fundamental solutions are:

[tex]e^{(-3+i)t}~ and~ e^{(-3-i)t}[/tex]

For root 1, the corresponding fundamental solutions are:

[tex]e^t~and~te^t[/tex]

For the root 0, the corresponding fundamental solutions are:

1, t, t²/2!, t³/3!, ..., [tex]t^8[/tex]/8!

For the roots (-5 + i) and (-5 - i), the corresponding fundamental solutions are:

[tex]e^{(-5+i)t} ~and~e^{(-5-i)t}[/tex]

Therefore,

The nine fundamental solutions to the differential equation are:

[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]

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The result of the expression (3B - 6A) for given matrices A and B will be:[tex]\begin{pmatrix}-36 \\-3 \\3 \\\end{pmatrix}[/tex]

A rectangular array of numbers, symbols, or expressions arranged in rows and columns is known in mathematics as a matrix..It is normally marked by a capital letter, and the matrix elements are usually wrapped in brackets or brackets. Matrices are useful in many mathematical subjects, including linear algebra, calculus, statistics, and computer science.

A matrix with "m" rows and "n" columns is said to have "mxn" dimensions (pronounced as "m by n").

For the given problem,

[tex]\[3B = \begin{pmatrix}3(-6) \\3(1) \\3(1) \\\end{pmatrix}= \begin{pmatrix}-18 \\3 \\3 \\\end{pmatrix}\][/tex]

[tex]\[6A = \begin{pmatrix}6(3) \\6(1) \\6(0) \\\end{pmatrix}= \begin{pmatrix}18 \\6 \\0 \\\end{pmatrix}\][/tex]

[tex]\[3B - 6A = \begin{pmatrix}-18 \\3 \\3 \\\end{pmatrix}- \begin{pmatrix}18 \\6 \\0 \\\end{pmatrix}= \begin{pmatrix}-18 - 18 \\3 - 6 \\3 - 0 \\\end{pmatrix}= \begin{pmatrix}-36 \\-3 \\3 \\\end{pmatrix}\][/tex]

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Answer:

The correct answer is: Yes; it is a reflection over the y-axis.

To see why, imagine folding the graph along the y-axis. Points on the right-hand side of the y-axis remain on the right-hand side, while points on the left-hand side move to the right. This transformation is equivalent to reflecting the original figure across the y-axis.

Step-by-step explanation:

Answer:

3344 cubic yards

Step-by-step explanation:

The volume of a rectangular prism is length x width x height.

If the area of the base is 152, that means the length x width = 152

So, 152 x 22 = 3344.

Answer:

sorry can't understand what your trying to say but i can help a lil if you translate it into english

Answer:

(the answer is y ≤ X + 1).....

The value of the area is πab which is enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.

To find the area enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.

To find the area of this ellipse, use the formula A = πab, where A is the area, a is the semi-major axis, and b is the semi-minor axis.

First, identify the values of a and b from the given equation.
In the equation (x²/a²) + (y²/b²) = 1, a² is the coefficient of x², and b² is the coefficient of y².

Now, calculate the area using the formula A = πab.
Plug the values of a and b into the formula and multiply them with π to find the area.

So, the area enclosed by the ellipse (x²/a²) + (y²/b²) = 1 is A = πab.

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The indefinite integral of 4t(1-16t^4) dt is: 2t^2 - (4/5)t^6 + c, Here, C is the constant of integration, which can be written as C = C1 + C2.

To find the indefinite integral of the given function, we'll integrate term by term. The given function is:

∫(4t - 16t^4) dt

Now we'll integrate each term:

∫4t dt - ∫16t^4 dt

For the first term, the power rule for integration states that ∫t^n dt = (t^(n+1))/(n+1) + C, where n is a constant:

∫4t dt = 4∫t^1 dt = 4(t^(1+1))/(1+1) + C1 = 4t^2/2 + C1 = 2t^2 + C1

For the second term, we'll apply the same rule:

∫16t^4 dt = 16∫t^4 dt = 16(t^(4+1))/(4+1) + C2 = 16t^5/5 + C2 = (16/5)t^5 + C2

Now combine the results:

∫(4t - 16t^4) dt = 2t^2 + (16/5)t^5 + C

Here, C is the constant of integration, which can be written as C = C1 + C2.

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