What is the concept of asymptotic stability?

Answer:

1. B

2.B

3.C It could also be d because when you round it it would give you      

        8.7. Also no one leaves a section of their fence not painted. (8.6602540378443864676372317075294) UNROUNDED

Step-by-step explanation:

Answer:

Conclusion:

Step-by-step explanation:

Given

We know that the diagonals of a parallelogram bisect each other.

Therefore,

Given RT = x and TP = 5x-28, so

x = 5x-28

5x = x+28

5x-x = 28

4x = 28

divide boh sides by 4

4x/4 = 28/4

x = 7

Thus, the value of x = 7

Similarly,

QT = TS

Given QT = 5y and TS = 2y+12, so

5y = 2y+12

5y-2y = 12

3y = 12

divide both sides by 3

3y/3 = 12/3

y = 4

Thus, the value of y = 4

Conclusion:

The completely factored form of the polynomial is 7x² (x + 6) (x - 4)

Given,

The polynomial ; 7x⁴ + 14x³ - 168x²

We have to find the complete factored form of this polynomial;

Polynomial;

An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable).

Here,

The polynomial is  7x⁴ + 14x³ - 168x²

Now,

Factorize the polynomial using common factors;

That is,

7x⁴ + 14x³ - 168x²

7x² (x² + 2x - 24)

Solve x² + 2x - 24 using quadratic formula;

That is,

\(\frac{-b(+-)\sqrt{b^{2}-4ac } }{2a}\) = \(\frac{-2(+-)\sqrt{2^{2} -4*1*-24} }{2*1}\) = \(\frac{-2(+-)\sqrt{4-96} }{2}\) = -2±√-92 / 2

Now,

Solve

-2 + √-92 / 2 = 3.79 ≈ 4

Solve

-2 - √-92 / 2 = -5.79 ≈ -6

Then,

The factors will be (x - 4) and (x + 6)

So,

7x² (x + 6) (x - 4) will be the factored form of the polynomial.

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The equation 3x+2y=14 can be written as in the slope-intercept form is y = (-3/2)x + 7.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

It is given that:

The equation of the line is:

3x + 2y = 14

The slope-intercept form is:

y = mx + c

Here m is the slope of the line and c is the y-intercept.

After arranging:

2y = -3x + 14

y = (-3/2)x + 7

Thus, the equation 3x+2y=14 can be written as in the slope-intercept form is y = (-3/2)x + 7.

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The distance it would take to travel across the river on the bridge than to take the ferry is 4√6 units.

In Mathematics and Geometry, the distance between two (2) end points that are on a coordinate plane can be calculated by using the following mathematical equation:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

x and y represent the data points (coordinates) on a cartesian coordinate.

By substituting the given end points into the distance formula, we have the following;

Distance AC = √[(-4 - 0)² + (2 - 2)²]

Distance AC = √[(-4)² + (0)²]

Distance AC = √[16 + 0]

Distance AC = √16

Distance AC = 4 units.

Distance AB = √[(2 - 0)² + (0 + 2)²]

Distance AB = √[(2)² + (2)²]

Distance AB = √[4 + 4]

Distance AB = √8

Distance AC = 2√2 units.

From Pythagorean Theorem, the length of BC is given by;

BC² = (2√2)² + 4²

c² = 8 + 16

c = √24

c = 4√6 units.

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Answer:

18

Step-by-step explanation:

Product in math terms means to multiply

When the term "a number" is used, you can just put a variable

"Is," tells you what the problem equals

With this information, we can create the equation:

(2/3x) + 6 = 12

Now that we have this equation, we can solve it

(2/3x) +6 = 12            We can subtract 6 from both sides.  

            -6     -6

2/3x = 12                   We can divide the 2/3 from each side.

/2/3         /2/3

x = 18

Hope this helps!

Answer:

1

Step-by-step explanation:

2/3x + 12 = 6

2/3x = -6

x = -6 divided by 2/3

(6 divided 2) divided 3 =1

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Answer:

C A person with an annual income of at least \$50{,}000$50,000dollar sign, 50, comma, 000 is more likely to be 454545 years old or above

A person in the 44 and under age group is more likely to have an annual income of at least $50,000​ is a correct statement.

We have given that,

A sociologist polled a random sample of people and asked them their age and annual income.

annual income is the amount of money you receive during the year into your bank account, before any deductions.

The two-way frequency table below shows the results.

Annual income vs. age group Less than $50,000 At least $50,000

Total Ages 44 and under 148 68 216 Ages 45 and above 38 94 132

Total 186 162 348

A person with an annual income of at least

\(=\$50000\\=$50,000dollar\)

Therefore statement b is true.

A person in the 44 and under age group is more likely to have an annual income of at least $50,000​.

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Answer:

-3/7

Step-by-step explanation:

1. The formula to find the slope of a line given two points is \(\frac{y_2-y_1}{x_2-x_1}\).

Therefore, the answer is -3/7.

The sum of the interior angle of a regular nonagon will be 1,260°.

The polygon is a 2D geometry that has a finite number of sides. And all the sides of the polygon are straight lines connected to each other side by side.

The interior angle of the polygon is given as,

Interior angle = 180° - 360° / n

And the sum of the interior angle is given as,

Sum of the interior angle = n x (Interior angle)

The number of sides of the regular nonagon is 9. Then the interior angle is given as,

Interior angle = 180° - 360° / 9

Interior angle = 180° - 40°

Interior angle = 140°

Then the sum of the interior angle of a regular nonagon will be given as,

Sum of the interior angle = 9 x 140°

Sum of the interior angle = 1,260°

The sum of the interior angle of a regular nonagon will be 1,260°.

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The FOIL method, which stands for First, Outer, Inner, Last, is a technique commonly used to multiply binomials.

It involves multiplying the First terms, then the Outer terms, then the Inner terms, and finally the Last terms, and then combining the results. While the FOIL method works well for multiplying binomials, it is not applicable for multiplying all polynomials.

The reason the FOIL method cannot be used to multiply all polynomials is that it only applies to the specific case of multiplying two binomials with two terms each. When dealing with polynomials that have more than two terms or polynomials of higher degrees, the FOIL method does not provide a systematic approach to handle the multiplication.

For example, consider multiplying the polynomial (x + 2) with the polynomial (x^2 + 3x - 4). Applying the FOIL method, we would only multiply the First terms (x * x^2), the Outer terms (x * 3x), the Inner terms (2 * x^2), and the Last terms (2 * -4). However, this approach overlooks the multiplication between the terms of different degrees (e.g., x * 3x or 2 * x^2) and fails to account for all possible combinations.

To multiply more complex polynomials, we typically use more advanced methods such as the distributive property, grouping, or the use of matrices. These methods provide a systematic and comprehensive approach to handle polynomial multiplication in general, accommodating polynomials with any number of terms or degrees.

In summary, while the FOIL method is a helpful technique for multiplying binomials, it cannot be used for multiplying all polynomials due to its limited applicability. For more complex polynomials, alternative methods are necessary to ensure accurate and comprehensive multiplication.

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The general solution of the given differential equation is:

y = (1/7) + C * \(exp(-x^7)\)

The exponential of the integral of the coefficient of y, in this case 7x6, provides the integrating factor. The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is 7x⁶.

The integrating factor is therefore exp(∫ 7x⁶ dx), which can be calculated as exp(x⁷/7).

Multiplying both sides of the differential equation by the integrating factor, we have:

exp(x⁷/7) * y' + 7x⁶ * exp(x⁷/7) * y = x⁶ * exp(x⁷/7)

Using the product rule on the left side, we can rewrite the equation as:

d/dx (exp(x⁷/7) * y) = x⁶ * exp(x⁷/7)

Integrating both sides with respect to x, we get:

exp(x⁷/7) * y = ∫ x⁶ * exp(x⁷/7) dx

The integral on the right side can be solved using integration by parts. Let's denote u = x⁶ and dv = exp(x⁷/7) dx. Then, du = 6x⁵ dx and v = 7/7 * exp(x⁷/7) = exp(x⁷/7).

Using the formula for integration by parts:

∫ u dv = uv - ∫ v du

We have:

∫ x⁶ * exp(x⁷/7) dx = x⁶ * exp(x⁷/7) - ∫ exp(x⁷/7) * 6x⁵ dx

Simplifying the integral on the right side, we obtain:

∫ exp(x⁷/7) * 6x⁵ dx = 6 * ∫ x⁵ * exp(x⁷/7) dx

We can apply integration by parts again to this integral, with u = x⁵ and dv = exp(x⁷/7) dx.

Continuing this process, we will eventually reach an integral of the form ∫ exp(x⁷/7) dx, which can be expressed in terms of special functions called exponential integrals.

Once we have the value of this integral, we can substitute it back into the expression for the integral of x⁶ * exp(x⁷/7) dx.

Finally, we divide both sides of the equation by exp(x⁷/7) and solve for y:

y = (1/exp(x⁷/7)) * (∫ x⁶ * exp(x⁷/7) dx)

The resulting expression will give the general solution to the given differential equation.

To solve this linear first-order ordinary differential equation, we can use an integrating factor. The integral of the coefficient of y's exponential integral, in this case 7x⁶, provides the integrating factor.

The integrating factor is therefore exp(∫ 7x⁶ dx), which can be calculated as exp((7/7) * x⁷) = exp(x⁷).

Multiplying both sides of the differential equation by the integrating factor, we have:

exp(x⁷) * y' + 7x⁶ * exp(x⁷) * y = x⁶ * exp(x⁷)

We can rewrite this equation as follows:

d/dx (exp(x⁷) * y) = x⁶ * exp(x⁷)

Integrating both sides with respect to x, we get:

exp(x⁷) * y = ∫ x⁶ * exp(x⁷) dx

To evaluate this integral, we can make a substitution. Let's substitute u = x⁷, then du = 7x⁶ dx.

The integral becomes:

(1/7) ∫ exp(u) du = (1/7) * exp(u) + C = (1/7) * exp(x⁷) + C

Now, dividing both sides of the equation by exp(x⁷), we have:

y = (1/7) + C * exp(-x⁷)

Therefore, the general solution of the given differential equation is:

y = (1/7) + C * exp(-x⁷)

where C is an arbitrary constant.

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The dim Nul A refers to the dimension of the null space of the matrix A, which is the number of free variables in the matrix.  dim Nul A = 6, and Col A ≠ R^2. The correct answer is B: No, because Col A is a subspace of R^7.

The rank-nullity theorem states that:
rank(A) + dim Nul A = number of columns in A
Given that matrix A is a 7x8 matrix with two pivot columns, we can calculate the dimension of its null space as follows:
1. Determine the rank of matrix A: Since there are two pivot columns, the rank of matrix A (rank A) is 2.
2. Apply the rank-nullity theorem: rank(A) + dim Nul A = number of columns in A
  In this case, 2 + dim Nul A = 8.
3. Solve for dim Nul A: dim Nul A = 8 - 2 = 6.
So, dim Nul A = 6.
To determine if Col A is equal to R^2, we need to consider the dimension of the column space of matrix A. The dimension of the column space is equal to the rank of matrix A:
dim Col A = rank A = 2.
However, Col A is a subspace of R^7 because the matrix A has 7 rows. Therefore, Col A cannot be equal to R^2, which is a subspace of R^2.

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Answer: 1406.25

Step-by-step explanation:

The soccer field at a park is 9,000 square yards. The soccer field takes up 6 2/5 square inches on the map of the park. How many square yards does 1 square inch on the map represent? Show your work. Help me pls

Given that:

Area of soccer field = 9000 sq yards

Soccer field takes up 6 2/5 square inches on the map of park

Hence,

9000 yd² = 6 2/5 in²

Let 1 in² on map = m

9000 yd² = 32/5 in² - - - (1)

m = 1 in² - - - (2)

Cross multiply ;

6.4 * m = 9000

m = 9000/6.4

m = 1406.25

Hence,

1 in² on map = 1406.25 feet²

900

The velocity v as a function of t is  4t³ - 4

Now, we are asked to find the velocity v as a function of t. Velocity is the rate of change of position with respect to time. Mathematically, we can express velocity as the derivative of position with respect to time, or v = ds/dt, where s is the position function and t is the time variable.

To find the velocity v as a function of t, we need to take the derivative of the position function s with respect to time. In this case, s = t⁴ - 4t, so we can use the power rule of differentiation to find ds/dt:

ds/dt = 4t³ - 4

Therefore, the velocity v as a function of time t is:

v(t) = ds/dt = 4t³ - 4

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Based on this sample of 17 adults, it appears that the mean annual salary in the town is significantly lower than the claimed value of $30,000. However, we should keep in mind that our conclusion is only based on a sample and may not necessarily hold true for the entire population.

We will test the claim that the mean annual salary for the adult population of a town is µ=$30,000 using the sample data provided.

Given:

- Population mean (µ) = $30,000

- Sample size (n) = 17

- Sample mean (X) = $22,298

- Sample standard deviation (s) = $14,200

- Significance level (α) = 0.05

Since we have a simple random sample from a normally distributed population, we can use a t-test to test the claim. Here are the steps:

1. State the null hypothesis (H₀) and alternative hypothesis (H₁):

  H₀: µ = $30,000 (claim)

  H₁: µ ≠ $30,000 (to test the claim)

2. Calculate the t-score using the sample data:

  t = (X - µ) / (s / √n)

  t = ($22,298 - $30,000) / ($14,200 / √17)

  t ≈ -2.056

Given that n=17, x(bar)=$22,298 and s=$14,200, we can calculate the t-statistic as follows:

t = (x(bar) - µ) / (s / sqrt(n))

t = ($22,298 - $30,000) / ($14,200 / sqrt(17))

t = -2.31

3. Determine the critical t-value (t_ critical) using the degrees of freedom (n - 1) and α:

  Degrees of freedom = 17 - 1 = 16

  Using a t-distribution table, with α/2 (0.025) and 16 degrees of freedom, we find that the t_ critical values are approximately ±2.12.

4. Compare the calculated t-score with the critical t-values:

  -2.056 lies within the range of -2.12 and 2.12.

5. Make a decision based on the comparison:

Since the calculated t-score is within the critical t-value range, we fail to reject the null hypothesis (H₀).

In conclusion, based on the sample data, we do not have sufficient evidence to reject the claim that the mean annual salary for the adult population of the town is $30,000 at the 0.05 significance level.

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Answer:

23 + \(\sqrt{29}\) + \(\sqrt{58}\)

Step-by-step explanation:

Answer:

Negative 2.5 greater-than negative 1.5

C

Step-by-step explanation:

-2.5 < -2

The particular solution to the given differential equation, x³y' + 2y = e^(1/x²), that satisfies the initial condition y(1) = e, is y = e.

To find the particular solution of the given differential equation, we can use the method of integrating factors. Let's break down the steps to solve it:

Rearrange the equation: We rewrite the given differential equation in the standard form:

y' + (2/x³)y = (e^(1/x²))/(x³)

Identify the integrating factor: The integrating factor (IF) is determined by multiplying the entire equation by x³. This results in:

x³y' + 2xy = e^(1/x²)

Apply the integrating factor: Multiplying the equation by the integrating factor x³ gives us:

(x⁶y)' = x³e^(1/x²)

Integrate both sides: Integrating both sides of the equation gives us:

x⁶y = ∫x³e^(1/x²) dx

Evaluate the integral: Unfortunately, the integral on the right side does not have an elementary function solution. Therefore, we cannot find an explicit expression for the integral.

However, we can still find the particular solution by applying the initial condition y(1) = e.

Solve for the particular solution: Using the initial condition, we substitute x = 1 and y = e into the equation:

1⁶ * e = ∫1³e^(1/1²) dx

e = ∫e dx

e = e

Since the left side and the right side are equal, the initial condition is satisfied.

We used the method of integrating factors to solve the differential equation and obtained an integral expression. Although we couldn't find an explicit solution for the integral, we were able to confirm that the initial condition y(1) = e satisfies the differential equation. This means that y = e is the particular solution that satisfies the given initial condition.

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The profit per month for the contractor is obtained as $9425.

Probability is the branch of Mathematics that deals with the measurement of the chance of occurrence of a random event.

The probability of any event always lie in the close interval of 0 and 1 [0,1].

The profit of the contractor per job is given as $3250.

As per the given table the expected value can be obtained as,

E(x) = \(\sum_{x = 1}^{4} xP(x)\)

⇒ E(x) = 1 × 0.1 + 2 × 0.2 + 3 × 0.4 + 4 × 0.3

⇒ 2.9

Then, the expected profit of is given as 3250 × 2.9 = $9425

Hence, the profit per month is obtained as $9425.

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Work Shown:

4/10 = 10/x

4x = 10*10

4x = 100

x = 100/4

x = 25

The first equation set up shown above is valid because we have similar triangles. Having similar triangles means the corresponding sides form equal proportions. After the equation is set up, we cross multiply (step 2) and isolate x.

A second linearly independent solution of y₂ is \(-\frac{1}{6x^3}\)

The general Equation is y" + P(x)y' + q(x)y = 0 ...............(i)

where P(x), Q(x) are continues in the internal I ≤ R.

If y₁(x) is a solution of equation 1 in I then y₁(x) ≠ 0.

Then y₂(x) = y₁(x)\(\int{\frac{e^{-\intP(x)dx}}{(y_{1}x)^2}}dx\) is another solution.

The differential equation is x²y" + xy' – 9y = 0 where x > 0.

As y₁(x) = x³ is one solution of differential equation.

Divide throughout by (x²) to given differential equation.

1/x² (x²y" + xy' – 9y = 0)

y" + (y'/x) – (9/x²)y = 0 ................(ii)

By comparing equation (i) & (ii) we get:

p(x)=1/x , q(x)= –are continuous for x>0

So, another solution,

y₂(x) = y₁(x)\(\int{\frac{e^{-\intP(x)dx}}{(y_{1}x)^2}}dx\)

Now putting the values of P(x) And Q(x)

y₂(x) = \(x^3\int\limits {\frac{e^{\int(1/x)dx} }{(x^3)^2}} \, dx\)

y₂(x) = \(x^3\int\limits {\frac{\frac{1}{x} }{x^6} }} \, dx\)

y₂(x) = \(x^3\int\limits {\frac{1}{x^7} }} \, dx\)

y₂(x) = \(x^3\int\limits {x^-7} } \, dx\)

y₂(x) = \(x^3\left[\frac{x^{-7+1}}{-7+1}\right]\)

y₂(x) = \(-\frac{1}{6}(x^3\times x^{-6})\)

y₂(x) = \(-\frac{1}{6x^3}\)

So, the answer of this question is \(-\frac{1}{6x^3}\).

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Answer:

Step-by-step explanation:

2z = 4a/m +3

 -3        -3

2z - 3 = 4a/m

 ×m        ×m

(2z -3)m = 4a

     ÷4      ÷4

[(2z -3)m]/4 = a

a = (2mz - 3m)/4

Answer:

The area of a square with side length of 2 inches is 4 square inches.

The area of a circle with radius 1.2 inches is:

\(\pi \times (1.2) ^{2} \\ = 3.14 \times 1.44 \\ = 4.5216 \: square \: inches\)

That is approximately 4.5 square inches.

Hence, the area of the square (4) is less than {<} that of the circle (4.5)!

The correlation coefficient is most directly related to the slope of the regression line. It measures the strength of the linear relationship between X and Y, while the slope of the regression line tells us how much Y changes for each unit change in X.

The correlation coefficient is a statistical concept that refers to the strength and direction of a linear relationship between two variables. It is a number that ranges from -1 to 1. A value of -1 indicates a perfect negative correlation, meaning that as one variable increases, the other decreases, while a value of 1 indicates a perfect positive correlation, meaning that as one variable increases, the other increases as well. A value of 0 indicates no correlation, meaning that there is no linear relationship between the two variables.

The correlation coefficient is most directly related to the slope of the regression line. The regression line is a line that best fits the data points in a scatterplot. It is used to estimate the value of one variable based on the value of another variable. The slope of the regression line is the change in Y for a unit change in X. It tells us how much Y changes for each unit change in X. The correlation coefficient measures the strength of the linear relationship between X and Y. If the correlation coefficient is positive, then the slope of the regression line is also positive, meaning that as X increases, Y also increases. If the correlation coefficient is negative, then the slope of the regression line is also negative, meaning that as X increases, Y decreases.

The mean of Y and the mean of X are not directly related to the correlation coefficient. The mean of Y is the average value of Y, while the mean of X is the average value of X. They are used to calculate the intercept of the regression line, which is the value of Y when X is 0. The regression constant is another name for the intercept of the regression line. It is also not directly related to the correlation coefficient.
The mean of Y, the mean of X, and the regression constant are not directly related to the correlation coefficient.

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i) Ratio of roses to total number of flowers is 4:12

ii) Ratio of camations to daffodils in three different ways are 5/3 , 5:3 and 5 to 3

iii)    Ratio of total number of flowers to camations is 12/5, 12:5 and 12to5

iii) Ratio of roses to daffodils are 4/3, 4:3 and 4to3

Given that,

Number of roses in bouquet = 4

Number of camations in bouquet = 5

Number of daffodils in bouquet =3

Thus total number of flowers in a bouquet = 4 + 5 + 3

                                                                            = 12

Therefore there are 12 flowers in bouquet.

a) The ratio of roses to the total number of flowers in the bouquet is ;

                               Number of roses :  total number of flowers

                                                          4 : 12

b) The three different ways of writing ratio of camations to daffodils are:

i) Fraction -  5/3

ii) Ratio using colon – 5:3

iii) Word form – 5 to 3

c) The three different ways of writing ratio of total number of flowers to camations are:

i) Fraction -  12/5

ii) Ratio using colon – 12:5

iii) Word form – 12 to 5

d) Ratio which compare flowers of one part of the bouquet to the another is given by:

Here we will compare number of roses to number of daffodils.

i) Word form-   4 to 3

ii) Ratio using colon -  4:3

iii) Fraction – 4/3

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Answer:

The domain should be option A

Step-by-step explanation:

Since the line has no end points it will go on forever, meaning that it has a domain of negative infinity to infinity

The mass of the 1-meter cylindrical bar with a constant density of 1 g/cm for its left half and 2 g/cm for its right half is 150 g.

To find the mass of the 1-meter cylindrical bar with a constant density of 1 g/cm for its left half and 2 g/cm for its right half, follow these steps:

1. First, convert the length of the bar from meters to centimeters, as the density is given in g/cm. There are 100 centimeters in a meter, so the length of the bar is 100 cm.

2. Since the bar is divided into two equal halves, each half has a length of 50 cm.

3. Calculate the mass of each half by multiplying the length by the respective density. For the left half, the mass is 50 cm × 1 g/cm = 50 g. For the right half, the mass is 50 cm × 2 g/cm = 100 g.

4. Add the mass of the two halves to get the total mass of the bar: 50 g + 100 g = 150 g.

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Answer: I would say 21 feet.

Step-by-step explanation:

Bernard is 12 feet ahead of Austin

So add 12 feet to get to Austin first

Then add the 9 feet that is apart from Austin and Bernard

12+9 = 21

21 feet would be the answer

Hope this helps ! ^^

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Answer: 62

Step-by-step explanation:

Answer:

(-2,1)

Step-by-step explanation:

Since this is an upwards opening parabola, the minimum point is at the vertex

We can find the x coordinate of the vertex by

y = 2x^2 +8x +9

where a=2  b =8 and c =9

x = -b/2a

x = -8/(2*2)

  = -8/4 = -2

The x coordinate is -2

Now substitute this in to find the y coordinate

y = 2(-2)^2 +8(-2)+9

  = 2*4 -16+9

  8-16+9

  =1

The vertex = (-2,1), which is the minimum point

Answer:

Step-by-step explanation:

y=2x²+8x+9

to find the minimum or maximum point x=-b/2a

x=-8/4=-2

y=2(-2)²+8(-2)+9

y=8-16+9

y=1

minimum point is (-2,-1)