What is the surface area of the cylinder with
height 7 cm and radius 8 cm? Round your
answer to the nearest thousandth.

The Taylor series of f(x)=1−x1​ centered at c=8 is:

1−x1​= n=0∑[infinity]​(−1) n+18 n+1(x−8) n

Simplifying the expression, we get:

1−x1​= n=0∑[infinity]​(−1) n(x−8) n7 n+1

And further simplifying, we get:

1−x1​= n=0∑[infinity]​(−1) n7 n(x−8) n+1

Finally, we get:

1−x1​= n=0∑[infinity]​(−1) n+17 n+1(x−8) n

The interval on which the expansion is valid can be found using the ratio test. Let a_n = (-1)^n*7^n*(x-8)^(n+1)/(n+1). Then we have:

|a_(n+1)/a_n| = 7|x-8|/(n+2)

For the series to converge, we need |a_(n+1)/a_n| < 1. This holds if 7|x-8| < n+2, or if x is in the interval (7/8, 9/8). Therefore, the expansion is valid on the interval (7/8, 9/8).

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The height of the tree is 6.08 meters.

We are given that;

Base to kays feet= 4.75m, kays feet to end of shadow=1.25m, kays height=1.60m

Now,

To find the height of the tree, you need to use similar triangles. The ratio of the corresponding sides of similar triangles is equal, so you can set up a proportion between the heights and the shadow lengths. You can write your solution as:

1.60/1.25 = h/4.75 h = 1.60/1.25 x 4.75 h = 6.08

Therefore, by the proportions the answer will be 6.08 meters.

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0.765

answer is 0.764758824 and because there is a 7 after the 4 we round up

Therefore, The exponential function approaches zero as the input approaches negative infinity, and as x increases towards infinity, the value of e−3x approaches zero.

(a) The right end behavior of y = log6(x) as x approaches infinity is that y approaches negative infinity. This is because as x increases towards infinity, the value of log6(x) becomes larger and larger negative values. Explanation: The logarithm function approaches negative infinity as the input approaches zero, and as x increases towards infinity, the value of log6(x) approaches negative infinity.
(b) The right end behavior of y = e−3x as x approaches infinity is that y approaches 0. This is because as x increases towards infinity, the exponent -3x becomes larger and larger negative values, making the value of e−3x approach zero.  

Therefore, The exponential function approaches zero as the input approaches negative infinity, and as x increases towards infinity, the value of e−3x approaches zero.

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Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

We have,

Function:

A function is a mathematical relationship or rule that assigns each input value (or element) from a set, called the domain, to a unique output value (or element) from another set, called the range.

Example:

Let's consider a function f(x) = 2x + 3.

This function takes an input value (x), multiplies it by 2, and then adds 3 to get the output value.

For example, if we input x = 4 into the function, we get f(4) = 2(4) + 3 = 11. So, the function maps the input value 4 to the output value 11.

Combined Function:

A combined function is formed by performing multiple operations on a given input value. It involves applying one function to the output of another function.

This allows us to express complex relationships between variables by combining simpler functions.

Example:

Let's consider two functions: f(x) = 2x and g(x) = x².

The combined function h(x) is formed by applying g(x) to the output of f(x). In other words, h(x) = g(f(x)).

If we input x = 3 into the combined function, we first evaluate f(x) = 2(3) = 6, and then evaluate g(6) = 6² = 36. So, h(3) = 36.

Quadratic Function:

A quadratic function is a type of function that can be represented by a polynomial equation of degree 2.

It has the general form f(x) = ax² + bx + c, where a, b, and c are constants.

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the value of the coefficient "a".

Example:

Let's consider the quadratic function f(x) = 2x² - 3x + 1.

This function has a coefficient of 2 for the x^2 term, -3 for the x term, and 1 for the constant term.

If we input x = 2 into the function,

We get f(2) = 2(2)² - 3(2) + 1 = 8 - 6 + 1 = 3.

So, the function maps the input value 2 to the output value 3.

The graph of this quadratic function is a parabola that opens upwards.

Thus,

Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

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A change in the position, size, or shape of a geometric figure is called a transformation. A transformation refers to any operation or change applied to a geometric figure that alters its position, size, or shape.

Transformations are fundamental concepts in geometry and are classified into various types, including translation, rotation, reflection, and dilation.

Translation involves moving a figure from one location to another without changing its size or shape.

Rotation refers to turning a figure around a fixed point by a certain angle.

Reflection is the flipping of a figure over a line to create a mirror image.

Dilation involves either enlarging or reducing the size of a figure proportionally.

These transformations are used to analyze and describe the behavior of geometric figures, explore symmetry and congruence, and solve various geometric problems. The term "transformation" encompasses all these types of changes in the position, size, or shape of a geometric figure.

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The correct answer is D, open market purchase, increasing the reserve requirement, and decreasing the discount rate.

When the Fed judges inflation to be the most significant problem in the economy and wishes to employ all three of its policy instruments, it will implement expansionary monetary policy.

This involves increasing the money supply in the economy to stimulate growth and reduce inflation.
To do this, the Fed will conduct open market operations, which involve purchasing government securities from banks.

This injects money into the banking system and increases the amount of reserves banks have available to lend out. This increase in lending stimulates economic growth and reduces inflation.
In addition to open market operations, the Fed will increase the reserve requirement, which is the amount of money that banks are required to hold in reserve.

This reduces the amount of money banks have available to lend out and helps to control inflation.
Finally, the Fed will decrease the discount rate, which is the interest rate at which banks can borrow money from the Fed.

This makes it cheaper for banks to borrow money and encourages them to lend more, stimulating growth and reducing inflation.
In summary, when the Fed judges inflation to be the most significant problem in the economy and wishes to employ all three of its policy instruments, it will implement expansionary monetary policy by conducting open market operations, increasing the reserve requirement, and decreasing the discount rate.

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The probability of getting a sum of 4, 5, or 6. is 0.4

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

Spinner = 1 to 5

Die = 6 sided

This means that the number of outcomes is

Outcomes = 6 * 5

Outcomes = 30

So, the sample space is

Die \ Spinner 1 2 3 4 5

1                       2      3    4      5      6

2                      3    4      5      6       7

3                       4      5      6       7    8

4                       5      6       7        8     9

5                       6       7        8     9      10

6                       7        8     9      10       11

The probability of getting a sum of 4, 5, or 6. is

P = Number/Sample size

From the table, we have

Number of sum of 4, 5, or 6 = 12

So, we have

P = 12/30

Evaluate

P = 0.4

Hence, the probability of getting a sum of 4, 5, or 6. is 0.4

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The work done by f in moving a particle once counterclockwise around the given curve is -15π.

The problem requires us to calculate the work done by the vector field f along a closed curve C, which is a circle centered at (5,5) with a radius of 5. To do this, we can use the line integral of f along C, which is given by:

∫C f · dr = ∫C (f(x,y) · T) ds

where T is the unit tangent vector to C and ds is the arc length element along C.

To parameterize the curve C, we can use the parametric equations:

x = 5 + 5cos(t)

y = 5 + 5sin(t)

with 0 ≤ t ≤ 2π. Then, the unit tangent vector T is given by:

T = (-sin(t), cos(t))

and the arc length element ds is given by:

ds = √(x'(t)² + y'(t)²) dt = 5 dt

Using these expressions, we can compute the line integral as:

∫C f · dr = ∫C [(x-3y)i + (3x-y)j] · (-sin(t)i + cos(t)j) 5 dt

After some algebraic manipulation, we obtain:

∫C f · dr = -15π

Therefore, the total work done by f in moving the particle once around the given curve counterclockwise is -15π.

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The work done by f in moving a particle once counterclockwise around the given curve is zero.

To find the work done by a vector field f in moving a particle along a curve C, we use the line integral formula. The line integral of a vector field f along a curve C is given by the formula ∫C f · dr, where dr is the differential of the position vector r(t) of the curve C. In this case, the vector field is f = (x - 3y)i + (3x - y)j and the curve is the circle (x - 5)² + (y - 5)² = 25 centered at (5,5) with radius 5. To evaluate the line integral, we need to parameterize the curve. Since the curve is a circle, we can use the parametrization r(t) = 5cos(t)i + 5sin(t)j, where t ranges from 0 to 2π. Then, dr = -5sin(t)dt i + 5cos(t)dt j.

Evaluating the line integral, we get ∫C f · dr = ∫0^2π f(r(t)) · dr/dt dt = ∫0^2π (-15sin²(t) + 15cos²(t))dt = 0. Therefore, the work done by f in moving a particle once counterclockwise around the given curve is zero.

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The critical value, tc, for a confidence level of c0.90 and sample size n=16 can be found using a t-distribution table. The degrees of freedom for this calculation is n-1, which in this case is 15. From the table, we find that the tc value is approximately 1.753. This means that if we take a sample of size 16 from a population and calculate a sample mean, we can be 90% confident that the true population mean falls within a range of ± tc multiplied by the standard error of the sample mean. The critical value tc is an important factor in calculating confidence intervals for sample means.

To find the critical value, we need to use a t-distribution table, which provides the t-scores for various levels of confidence and degrees of freedom. The degrees of freedom for this calculation is n-1, which is 16-1=15. We look for the row in the table that corresponds to 15 degrees of freedom and then find the column that corresponds to a confidence level of 0.90. The value at the intersection of this row and column is the critical value, which in this case is approximately 1.753.

The critical value tc for a confidence level of c0.90 and sample size n=16 is approximately 1.753. This value is important in calculating confidence intervals for sample means, which allows us to estimate the range within which the true population mean is likely to fall with a certain level of confidence.

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If a = 4, then a 2 · a 3 is equivalent to all of the following except _ 4^2 · 4^3 = 1,024

Noted that Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

We are given that a = 4, then the expression could be;

a^2 · a ^3

Substitute the values;

a^2 · a ^3  = 4^2 · 4^3

= 16 . 64

= 1,024

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The no of solutions for the equations given in the question which comes out to be as final answer is c. 746,858,751.

To solve this problem, we can use the stars and bars method. We want to find the number of non-negative integer solutions to the equation a+b+c+d+e=500, where each variable is at least 10.

First, we can subtract 10 from each variable to get a new equation a'+b'+c'+d'+e'=450, where each variable is non-negative. Then, we can use the stars and bars method to find the number of solutions.

We need to place 4 bars among the 450 stars to separate the stars into 5 groups. This can be done in (450+4) choose 4 ways, which simplifies to (454 choose 4). However, this counts solutions where some variables are less than 10.

To count the number of solutions where some variables are less than 10, we can use inclusion-exclusion. There are 5 ways to choose 1 variable to be less than 10, 10 choose 2 ways to choose 2 variables to be less than 10, and so on. Using the principle of inclusion-exclusion, the number of solutions with at least one variable less than 10 is:

5(440 choose 4) - 10(430 choose 4) + 10(420 choose 4) - 5(410 choose 4) = 10,316,800

Therefore, the final answer is (454 choose 4) - 10,316,800 = 746,858,751.

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To write csc(35π/18) in terms of the cosecant of a positive acute angle, we need to find a reference angle for 35π/18 in the first quadrant.

First, we can simplify 35π/18 by noting that it is equivalent to 70π/36, since 35 and 18 share a common factor of 5 and we can simplify π/2 - π/36 to π/36.

Next, we can find a reference angle for 70π/36 by subtracting the nearest multiple of π (which is 2π) and taking the absolute value.

|70π/36 - 2π| = |16π/36| = 4π/9

Therefore, we have:

csc(35π/18) = csc(70π/36) = csc(2π - 4π/9)

Since the cosecant function is periodic with period 2π, we can add or subtract any multiple of 2π to the argument without changing the value of the function. In particular, we can add 4π/9 to 5π/9 (which is in the first quadrant) to get:

2π - 4π/9 = 2π - (5π/9 - 4π/9) = π + π/9

Therefore, we have:

csc(35π/18) = csc(2π - 4π/9) = csc(π + π/9)

Now, we can use the fact that the cosecant function is odd (i.e., csc(-x) = -csc(x)) to write:

csc(π + π/9) = -csc(-π/9)

Finally, since π/9 is an acute angle in the first quadrant, we have:

csc(-π/9) = -csc(π/9)

Putting it all sum together, we have:

csc(35π/18) = -csc(π/9)

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We would need a sample size of approximately 598 men to have a 90% chance of detecting a true proportion of 8% with a significance level of 0.1.

To determine the sample size required, we need to perform a hypothesis test. The null hypothesis is that the proportion of men playing golf is 5%, and the alternative hypothesis is that it is greater than 5%.

We want to have a significance level (alpha) of 0.1, which corresponds to a confidence level of 0.9, and we also want a statistical power of 0.9. Assuming a one-tailed test, we can use a z-test to calculate the sample size needed.

Using a statistical calculator, we find that the critical value of z for a significance level of 0.1 is 1.28, and the critical value of z for a power of 0.9 is 1.28 + 1.28 = 2.56. The effect size is 0.03, which is the difference between the hypothesized proportion of 0.05 and the true proportion of 0.08. Plugging these values into the sample size formula for a z-test, we get:

n = ((1.28 + 2.56) / 0.03)² = 597.3

Therefore, we would need a sample size of approximately 598 men to have a 90% chance of detecting a true proportion of 8% with a significance level of 0.1.

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Answer:  201.06 square feet

Step-by-step explanation:

The radius of the circle is half of the diameter, so it is 8ft.

Then, we can use the area formula for a circle: 2πr

A = 64π square feet

A ≈ 201.06 square feet (rounded to the nearest hundredth)

Answer:

A ≈ 201.06 ft.

Step-by-step explanation:

To find the area, you use the formula A = [tex]\pi[/tex]d².

In this case, it would be A = [tex]\frac{1}{4}[/tex] · 3.14 · 16² or A = [tex]\frac{1}{4}[/tex] · 3.14 · 256 = 201.06.

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When a tax is imposed, some of the lost surplus is converted to tax revenue, while the rest is deadweight loss.

A tax creates a wedge between the price paid by buyers and the price received by sellers, reducing the quantity of goods traded in the market. This reduction in quantity causes a loss in surplus, which is the sum of consumer surplus and producer surplus. However, some of this lost surplus is converted to tax revenue, which is the amount of money collected by the government from the tax. The amount of lost surplus converted to tax revenue depends on the price elasticity of demand and supply in the market. If the demand and supply are relatively inelastic, a larger share of the lost surplus is converted to tax revenue. On the other hand, if the demand and supply are relatively elastic, a smaller share of the lost surplus is converted to tax revenue. The rest of the lost surplus that is not converted to tax revenue is called deadweight loss, which represents the reduction in economic welfare that is not compensated by the tax revenue. Deadweight loss occurs because the tax creates a distortion in the market that reduces the efficient allocation of resources, leading to inefficiencies in production and consumption.

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We can use the power series representation sum of the function f(x) = (1+x)^3 to find a closed-form expression for the series x[infinity] n=1 (-1)^n (2n+1)^3n.

Specifically, we have:

f(x) = (1+x)^3 = 1 + 3x + 3x^2 + x^3

Taking the cube of this expression gives:

f(x)^3 = (1 + 3x + 3x^2 + x^3)^3

Expanding this out using the binomial theorem gives:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + 126x^4 + 126x^5 + 84x^6 + 36x^7 + 9x^8 + x^9

We can rewrite the terms with even powers of x as:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + x^4 (126 + 126x + 84x^2 + 36x^3 + 9x^4)

Note that the expression in parentheses is just the power series representation of (1+x)^4. Therefore, we can simplify the above expression to:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + x^4 (1+x)^4

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[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{37}\\ a=\stackrel{adjacent}{x}\\ o=\stackrel{opposite}{35} \end{cases} \\\\\\ x=\sqrt{ 37^2 - 35^2}\implies x=\sqrt{ 1369 - 1225 } \implies x=\sqrt{ 144 }\implies x=12[/tex]

a) The firm’s long-run total, average, and marginal cost functions is:

C = wl + vk = (0.1w + 0.2v)q

AC = C/q = 0.1w + 0.2v

MC = dC/dq = 0.1w + 0.2v

b) The firm’s short-run total, average, and marginal cost functions:

C = wl + 10v = 0.1wq + 10v

AC = C/q = 0.1w + 10v/q

MC = dC/dq = 0.1w

c) This firm’s long-run and short-run average and marginal cost curves.

Long run cost:

AC = 0.3 + 0.2 = 0.5

MC = 0.3 + 0.2 = 0.5

Short run:

AC = 0.3 + 10/q

MC = 0.3

Cost function shows the relationship between the cost of production and the level of output. In the short run a portion of the total cost is fixed but, in the long run, all cost are variable. Average cost equals the cost per unit (i.e., total cost divided by output) and the marginal cost equals the change in cost per unit change in output.

The producer used k and l such that 5k = 10l

The output q, then, is

q = 5k = 10l

i.e., k = 1/5q = 0.2q and l = 1/10q = 0.1q

Long run cost:

C = wl + vk = (0.1w + 0.2v)q

AC = C/q = 0.1w + 0.2v

MC = dC/dq = 0.1w + 0.2v

b) Suppose that k is fixed at 10 in the short run. Calculate the firm's short-run total, average and marginal cost functions.

b) K= 10,

Short run cost:

C = wl + 10v = 0.1wq + 10v

AC = C/q = 0.1w + 10v/q

MC = dC/dq = 0.1w

c) Long run cost:

AC = 0.3 + 0.2 = 0.5

MC = 0.3 + 0.2 = 0.5

Short run:

AC = 0.3 + 10/q

MC = 0.3

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Rounded to the nearest ten-thousandth, the probability is 0.9375.

Probability is a branch of mathematics in which the chances of experiments occurring are calculated. It is by means of a probability, for example, that we can know from the chance of getting heads or tails in the launch of a coin to the chance of error in research.

Let's assume that the probability of a randomly selected motorist driving more than 5 miles per hour over the speed limit is p. Then, the probability of a motorist not driving more than 5 miles per hour over the speed limit is 1-p.

The probability of at least one motorist driving more than 5 miles per hour over the speed limit can be found by using the complement rule. That is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - P(no motorist driving more than 5 miles per hour over the speed limit)

The probability of no motorist driving more than 5 miles per hour over the speed limit can be found by using the binomial distribution. Since there are 4 motorists and each one has a probability of 1-p of not driving more than 5 miles per hour over the speed limit, the probability is:

P(no motorist driving more than 5 miles per hour over the speed limit) = (1-p)⁴

Therefore, the probability of at least one motorist driving more than 5 miles per hour over the speed limit is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - (1-p)⁴

We are not given a specific value for p, so we cannot calculate the probability exactly. However, if we assume that p = 0.5 (i.e., there is a 50-50 chance of a randomly selected motorist driving more than 5 miles per hour over the speed limit), then the probability of at least one motorist driving more than 5 miles per hour over the speed limit is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - (1-0.5)⁴ = 0.9375

Rounded to the nearest ten-thousandth, the probability is 0.9375. However, if we assume a different value for p, the probability will be different.

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There are 30 different sandwiches possible at the delicatessen, assuming that one item is used out of each category.

To calculate the number of different sandwiches possible, we need to multiply the number of options for each category. Since we are choosing one item from each category, we can use the multiplication principle.

The delicatessen offers 3 kinds of bread, 5 kinds of meat, and 2 kinds of vegetables. Using the multiplication principle, we can find the total number of different sandwiches possible as follows:

Number of different sandwiches = number of options for bread × number of options for meat × number of options for vegetables

= 3 × 5 × 2

= 30

It is important to note that this assumes that all combinations of the options are allowed.

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the lateral area of the figure to the right is 160 m². Option C

From the information given, the shape is a triangular prism.

Thus, the formula for calculating the lateral area of a triangular prism is expressed as;

A = (a + b + c)h

Such that the parameters of the formula are;

Substitute the values, we have;

Lateral area = 16(10)

Multiply the values

Lateral area = 180 m²

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The correct statement regarding the number of solutions for each system is given as follows:

The first system of equations is defined as follows:

4(3 + y) - y= 6 + 3(y + 2)

Applying the distributive property and then combining the like terms, the solution is obtained as follows:

12 + 4y - y = 6 + 3y + 6

12 + 3y = 12 + 3y.

The two sides are equal, hence the system has an infinite number of solutions, that is, all real numbers are solutions.

The second equation is given as follows:

-8(w + 1) = 2(1 - 4w) - 9

Hence:

-8w - 8 = 2 - 8w - 9

-8w - 8 = -8w - 7

0w = 1

Division by zero, hence the system has no solution.

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The statement is true. With a classic update using linear function approximation, we will always converge to some values, but they may not be optimal. This is because linear function approximation only allows for a limited representation of the value function, and the approximated function may not capture the true underlying structure of the problem.

Linear function approximation is commonly used in reinforcement learning to estimate the value function. The idea is to approximate the value function using a linear combination of features. During the learning process, the weights of the linear combination are updated using the classic update rule. While this approach is computationally efficient, it can result in suboptimal policies. The reason for this is that the approximated function may not be able to capture the complexity of the problem. This can lead to inaccuracies in the value function estimates, which in turn can result in suboptimal policies. To address this issue, more advanced function approximation methods, such as neural networks, can be used to approximate the value function. These methods can capture more complex relationships in the data and provide more accurate estimates of the value function.

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Answer:

Step-by-step explanation:

To find the eigenvalues of a symmetric matrix, we can first compute the characteristic polynomial, which is the determinant of the matrix minus λ times the identity matrix.

For the given matrix, the characteristic polynomial is λ^2 - 6λ + 8, which can be factored as (λ - 2)(λ - 4). Thus, the eigenvalues are λ = 2 and λ = 4. Since the matrix is symmetric, we know that its eigenvalues are real and its eigenvectors can be chosen to be orthogonal. This property makes symmetric matrices particularly useful in many applications, such as in linear algebra, physics, and engineering.

To find the eigenvalues of the symmetric matrix:

| 3 1 |

| 1 3 |

We can start by finding the characteristic polynomial, which is the determinant of the matrix minus the eigenvalue λ times the identity matrix:

| 3-λ 1 |

| 1 3-λ |

(3-λ)(3-λ) - 1 = λ^2 - 6λ + 8 = (λ-2)(λ-4)

Setting this polynomial equal to zero, we get the two eigenvalues:

λ = 2, 4

Therefore, the eigenvalues of the symmetric matrix are 2 and 4.

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Factored 25x^2 - 36 as the product of (5x + 6) and (5x - 6). To factor completely 25x^2 - 36, we first note that both 25 and 36 are perfect squares. Specifically, 25 = 5^2 and 36 = 6^2.

Using the difference of squares identity, we can write:

25x^2 - 36 = (5x)^2 - 6^2

Now, we can use the difference of squares formula again to obtain:

25x^2 - 36 = (5x + 6)(5x - 6)

In general, when factoring a quadratic expression of the form ax^2 + bx + c, where a, b, and c are constants, it is helpful to look for common factors or perfect squares first. The difference of squares formula can also be a useful tool in factoring quadratic expressions.

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The length of the curve   r(t) = 4t, t2, 1 6 t3 , 0 ≤ t ≤ 1 is approximately 3.022 units.

A curve is a shape or a line that is smoothly drawn in a plane having a bent or turns in it

[tex]\int (\dfrac{dx}{dt})^2+ (\dfrac{dy}{dt})^2 +(\dfrac{dz}{dt}^2 dt)[/tex]

where[tex]r(t) = x(t)i + y(t)j + z(t)k.[/tex]

In this case, we have:

[tex]x(t) = 4t\\y(t) = t^2\\z(t) =\dfrac{1}{6} t^3[/tex]

So, we need to find[tex]:\dfrac{dx}{dt} \dfrac{dy}{dt} \dfrac{dz}{dt}[/tex]

[tex]\dfrac{dx}{dt}[/tex]= 4

[tex]\dfrac{dy}{dt}[/tex] = 2t

[tex]\dfrac{dz}{dt}[/tex]=[tex]1/2 t^2[/tex]

Now we can plug these into the arc length formula:

[tex]\int (4)^2 +(2t)^2 + (\frac{1}{2t^2^})^2 dt[/tex] dt from 0 to 1

Simplifying under the square root:

[tex]\int 16+ 4t^2 +\frac{1}4t^4)[/tex] from 0 to 1

This integral is difficult to solve analytically, so we can use numerical methods to approximate the value. One way is to use Simpson's rule, simplifying further:

L= [tex]\dfrac{1}{3}[\sqrt{16+\sqrt{16} +\sqrt{16.111} +\sqrt[2]{16.222} +\sqrt[2]{16.4167}+\sqrt{17.1111} + \sqrt[2]{17.6944} +\sqrt{20}][/tex]

Therefore, the length of the curve is approximately 3.022 units.

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Answer:

a) 1/6

b)1/2

c)5/6

d)1/6

Step-by-step explanation:

because the probability of rolling one number is one in six we can add however many other probabilities