Consider & system described by the following third order differential equation; dy (t) dy (t) dy (t) + 6- +11 6y (t) = u (t) _ dt3 dt2 dt Compute the zero-state response of the system for the input u (t) = e 'p (t) . (Hint: 6s2 _ Is+6 = (s+1)(s+2)(s+3)) Compute the zero input response of the system for t 2 0, given that (0-) =1, y (07) = -1, i (07) = 1 Compute the output when the input is u (t) ~e-4'p (t) and the initial conditions are the same as those specified in part (b).

The amount of detergent left after 2 uses is less than 0.5 ounces. Therefore, Mya will throw away her laundry detergent bottle first. Given equations are: y = -1.6x + 50 and y = 50(0.75)^x

Let’s find out when each of them will throw away their laundry detergent bottle.

To do that, we need to find the point at which the amount of detergent is 0.5 ounces.

1.6x = 50 – y (from equation 1)

y = 50(0.75)^x

Substitute for y from equation 2 into equation 1.1.6x = 50 – 50(0.75)^x

Simplify: 1.6x = 50(1 – 0.75^x)

Now, we can solve for x using trial and error method, keeping in mind that x has to be a positive integer.

We’ll start with x = 1.

Using x = 1,

we get: 1.6(1) = 50(1 – 0.75)≈ 8.2

The amount of detergent left after 1 use is greater than 0.5 ounces. We need to try with a larger value of x.

Using x = 2,

we get: 1.6(2) = 50(1 – 0.75^2)≈ 5.8

The amount of detergent left after 2 uses is less than 0.5 ounces. Therefore, Mya will throw away her laundry detergent bottle first.

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The probability of low sales for the next three days, given that sales were low today, is 1.0 or 100%.

To find the transition matrix for the Markov chain, we can represent it as follows:

     |  P(1 → 1)  P(1 → 2)  P(1 → 3) |

     |  P(2 → 1)  P(2 → 2)  P(2 → 3) |

     |  P(3 → 1)  P(3 → 2)  P(3 → 3) |

From the given information, we can determine the transition probabilities as follows:

P(1 → 1) = 1 (since if sales are low one day, they are always low the next day)

P(1 → 2) = 0 (since if sales are low one day, they can never be average the next day)

P(1 → 3) = 0 (since if sales are low one day, they can never be high the next day)

P(2 → 1) = 0.1 (10% chance of going from average to low)

P(2 → 2) = 0.4 (40% chance of staying average)

P(2 → 3) = 0.5 (50% chance of going from average to high)

P(3 → 1) = 0.7 (70% chance of going from high to low)

P(3 → 2) = 0 (since if sales are high one day, they can never be average the next day)

P(3 → 3) = 0.3 (30% chance of staying high)

The transition matrix is:

     |  1.0  0.0  0.0 |

     |  0.1  0.4  0.5 |

     |  0.7  0.0  0.3 |

To find the probability of low sales for the next three days, we can calculate the product of the transition matrix raised to the power of 3:

     |  1.0  0.0  0.0 |³

     |  0.1  0.4  0.5 |

     |  0.7  0.0  0.3 |

Performing the matrix multiplication, we get:

     |  1.0  0.0  0.0 |

     |  0.1  0.4  0.5 |

     |  0.7  0.0  0.3 |

So, the probability of low sales for the next three days, given that sales were low today, is 1.0 or 100%.

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The complete question :

The Creamlest Cone, a local ice cream shop, classifies sales each day as "Tow." average,"or "high. "if sales are low one day, then they are always low the next day if sales are average one day, then there is a 10% chance they will be low the next day, a 4090 chance they wal be average the next day and a 50% chance they will be high the next day. If sales are high one day, then there is a 70% chance they wil be low the next day and a 30% chance they will be high the next day if state 1 = ow sales, state 2 average sales, and state 3 high sales, find the transition matnx for the Markov chain write entries as integers or decimals. If sales were low today, what is the probability that they will be low for the next three days? Write answer as an integer or decimal

The frequency of the note A\#, which is 1 half step above A3 Round to the nearest whole number is approximately 220 Hz.

To find the frequency of the note A# (A sharp), which is 1 half step above A3, we can use the given function:

F(x) = F0 * (1.059463)^x

Here, F(x) represents the frequency at a certain number of half steps above the reference frequency F0.

Given that the frequency of the note A3 is 220 Hz, we can set up the equation:

220 = F0 * (1.059463)^x

Now, we need to find the value of x for A# (1 half step above A3). Since each half step represents a change of 1 in x, we have x = 1.

Substituting x = 1 into the equation, we get:

220 = F0 * (1.059463)^1

220 = F0 * 1.059463

Dividing both sides by 1.059463 to isolate F0:

F0 = 220 / 1.059463

F0 ≈ 207.65

Now, we can find the frequency of the note A# by plugging in F0 and x = 1 into the original equation:

F(A#) = F0 * (1.059463)^x

      = 207.65 * (1.059463)^1

Calculating this expression:

F(A#) ≈ 207.65 * 1.059463

     ≈ 220.50

Rounding this value to the nearest whole number, we get:

F(A#) ≈ 220

Therefore, the frequency of the note A# (1 half step above A3) is approximately 220 Hz.

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Answer:

P = 20/7

Step-by-step explanation:

P/4 = 5/7

Multiply by 4 on both sides.

P = 20/7

The right response is that B is 20% larger than C.

Let's analyze the given information and the statements:

Given:

A is 40% smaller than B.

C is 20% bigger than A.

Statement 1: If B decreases by 20%, it will be the same value as C.

This statement cannot be determined based on the given information. We don't have the exact values of B and C, so we cannot make a conclusive comparison.

Statement 2: C is 20% smaller than B.

This statement cannot be true because it contradicts the given information that C is 20% bigger than A. If C were 20% smaller than B, it would mean C is smaller than A.

Statement 3: If C increases by 20%, it will be the same value as B.

This statement cannot be determined based on the given information. We don't have the exact values of B and C, so we cannot make a conclusive comparison.

Statement 4: B is 20% bigger than C.

This statement is consistent with the given information that A is 40% smaller than B, and C is 20% bigger than A. If A is smaller than B, and C is bigger than A, then it follows that B is bigger than C.

Based on the analysis, the only statement that is true is Statement 4: B is 20% bigger than C.

Therefore, the correct answer is: B is 20% bigger than C.

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a. X follows a Poisson distribution with mean 8, P(X = 5) = 0.1042.

b. Using Poisson distribution with mean 4, P(X > 2) = 0.7576.

c. T follows an exponential distribution with rate λ = 8, P(T < 10) = 0.4519.

d. Using Poisson distribution with mean 4, P(X = 0) = 0.0183.

a. The distribution of X, the number of flights arriving in the next hour, is a Poisson distribution with a mean of 8. To calculate the probability of exactly 5 flights arriving, we use the Poisson probability formula:

[tex]P(X = 5) = (e^(-8) * 8^5) / 5![/tex]

b. To calculate the probability of more than 2 flights arriving in the next 30 minutes, we use the Poisson distribution with a mean of 4 (half of the mean for an hour). We calculate the complement of the probability of at most 2 flights:

P(X > 2) = 1 - P(X ≤ 2).

c. The distribution of T, the time between two flight arrivals, follows an exponential distribution. The mean time between arrivals is 1/8 of an hour (λ = 1/8). To calculate the probability of the time between arrivals being less than 10 minutes (1/6 of an hour), we use the exponential distribution's cumulative distribution function (CDF).

d. To calculate the probability of no flights arriving in the next 30 minutes, we use the Poisson distribution with a mean of 4. The probability is calculated as

[tex]P(X = 0) = e^(-4) * 4^0 / 0!.[/tex]

Therefore, by using the appropriate probability distributions, we can calculate the probabilities associated with the number of flights and the time between arrivals. The Poisson distribution is used for the number of flight arrivals, while the exponential distribution is used for the time between arrivals.

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the function, f(x) = x³/x, the first step is to find its derivative f′(x) which will help in finding the slope of the tangent at point (8,3/8).

Hence, f′(x) can be found as follows; f(x) = x³/x

ƒ(x) = x²

ƒ′(x) = 2x

Taking the point given, (8,3/8), and substituting it in the function to get the slope of the tangent;

ƒ′(8) = 2(8)

= 16

At point (8,3/8), the slope of the tangent is 16.Using the point-slope form of a linear equation, y - y₁ = m(x - x₁)

We know the point (x₁,y₁) = (8,3/8)

and the slope m = 16

Substituting into the equation, we get y - 3/8 = 16(x - 8)

Multiplying through by the common denominator of 8,y - 3 = 16x - 128

Rearranging the equation, we get y = 16x - 125

The equation of the tangent line is y = 16x - 125.

The equation of the tangent line is y = 16x - 125. Given the function, f(x) = x³/x, the first step is to find its derivative f′(x) which will help in finding the slope of the tangent at point (8,3/8).

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Based on the scatter plot given below, we can say that statement C is correct. the given scatter plot shows a positive association.

A scatter plot is the graph that shows relationship between two variables. The independent variable is plotted on x axis and dependent variable on y axis.

Statement A is false. the point (18,2) does not lie on the scatter plot, let alone be an outliner.

Statement B is false as well. The scatter plot shows a quadratic association forming shape of a half parabola.

Statement C is correct. There is a positive association in X and Y. The scatter points are going in upward direction, i.e., as x increases, y increases.

Statement D is false. There is an association between the two variables plotted, as clearly the points are clustered and not scattered.

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For any n, applying the Euclidean algorithm to consecutive Fibonacci numbers takes just one step.

In formulaic terms, the number of steps required is a constant 1, independent of the value of n.

When applying the Euclidean algorithm to a pair of consecutive Fibonacci numbers, let's say fₙ and fₙ₊₁, the algorithm can be written as follows:

Step 1: Divide fₙ₊₁ by fₙ and obtain the quotient q₁ and remainder r₁.

  fₙ₊₁ = q₁ * fₙ + r₁

Step 2: If r₁ is not zero, repeat the division with fₙ as the divisor and r₁ as the dividend, obtaining quotient q₂ and remainder r₂.

  fₙ = q₂ * r₁ + r₂

Step 3: Continue this process, dividing the previous remainder by the current remainder, until reaching a remainder of zero.

  r₁ = qₙ * rₙ₋₁ + 0

The number of steps it takes to reach a remainder of zero corresponds to the number of iterations required by the Euclidean algorithm. Each iteration involves performing a division operation and obtaining a new remainder.

In the case of Fibonacci numbers, it can be shown that the Euclidean algorithm applied to consecutive Fibonacci numbers terminates in exactly one step. This is because consecutive Fibonacci numbers are always coprime, meaning they have no common factors other than 1.

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A quadratic function is not an example of a linear function because it does not satisfy the criteria of being a straight line when graphed.

A quadratic function is not an example of a linear function.

In mathematics, a linear function is a function that can be represented by a straight line when graphed. It has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept. The graph of a linear function is always a straight line.

On the other hand, a quadratic function is a polynomial function of degree 2. It has the general form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a curve called a parabola, not a straight line. The shape of the parabola depends on the values of the coefficients a, b, and c.

Therefore, a quadratic function is not an example of a linear function because it does not satisfy the criteria of being a straight line when graphed.

A linear function is a mathematical function that can be represented by a straight line when graphed. It has the general form:

f(x) = mx + b

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Given information: Sample size of first population, n1 = 14Sample mean of first population, X1 = 60.4Standard deviation of first population, s1 = 12.8Sample size of second population, n2 = 13Sample mean of second population, X2 = 43.4Standard deviation of second population, s2 = 16.5Level of significance, α = 0.10

(a) The test statistic can be calculated using the formula below :t = (X1 - X2)/[sqrt(s1^2/n1 + s2^2/n2)]Where,X1 and X2 are the sample means of the first and second populations respectively.s1 and s2 are the sample standard deviations of the first and second populations respectively.n1 and n2 are the sample sizes of the first and second populations respectively. Substituting the given values, we get: t = (60.4 - 43.4)/[sqrt((12.8^2/14) + (16.5^2/13))]t = 3.069Therefore, the test statistic is 3.069.(b) The p-value can be found using the t-distribution table. With the calculated test statistic, the degrees of freedom can be calculated as follows: d f = n1 + n2 - 2df = 14 + 13 - 2df = 25With a level of significance, α = 0.10 and degrees of freedom, df = 25, the p-value is 0.005.Therefore, the p-value is 0.005.(c) The null hypothesis is:H0: μ1 - μ2 = 0Where, μ1 is the mean of the first population.μ2 is the mean of the second population .The alternative hypothesis is: Ha: μ1 - μ2 ≠ 0As the calculated p-value is less than the level of significance, α = 0.10, we reject the null hypothesis and conclude that there is evidence to conclude that the first population mean is not equal to the second population mean. Therefore, the answer is "Reject" the null hypothesis. Evidence to conclude the first population mean is not equal to the second.(d) There is a population mean difference between the two populations.

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The cost per pound of cinnamon and spice tea will be calculated in this question. Cinnamon tea costs 4 dollars per pound and spice tea costs 3 dollars per pound is found by solving linear equations. The detailed solution of the question is provided below.

A merchant mixed 12 lb of cinnamon tea with 2 lb of spice tea to produce a 14-pound mixture that cost $15. Another mixture included 14 lb of cinnamon tea and 12 lb of spice tea to produce a 26-pound mixture that cost $32. Now we have to calculate the cost per pound of cinnamon tea and spice tea.

There are different ways to approach mixture problems, but the most common one is to use systems of linear equations. Let x be the price per pound of the cinnamon tea, and y be the price per pound of the spice tea. Then we have two equations based on the given information:

12x + 2y = 15 (equation 1)

14x + 12y = 32 (equation 2)

We can solve for x and y by using elimination, substitution, or matrices. Let's use elimination. We want to eliminate y by

multiplying equation 1 by 6 and equation 2 by -1:

72x + 12y = 90 (equation 1 multiplied by 6)

-14x - 12y = -32 (equation 2 multiplied by -1)

58x = 58

x = 1

Now we can substitute x = 1 into either equation to find y:

12(1) + 2y = 15

2y = 3

y = 3/2

Therefore, the cost per pound of cinnamon tea is $1, and the cost per pound of spice tea is $1.5.

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To solve the given differential equation:

d²y/dx² - 6x² = 2, we can integrate the equation twice to find the general solution. Integrating the equation once will give us:

dy/dx = ∫(6x² + 2) dx

= 2x³ + 2x + C₁,

where C₁ is the constant of integration.

Integrating once again will give us:

y = ∫(2x³ + 2x + C₁) dx

= (2/4)x⁴ + (2/2)x² + C₁x + C₂

= 1/2 x⁴ + x² + C₁x + C₂,

where C₂ is another constant of integration.

Now, we can apply the initial condition y(1) = 6 to find the values of C₁ and C₂.

Substituting x = 1 and y = 6 into the equation:

6 = 1/2 (1)⁴ + (1)² + C₁(1) + C₂

= 1/2 + 1 + C₁ + C₂.

Simplifying the equation, we have:

6 = 3/2 + C₁ + C₂.

Rearranging the equation, we get:

C₁ + C₂ = 6 - 3/2

= 12/2 - 3/2

= 9/2.

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The probability that the average vitamin content is between 59.9 and 60.15 mg is approximately 0.7745 or 77.45%.

To solve this problem, we can use the properties of the sampling distribution of the sample mean.

Population mean (μ) = 60.0 mg

Population standard deviation (σ) = 0.5 mg

Sample size (n) = 25

We need to find the probability that the average vitamin content (sample mean) is between 59.9 and 60.15 mg.

First, we calculate the standard error of the mean (SE), which is the standard deviation of the sampling distribution:

SE = σ / √n

SE = 0.5 / √25 = 0.5 / 5 = 0.1 mg

Next, we can convert the values 59.9 and 60.15 to z-scores using the formula:

z = (x - μ) / SE

For 59.9 mg:

z1 = (59.9 - 60.0) / 0.1 = -1

For 60.15 mg:

z2 = (60.15 - 60.0) / 0.1 = 1.5

Now, we can find the probability using the z-table or calculator.

P(59.9 < x < 60.15) = P(-1 < z < 1.5)

Using the z-table, we can find the corresponding probabilities for z = -1 and z = 1.5 and then subtract the smaller probability from the larger probability to find the desired probability.

P(-1 < z < 1.5) ≈ P(z < 1.5) - P(z < -1)

Looking up the values in the z-table, we find:

P(z < 1.5) = 0.9332

P(z < -1) = 0.1587

Therefore,

P(-1 < z < 1.5) ≈ 0.9332 - 0.1587 = 0.7745

So, the probability that the average vitamin content is between 59.9 and 60.15 mg is approximately 0.7745 or 77.45%.

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After the first iteration, the new estimate x₁ is 1/3. The correct answer is B: 1/3.

To find the new estimate x₁ using the Newton-Raphson method, we need to apply the following iteration formula:

x₁ = x₀ - f(x₀) / f'(x₀)

In this case, the given equation is x⁴ - 3x + 1 = 0. To find the root using the Newton-Raphson method, we need to find the derivative of the function, which is f'(x) = 4x³ - 3.

Given that the initial guess is x₀ = 0, we can substitute these values into the iteration formula:

x₁ = 0 - (0⁴ - 3(0) + 1) / (4(0)³ - 3)

= -1 / -3

= 1/3

Therefore, after the first iteration, the new estimate x₁ is 1/3.

The correct answer is B: 1/3.

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The probability that the whole shipment will be accepted is approximately 0.9999. Based on this probability, it is highly likely that almost all shipments will be accepted.

To calculate the probability that the whole shipment will be accepted, we need to consider the rate of defects and the acceptance criteria.

Given:

Defect rate (p) = 3% = 0.03

To determine if the shipment will be accepted, we need to determine the number of defective tablets in the shipment. If the number of defective tablets is below a certain threshold, the shipment will be accepted.

Assuming the shipment contains a large number of tablets, we can approximate the number of defective tablets using a binomial distribution. The probability of accepting the shipment is equal to the probability of having fewer than the acceptance threshold number of defective tablets.

To calculate this probability, we sum the probabilities of having 0, 1, 2, ..., (threshold-1) defective tablets.

Let's assume the acceptance threshold is set at k defective tablets (where k is determined by the buyer). In this case, we need to calculate the probability of having fewer than k defective tablets.

Using the binomial probability formula, the probability of having exactly x defective tablets in the shipment is given by:

P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)

where n is the total number of tablets in the shipment.

In our case, we want to find the probability of having fewer than k defective tablets:

P(X < k) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = k-1)

For simplicity, let's assume the shipment contains 100 tablets (n = 100) and the acceptance threshold is set at 5 defective tablets (k = 5).

Using the binomial probability formula, we can calculate the probabilities for each value of x and sum them up:

P(X = 0) = C(100, 0) * (0.03)^0 * (1 - 0.03)^(100 - 0)

P(X = 1) = C(100, 1) * (0.03)^1 * (1 - 0.03)^(100 - 1)

P(X = 2) = C(100, 2) * (0.03)^2 * (1 - 0.03)^(100 - 2)

...

P(X = 4) = C(100, 4) * (0.03)^4 * (1 - 0.03)^(100 - 4)

The probability that the whole shipment will be accepted is:

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

Calculating the probabilities and summing them up, we find:

P(X < 5) ≈ 0.9999

Therefore, the probability that the whole shipment will be accepted is approximately 0.9999 (rounded to four decimal places).

Based on this probability, it is highly likely that almost all shipments will be accepted.

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After solving the equation 1+2+3+....+n+(n+1)= [(n+2)/2](n+1) we can conclude our results.

The statement is the addition of all natural numbers.  Let us suppose that 1+2+3+....+n= sum [n(n+1)]/2 for some positive integer n. Adding (n+1) to both the sides of the equation,1+2+3+....+n+(n+1)= sum [n(n+1)]/2 +(n+1)

Using the formula for the sum of natural numbers 1+2+3+....+n, we can substitute, sum [n(n+1)]/2= [n/2](n+1)1+2+3+....+n+(n+1)= [n/2](n+1) +(n+1)

On simplifying, we get, 1+2+3+....+n+(n+1)= [(n+2)/2](n+1)

Now, we know that the sum of natural numbers 1+2+3+....+n+(n+1) is [n(n+1)]/2 + (n+1).

We have to equate it to [(n+2)/2](n+1).

Therefore, equating these two sums, [n(n+1)]/2 + (n+1) = [(n+2)/2](n+1)2n+2 = n² + 3n + 22n = n² + 3n + 2(n² - 2n) = 0(n-1) (n-2) = 0 n = 1, 2

This is the required proof for the statement using direct proof.

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The equation for this case is:

N = 12 + (2/3)*18

We know that the phone comes with 18 aplications installled, and she uses 2/3 of these 18 aplications.

We also know that she installed another 12, that she uses regularly.

Then the total number N of applications that she uses is given by the equation:

N = 12 + (2/3)*18

That is, the 12 she installed, plus two third of the original 18 that came with the phone.

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Given that f(t) = √12-9 The value under the radical sign must be non-negative for the result to be a real number. Hence, we have to check if: 12-9 >= 0 is true or not.

This is true. Therefore, for every value of t, f(t) is a real number. To evaluate the real values of t for the given function f(t) = √12-9, we have to evaluate the values of t for which the function returns a real number. For the function, we know that f(t) is real when the expression under the radical is greater than or equal to zero.

So,12 - 9 ≥ 0 → 3 ≥ 0.This is a true statement. Therefore, the given function f(t) is always a real number for any value of t.For this reason, we can say that the domain of the given function f(t) is all real numbers. Therefore, we can say that f(t) is defined for all values of t which belong to the set of all real numbers [t∈R].

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The given sin A = 63/65 and that angle A is in Quadrant I. We need to find the exact value of cos A in simplest radical form using a rational denominator. In the simplest radical form using a rational denominator is (46/65) which cannot be further simplified since 46 and 65 have no common factors other than 1.

Given: sinA = 63/65 and A is in Quadrant I.In a right triangle, sinA = Opposite/Hypotenuse = BC/AC.Let BC = 63, AC = 65 and AB = √(AC² - BC²) = √(65² - 63²) = √(2116) = 46.So, cosA = Base/Hypotenuse = AB/AC = 46/65. Therefore, the exact value of cosA in simplest radical form using a rational denominator is (46/65) which cannot be further simplified since 46 and 65 have no common factors other than 1. The given problem is to find the exact value of cosA in simplest radical form using a rational denominator, given that sinA = 63/65 and angle A is in Quadrant I.In a right-angled triangle, the opposite side is the side that is opposite to the angle of interest and the hypotenuse is the longest side of the right-angled triangle, and it is always opposite to the right angle, while the adjacent side is the side adjacent to the angle of interest.

To calculate the cosine of an angle in a right-angled triangle, we need to calculate the ratio of the adjacent side to the hypotenuse, using the following formula: cosA = Base/Hypotenuse = AB/AC. In the given question, we are given that sinA = 63/65 and that angle A is in Quadrant I. In Quadrant I, all trigonometric functions are positive, i.e., sinA = BC/AC > 0. This implies that the length of the opposite side (BC) is positive, and the length of the hypotenuse (AC) is also positive. Using the Pythagorean theorem, we can calculate the length of the base (AB).We have AB = √(AC² - BC²).Substituting the values, we get AB = √(65² - 63²) = √(2116) = 46.Now, we can calculate the cosine of angle A using the formula mentioned above. cosA = AB/AC = 46/65.Therefore, the exact value of cosA in the simplest radical form using a rational denominator is (46/65) which cannot be further simplified since 46 and 65 have no common factors other than 1.

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Answer:

Given:[tex]x^(3) + 11x^(2) + 20x + 7[/tex]is divided by x+1 We are to find the result using synthetic division. Step 1:  Set up the synthetic division table with the coefficients of the polynomial being divided by the divisor as follows.

In this case, we do not have a remainder. Therefore, the answer is simply x² + 10x - 3, which is the quotient obtained using synthetic division. Note: The process of synthetic division is just an algorithm to divide polynomials with the help of the factor theorem.

[tex]-1 | 1 11 20 7         | -1|   -10  -10  -10         |   0 1  10  -3[/tex]Step 5:  Rewrite the polynomial whose coefficients are in the bottom row of the table.  Therefore,  x³+11x²+20x+7 when divided by x+1 gives  x² + 10x - 3.  The quotient is [tex]x² + 10x - 3.[/tex]If there is a remainder, we express the result in the form[tex]q(x)+(r(x))/(b(x)).[/tex]

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The maimum values of the function ximum and min on the interval [-2, 4] are as follows: Absolute Maximum = 146 at x = 3.Absolute Minimum = 2 at x = -2 and x = -1.

The given function is,

[tex]f(x) = 4x³ − 12x² − 36x + 2,[/tex]

on the interval [-2, 4]Step 1To find the absolute maximum and minimum values of f, we need to follow these steps:

The absolute maximum and minimum values of f can occur either at a critical point inside the interval or at an endpoint of the interval. We begin by finding the derivative of f.

[tex]f′(x) = 12x² − 24x − 36[/tex]

= [tex]12(x² − 2x − 3)[/tex]

= [tex]12(x − 3)(x + 1)[/tex]

Step 2We solve [tex]f′(x) = 0[/tex] to obtain the critical numbers.

12(x − 3)(x + 1) = 0

⇒ [tex]x = -1, 3,[/tex]

are the critical numbers. Now, we find the function values at the critical numbers and endpoints of the interval [-2, 4].

[tex]f(−2) = 2,[/tex]

[tex]f(-1) = 2,[/tex]

[tex]f(3) = 146,[/tex]

[tex]f(4) = 6[/tex]

Therefore, the maimum values of the function ximum and min

on the interval [-2, 4] are as follows:

Absolute Maximum = 146

at x = 3.

Absolute Minimum = 2 at

x = -2

and x = -1.

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The correct option is the first one, any integer less than -3

Let's define our integer as n.

We want to find the possible values of n such that:

n + 3 < 0

Let's solve that inequality for the variable n, we can do that by subtracting 3 in both sides, then we will get:

n < -3

So any integer less than -3 works fine, the correct option is F.

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Answer:

$135

Step-by-step explanation:

Assuming that the 70% discount was applied on the original price of $450,

you multiply 0.7 (70%) to 450 and subtract that value from the original price. Basically, you are left with 30% of the original price.

Imagine taking the full price of the camera and subtracting 70% off that original price.

450 - (0.7)(450) = $135

The price of the camera after the 70% off discount is $135.

The exact solutions to the equation (x - 8)^2 = 45 are x = 8 + √45 and x = 8 - √45. The approximate solutions to three decimal places are x ≈ 11.873 and x ≈ 4.127.

To solve the equation (x - 8)^2 = 45, we start by taking the square root of both sides to eliminate the square. This gives us x - 8 = ±√45.  Taking the positive square root, we have x - 8 = √45. Adding 8 to both sides, we get x = 8 + √45. This is one solution.

Taking the negative square root, we have x - 8 = -√45. Adding 8 to both sides, we get x = 8 - √45. This is the second solution. To find the approximate solutions to three decimal places, we evaluate the square root of 45, which is approximately 6.708.

For x = 8 + √45, we get x ≈ 8 + 6.708 ≈ 14.708.

For x = 8 - √45, we get x ≈ 8 - 6.708 ≈ 1.292.

Therefore, the exact solutions are x = 8 + √45 and x = 8 - √45, and the approximate solutions to three decimal places are x ≈ 11.873 and x ≈ 4.127.

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Answer:

h(t) = -8t - 5

Step-by-step explanation:

Since h(t) decreases 8 units for each unit interval in t, the slope is -8.

At t = 0, h(t) = -5, so the y-intercept is -5.

y = mx + b

h(t) = -8t - 5

        The angle between an edge of a cube and its diagonal is:

        θ  =  arccos 1/√3

Step-by-step explanation:

Theta  Symbol: (θ), Square-root Symbol: (√):

        Label the Points:

        A(0, 0, 0)

        B(1, 0, 0)

        C(1, 1, 1)

                    AB  is the Edge

                    AC is the Diagonal

       The Lenth of the Edge AB  is (1) Since it's the side length of the cube.

       AC = √(1 - 0)^2 + (1 - 0)^2 + (1 - 0)^2 = √3

        The Dot Product Formula:

        u * v  =   |u| |v| cos  θ, Where θ is the angle between the vectors:

        AB  = (1, 0, 0 )

        AC  = (1, 1, 1 )

        AB * AC = (1)(1)   + (0)(1)  + (0)(1)  =  1

        1  =  (1)(√3)  cos  θ

        Divide both sides by √3

        cos  θ  = 1/√3

       θ  =  arccos 1/√3

     Therefore,  The angle between an edge of a cube and its diagonal is:

        θ  =  arccos 1/√3

I  hope this helps!

Hence, the probability of getting an average X greater than 75 is 0.6915.

Given:

Sample size (N) = 64

Sample mean (X) = 76

Standard deviation (σ) = 16

To find:

The probability of getting the average X greater than 75, P(X> 75)

We can find the probability as follows:

P(X > 75) = P(Z > (75 - 76) / (16 / √64)) = P(Z > -0.5)

We know that the standard normal distribution is symmetric about its mean, which is 0. So, the area of interest is P(Z > -0.5), which is equivalent to the area P(Z < 0.5) using the symmetry of the standard normal distribution.

Using the standard normal distribution table, we find that P(Z < 0.5) = 0.6915.

Therefore, P(X > 75) = P(Z > -0.5) = P(Z < 0.5) = 0.6915.

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The probability that a randomly selected man from the certain age group runs slower than 7.1 miles per hour is 0.9332.

Given: Running speed for adult men of a certain age group is known to follow a normal distribution, with mean 5.6 miles per hour and standard deviation

To find: What speed would he need to be able to run for this to be the case?

First we find the z score corresponding to 80% probability.

Using standard normal table, we get the corresponding z-score for 0.8 is 0.84.

z = (x - μ)/ σ

0.84 = (x - 5.6) / 1

x - 5.6 = 0.84

x = 5.6 + 0.84

x = 6.44 miles per hour (2 decimal places)

Therefore, Jim needs to run at least 6.44 miles per hour to be able to run faster than 80% of adult men in this age group.

Probability that a randomly selected man from the certain age group runs slower than 7.1 miles per hour is:

P (x < 7.1) = P (z < (7.1 - 5.6) / 1) = P (z < 1.5) = 0.9332 (using standard normal table)

Hence, the probability that a randomly selected man from the certain age group runs slower than 7.1 miles per hour is 0.9332.

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