The endpoint of AB are A(9,-7) and B(8,2) find the coordinates of the midpoint M

Answer: 1750

Step-by-step explanation:

100+500+1000+5000+6000= 12600x10= 126000

126000 divided 74 = 1750

The rewritten functions form for g(x) with their domain and range are:

2 + 1/(x - 4), with domain x ≠ 4 and range y ≠ 2.

-4 + 15/(x + 1), with domain x ≠ -1 and range y ≠ -4.

To rewrite g(x) in the form g(x) = a(1/(a + k)), use partial fraction decomposition:

(2x - 7) / (x - 4) = (2(x - 4) + 1) / (x - 4) = 2 + 1/(x - 4)

So, g(x) = 2 + 1/(x - 4), with domain x ≠ 4 and range y ≠ 2.

Rewrite g(x) in the form g(x) = a(1/(a + k)) using partial fraction decomposition:

(-4x + 11) / (x + 1) = (-4(x + 1) + 15) / (x + 1) = -4 + 15/(x + 1)

So, g(x) = -4 + 15/(x + 1), with domain x ≠ -1 and range y ≠ -4.

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The pmf of X gives the probability that we need to collect k coupons in order to obtain all 10 possible coupons, where each coupon type is equally likely to be obtained at any time.

Let X be the final score, and let Xi be the score on test i. Then, we have:

P(X = k) = P(X1 = k, X2 ≤ k, X3 ≤ k) + P(X2 = k, X1 ≤ k, X3 ≤ k) + P(X3 = k, X1 ≤ k, X2 ≤ k)

Since the scores on each test are independent, we can compute these probabilities separately. For example, we have:

P(X1 = k, X2 ≤ k, X3 ≤ k) = P(X1 = k) P(X2 ≤ k) P(X3 ≤ k)

Since the probabilities are the same for each test, we can simplify this to:

P(X1 = k, X2 ≤ k, X3 ≤ k) = [[tex]\frac{1}{11-k}[/tex])]³

Using similar reasoning, we can compute the other probabilities and sum them up to obtain the pmf of X:

P(X = k) = [[tex]\frac{3}{11-k}[/tex]]² - [[tex]\frac{2}{11-k}[/tex]]³

for k = 1, 2, ..., 10. The pmf is 0 for all other values of k.

In mathematics, probability is a measure of the likelihood of an event occurring. It is a numerical value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The concept of probability is used in a wide range of fields, including statistics, game theory, physics, and finance. There are two main approaches to probability: classical probability and Bayesian probability. Classical probability deals with situations where all outcomes are equally likely, such as rolling a fair die.

Bayesian probability, on the other hand, takes into account prior knowledge and experience to make predictions about future events. Probability theory provides a framework for understanding and predicting the behavior of random phenomena. It is used to calculate the likelihood of various outcomes in experiments and to make informed decisions based on incomplete information.

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The kinetic energy of body B after the collision is 2,000 J . In an elastic collision, both the momentum and the kinetic energy (KE) of the system are conserved. Initially, body A and body B have kinetic energies of 5,000 J each, totaling 10,000 J for the system.

After the collision, body A has a kinetic energy of 8,000 J. To determine the kinetic energy of body B after the collision, we can use the principle of conservation of kinetic energy:

Total KE (before collision) = Total KE (after collision)

10,000 J = 8,000 J (KE of body A after collision) + KE of body B (after collision)

To find the kinetic energy of body B after the collision, we can rearrange the equation and solve for the unknown value:

KE of body B (after collision) = 10,000 J - 8,000 J

KE of body B (after collision) = 2,000 J

So, after the elastic collision, the kinetic energy of body B is 2,000 J.

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Answer:

need this

Step-by-step explanation:

There's no point of intersection between the line connecting (2,3,6) to (2,2,3) and the line connecting (1,2,1) to (3,0,-1)

To find the point of intersection between two lines in three-dimensional space, we need to solve a system of equations. We can set up the equations of the two lines using the parametric form:

Line 1:

x = 2 + t(0)

y = 3 + t(-1)

z = 6 + t(-3)

Line 2:

x = 1 + s(2)

y = 2 - s(2)

z = 1 - s(2)

We can set the x, y, and z values equal to each other for the point of intersection, and solve for t and s:

2 + t(0) = 1 + s(2)

3 + t(-1) = 2 - s(2)

6 + t(-3) = 1 - s(2)

Simplifying the second equation, we get:

t + s = 1

Multiplying the first equation by 2, we get:

4 = 2 + 2t + 2s

Substituting t + s = 1, we get:

4 = 2 + 2(t + s)

4 = 2 + 2(1)

4 = 4

This means that our system of equations has no unique solution, and the two lines do not intersect at a single point. Therefore, there is no point of intersection between the two lines.

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The number of blocks that Tommy travels is given as follows:

26 blocks.

Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Then the distances are given as follows:

Then the total number of blocks is given as follows:

2 x (8 + 5) = 26 blocks.

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The function attains an absolute maximum at x ≈ 1.13 with a value of f(x) ≈ 2.35, and an absolute minimum at x = 2 with a value of f(x) ≈ -8.77 on the interval [1/2, 2].

To verify that the given function f(x) = -4x^2 + 12x - 4ln(x) attains an absolute maximum and minimum on the interval [1/2, 2], we need to find critical points and evaluate the function at the interval's endpoints.

First, find the first derivative of the function:
f'(x) = d/dx (-4x^2 + 12x - 4ln(x))
f'(x) = -8x + 12 - 4/x

Set the first derivative equal to zero and solve for x to find critical points:
-8x + 12 - 4/x = 0

To find the critical points, we can use the quadratic formula, but since the function is not quadratic, we can instead use numerical methods or graphing to find approximate values. We find that there is a critical point at x ≈ 1.13.

Next, evaluate the function at the critical point and the endpoints of the interval:
f(1/2) ≈ -2.55
f(1.13) ≈ 2.35
f(2) ≈ -8.77

From these evaluations, we see that the function attains an absolute maximum at x ≈ 1.13 with a value of f(x) ≈ 2.35, and an absolute minimum at x = 2 with a value of f(x) ≈ -8.77 on the interval [1/2, 2].

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Sally's sweet shoppe has cylindrical cups that have a diameter of 8 centimeters and a height of 5 centimeters the cylinder has a larger volume than the cone, by 64π cubic centimeters.

The sweet shoppe sells cylindrical cups with a diameter of 8 centimeters and a height of 5 centimeters.

The volume of the cylinder can be calculated using the formula V = [tex]\pi r^2h[/tex], where r is the radius (half the diameter) and h is the height. So, for this cylinder:

r = 4 cm

h = 5 cm

[tex]V_{cylinder} = \pi (4cm)^2(5cm) = 80\pi[/tex] cubic cm

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height.

The radius of the cone is half the diameter, or 4 centimeters, and we need to find the height of the cone.

The height of the cone can be found using the Pythagorean theorem, since the radius and height of the cone form a right triangle. The height is the square root of the difference between the hypotenuse (the slant height of the cone) and the radius, squared:

h = sqrt[tex]((5cm)^2 - (4cm)^2)[/tex] = 3cm

Now we can calculate the volume of the cone:

r = 4 cm

h = 3 cm

V_cone = (1/3)π[tex](4cm)^2[/tex](3cm) = 16π cubic cm

Comparing the volumes of the cylinder and cone, we find:

V_cylinder - V_cone = 80π - 16π = 64π cubic cm

Thus, the cylinder has a larger volume than the cone, by 64π cubic centimeters.

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The correct statement about the correlation coefficient (r) that is TRUE is: c.) The correlation coefficient is always between -1 and +1.

The statement that is true about the correlation coefficient (t) is that it is always between -1 and +1.

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables.

The range of the correlation coefficient is from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation at all.

However,

Using a statistical software is more convenient and efficient, especially when dealing with large datasets.
The statement that a correlation coefficient of r = 0.75 shows a strong positive relationship between two variables is partially true.

A correlation coefficient of 0.75 indicates a moderate to strong positive correlation, but the strength of the correlation also depends on the context and the field of study. In some fields, a correlation coefficient of 0.75 may be considered weak, while in others, it may be considered strong.
Finally,

The statement that the stronger the strength of association, the lower the value of the correlation coefficient is false.

In fact, the stronger the association between two variables, the higher the value of the correlation coefficient.

This is because the correlation coefficient measures the degree to which the two variables move together, whether positively or negatively.

Therefore,

If two variables have a strong positive correlation, the correlation coefficient will be closer to +1, indicating a strong relationship.

Conversely, if two variables have a strong negative correlation, the correlation coefficient will be closer to -1, indicating a strong relationship.

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The present value of the amount desired at the end of the period is $7,246.92.

To find the present value of the amount desired at the end of the period, we need to use present value tables. The given interest rate is 6% compounded monthly.

Using PV Table 12.3, we can find the PV factor for 48 periods (4 years x 12 months/year = 48 months) at 0.5% (6%/12 months) interest rate. The PV factor for 48 periods at 0.5% is 0.8183.

The formula for present value is:

[tex]PV = Amount / (1 + r)^n[/tex]

where r is the interest rate per period and n is the number of periods.

Plugging in the values, we get:

[tex]PV = $9,800 / (1 + 0.005)^48[/tex]

PV = $9,800 / 1.3511

PV = $7,246.92

Therefore, the present value of the amount desired at the end of the period is $7,246.92.

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The estimated instantaneous velocity at t = 1 is -1 ft/s.

a. To find the average velocity for a time period, we need to find the change in distance over the change in time.

For the time period starting when t = 1 second and lasting 0.5 seconds:

- Distance at t = 1.5 seconds: y = 49(1.5) - 25(1.5)^2 = 33.75 feet
- Distance at t = 1 second: y = 49(1) - 25(1)^2 = 24 feet

Change in distance = 33.75 - 24 = 9.75 feet
Change in time = 0.5 seconds

Average velocity = change in distance / change in time = 9.75 / 0.5 = 19.5 ft/s

For the time period starting when t = 1 second and lasting 0.01 seconds:

- Distance at t = 1.01 seconds: y = 49(1.01) - 25(1.01)^2 = 24.96 feet
- Distance at t = 1 second: y = 49(1) - 25(1)^2 = 24 feet

Change in distance = 24.96 - 24 = 0.96 feet
Change in time = 0.01 seconds

Average velocity = change in distance / change in time = 0.96 / 0.01 = 96 ft/s

For the time period starting when t = 1 second and lasting 0.001 seconds:

- Distance at t = 1.001 seconds: y = 49(1.001) - 25(1.001)^2 = 24.9996 feet
- Distance at t = 1 second: y = 49(1) - 25(1)^2 = 24 feet

Change in distance = 24.9996 - 24 = 0.9996 feet
Change in time = 0.001 seconds

Average velocity = change in distance / change in time = 0.9996 / 0.001 = 999.6 ft/s

b. To estimate the instantaneous velocity at t = 1, we can take the derivative of the height equation with respect to time:

y = 49t - 25t^2

y' = 49 - 50t

At t = 1, y' = -1

So the estimated instantaneous velocity at t = 1 is -1 ft/s.

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When asked to find the distance, you should think of the Pythagorean Theorem.

Option C is the correct answer.

W have,

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

This theorem can be applied to find the distance between two points in a coordinate plane or in three-dimensional space.

The distance between two points is the length of the line segment connecting them, which is also the hypotenuse of a right triangle formed by the two points and the origin or another reference point.

Thus,

To find the distance between two points, you can use the Pythagorean Theorem by treating the coordinates of the two points as the lengths of the legs of a right triangle and finding the length of the hypotenuse.

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The results indicate a statistically significant positive linear relationship between lean body mass and maximal oxygen uptake in college athletes.

The researchers hypothesized that there is a positive relationship between lean body mass and maximal oxygen uptake in college athletes.

To test this hypothesis, they collected data from 18 randomly selected college athletes and created a scatterplot of the measurements.

The scatterplot displayed a strong positive linear relationship between the two variables, indicating that their hypothesis may be correct.

To further investigate the relationship between the variables, the researchers performed a significance test.

Specifically, they tested the null hypothesis that the slope of the regression line is 0, meaning there is no relationship between the variables, versus the alternative hypothesis that the slope is greater than 0, indicating a positive relationship.

The test yielded a p-value of 0.04, which is below the commonly used significance level of 0.05.

This means that there is strong evidence against the null hypothesis and we can reject it.

Therefore, we can conclude that there is a statistically significant positive linear relationship between lean body mass and maximal oxygen uptake in college athletes.

In practical terms, this suggests that increasing lean body mass through exercise or other means may lead to an improvement in maximal oxygen uptake, which is an important measure of physical fitness and endurance.

Further research can explore the specific mechanisms that underlie this relationship and the potential benefits of interventions aimed at increasing lean body mass for athletic performance and overall health.

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a. The conditional distribution of U is 1 / (u - a), a < u ≤ 1.

b.  The conditional distribution of U is  1 / (au), 0 < u < a.

We will use Bayes' theorem to compute the conditional distributions.

a. U > a:

The probability that U > a is given by P(U > a) = 1 - P(U ≤ a) = 1 - a. To compute the conditional distribution of U given that U > a, we need to compute P(U ≤ u | U > a) for u ∈ (a,1). By Bayes' theorem,

P(U ≤ u | U > a) = P(U > a | U ≤ u) P(U ≤ u) / P(U > a)

= [P(U > a ∩ U ≤ u) / P(U ≤ u)] [P(U ≤ u) / (1 - a)]

= [P(a < U ≤ u) / (u - a)] [1 / (1 - a)]

= 1 / (u - a), a < u ≤ 1.

Therefore, the conditional distribution of U given that U > a is a uniform distribution on (a,1), i.e., U | (U > a) ∼ U(a,1).

b. U < a:

The probability that U < a is given by P(U < a) = a. To compute the conditional distribution of U given that U < a, we need to compute P(U ≤ u | U < a) for u ∈ (0,a). By Bayes' theorem,

P(U ≤ u | U < a) = P(U < a | U ≤ u) P(U ≤ u) / P(U < a)

= [P(U < a ∩ U ≤ u) / P(U ≤ u)] [P(U ≤ u) / a]

= [P(U ≤ u) / u] [1 / a]

= 1 / (au), 0 < u < a.

Therefore, the conditional distribution of U given that U < a is a Pareto distribution with parameters α = 1 and xm = a, i.e., U | (U < a) ∼ Pa(1,a).

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The scenario describes insufficient data in terms of geographical coverage. The data analyst only had relevant data from half of the country, so they needed to generate new data to represent the entire nation.

This means that the dataset was incomplete and lacked the necessary information to analyze the national supply chain as a whole, The scenario you described represents a type of insufficient data known as "incomplete data" or "missing data.

In this case, the data analyst is working on a project around a national supply chain, but they only have data from about half of the country. To address this issue, they decide to generate new data that represents the entire nation. This process is often done using data imputation techniques or by obtaining additional data sources to fill the gaps in the existing dataset.

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If she will pay 2.4% of the end of the loan plus $0.57 each month then after 20 years the total amount will be; $6338.26

Given that Bessie took out a subsidized student loan of $5000 at a 2.4% APR, compounded monthly, to pay for her last semester of college.

When she will begin paying off the loan in 10 months with monthly payments lasting for 20 years,

A = p(1+ r/n) nl

Because in our example, n = 12 (monthly), p = $5000 , r = 2.4% = = 0.024, and t = 20 years.

A = $69457.89.

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a) Method of moments can be used to estimate α by equating the first two moments (sample mean and variance) with their theoretical counterparts and solving for α.

b) The MLE of α satisfies the equation: Ψ(3α) − Ψ(α) + 2nΣ[ln(Xi) − ln(1 − Xi)] = 0, where Ψ is the digamma function.

c) The asymptotic variance of the MLE is (9n[Ψ'(3α) − Ψ'(α)])^(-1), where Ψ' is the trigamma function.

a) The method of moments involves equating the first two moments of the distribution with their sample counterparts and solving for the parameter α. Setting the theoretical mean and variance of the given distribution equal to their sample counterparts and solving for α, we get α = (4n − 1)/(9n − 2).

b) The log-likelihood function for the given distribution is l(α) = n[ln(Γ(3α)) − ln(Γ(α)) − ln(Γ(2α))] + (α − 1)Σ[ln(Xi) + 2ln(1 − Xi)]. Taking the derivative of l(α) with respect to α and equating it to zero, we get the MLE of α as the solution to the equation: Ψ(3α) − Ψ(α) + 2nΣ[ln(Xi) − ln(1 − Xi)] = 0, where Ψ is the digamma function.

c) The asymptotic variance of the MLE can be found using the Fisher information. The Fisher information is given by I(α) = −n[Ψ''(α) + 2Ψ''(2α)], where Ψ'' is the polygamma function. The asymptotic variance of the MLE is then (I(α)^(-1)), which simplifies to (9n[Ψ'(3α) − Ψ'(α)])^(-1), where Ψ' is the trigamma function.

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Experimental Probability = (1/10) + (4/10) - (1/10) = 4/10 = 2/5

Theoretical Probability = (3/8) + (4/8) - (2/8) = 5/8

1.

Event = Land on Yellow

Notation = Y

Experimental Probability = 3/10

Here, Number of ways to have yellow = 3 [8Y, 2Y, 8Y]

Theoretical Probability = 2/8 = 1/4

Here, Total possible ways = 8; Number of ways to have yellow = 2 [2, 8]

2.

Event = Land on blue and 3

Notation = 3B

Experimental Probability = 1/8 [3B]

Theoretical Probability = 1/8

3.

Event = Landing on a Blue or a Pink

Notation = B or P

Experimental Probability = 7/10

Theoretical Probability = 6/8 = 3/4

4.

Event = Landing on a pink or an odd number

Notation = P or odd

Experimental Probability = (1/10) + (4/10) - (1/10) = 4/10 = 2/5

Theoretical Probability = (3/8) + (4/8) - (2/8) = 5/8

5.

Event = Landing of yellow and odd

Notation = Y and Odd

Experimental Probability = 0

Theoretical Probability = 0

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(a) The probability that the sample proportion will be within +0.03 of the population proportion is 0.7242.

(b) The probability that the sample proportion will be within +0.05 of the population proportion is 0.9312.

(a) The standard error of the sample proportion is given by:

SE = √[p(1-p)/n]

where p = population proportion, n = sample size

SE = √[0.7(1-0.7)/300] = 0.0274

To find the probability that the sample proportion will be within +0.03 of the population proportion, we need to find the z-scores for the upper and lower limits of the interval and then find the probability between those z-scores using a standard normal distribution table. The z-score for +0.03 is:

z = (0.03)/0.0274 = 1.09

The z-score for -0.03 is -1.09 (since it is the same distance from the mean but in the opposite direction). Thus, we need to find the probability between -1.09 and 1.09:

P(-1.09 < z < 1.09) = P(z < 1.09) - P(z < -1.09)

Using a standard normal distribution table, we find:

P(z < 1.09) = 0.8621

P(z < -1.09) = 0.1379

Therefore, the probability that the sample proportion will be within +0.03 of the population proportion is:

0.8621 - 0.1379 = 0.7242 (rounded to four decimal places)

(b) Using the same formula for standard error, we get:

SE = √[0.7(1-0.7)/300] = 0.0274

To find the probability that the sample proportion will be within +0.05 of the population proportion, we need to find the z-scores for the upper and lower limits of the interval and then find the probability between those z-scores using a standard normal distribution table. The z-score for +0.05 is:

z = (0.05)/0.0274 = 1.82

The z-score for -0.05 is -1.82 (since it is the same distance from the mean but in the opposite direction). Thus, we need to find the probability between -1.82 and 1.82:

P(-1.82 < z < 1.82) = P(z < 1.82) - P(z < -1.82)

Using a standard normal distribution table, we find:

P(z < 1.82) = 0.9656

P(z < -1.82) = 0.0344

Therefore, the probability that the sample proportion will be within +0.05 of the population proportion is:

0.9656 - 0.0344 = 0.9312 (rounded to four decimal places)

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Maximum Profit = P(x). Number of Units = x * 1000. To find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, we need to use the profit equation: Profit = Revenue - Cost

Let's assume that the profit equation is given by:
P(x) = R(x) - C(x)
where x is the number of units produced and sold in thousands of units.
To find the maximum profit, we need to find the value of x that maximizes the profit function P(x). This can be done by taking the derivative of P(x) with respect to x and setting it equal to zero:
P'(x) = R'(x) - C'(x) = 0
where R'(x) and C'(x) are the first derivatives of R(x) and C(x), respectively.
Solving for x, we get:
x = (R'(x) - C'(x)) / (2C''(x))
where C''(x) is the second derivative of C(x).
Once we have found the value of x that maximizes the profit function, we can find the maximum profit by plugging it back into the profit equation:
Maximum Profit = P(x)
To find the number of units that must be produced and sold in order to yield the maximum profit, we simply need to plug the value of x into the production function:
Number of Units = x * 1000 (since x is in thousands of units)
So, to summarize: To find the maximum profit, we need to take the derivative of the profit function, set it equal to zero, and solve for x. Then we plug this value of x back into the profit function to get the maximum profit. To find the number of units that must be produced and sold in order to yield the maximum profit, we simply need to multiply the value of x by 1000.

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Answer: 130 miles every week

Step-by-step explanation:

26*5=130

Round trip means from home to work and back home

he

volume

of the parallelepiped is 235 cubic units.

V = |u · (v × w)|

where · represents the dot product and × represents the

cross product

.

First, we need to find the cross product of v and w:

v × w = (i+j+9k) × (i+4j+9k)
     = (36i - 7j - 3k)

Next, we take the dot product of u with the cross product of v and w:

u · (v × w) = (7i+2j+k) · (36i - 7j - 3k)
           = 252 - 14 - 3
           = 235

Finally, we take the absolute value of this result to get the volume:

V = |u · (v × w)| = |235| = 235 cubic units.

Therefore, the volume of the parallelepiped is 235 cubic units.

To find the volume of the

parallelepiped

with adjacent edges u, v, and w, you need to calculate the scalar triple product of these vectors. The scalar triple product is the absolute value of the

determinant

of the matrix formed by the components of the three vectors.

Given vectors:
u = 7i + 2j + k
v = i + j + 9k
w = i + 4j + 9k

Step 1: Write the matrix using the components of u, v, and w:
| 7  2  1 |
| 1  1  9 |
| 1  4  9 |

Step 2: Calculate the determinant of the matrix:
7 * (1 * 9 - 4 * 9) - 2 * (1 * 9 - 1 * 9) + 1 * (1 * 4 - 1 * 1)

Step 3: Simplify the expression:
7 * (9 - 36) - 2 * (9 - 9) + (4 - 1)

Step 4: Calculate the result:
7 * (-27) - 0 + 3

Step 5: Find the absolute value of the result:
|-189 + 3| = |-186| = 186

The volume of the parallelepiped is 186 cubic units.

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To solve this problem, we can use the principle of inclusion-exclusion. First, we can count the total number of arrangements of the letters in "tamely," which is 6! = 720.

Next, we can count the number of arrangements where t is before a, which is 5! (since we treat ta as a single unit) multiplied by the 2 ways to arrange the remaining letters, which is 2*4! = 48. Similarly, we can count the number of arrangements where a is before m or m is before e, which is also 48.

However, we have double-counted the arrangements where both t is before a and a is before m, or where both t is before a and m is before e, or where both a is before m and m is before e.

Each of these arrangements can be counted as 4! = 24. Therefore, the total number of arrangements that satisfy the conditions is 48+48+48-24-24-24+0 = 72. In summary, there are 72 arrangements of "tamely" with either t before a, or a before m, or m before e.

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A graph of the triangle after a dilation by scale factor 3 using the blue dot as the centre of enlargement is shown below.​

In Mathematics and Geometry, a dilation is a type of transformation which typically changes the size of a geometric object, but not its shape.

In order to dilate the coordinates of the preimage (right-angled triangle) by using a scale factor of 3 centered at the blue dot, the transformation rule would be represented this mathematical expression:

(x, y)  →  (k(x - a) + a, k(y - b) + b)

(x, y)  →  (3(x - a) + a, 3(y - b) + b)

In this scenario, the intersection of the three (3) medians would represent the centre of the given traingle;

AO ≅ 20D

BO ≅ 20E

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The values of x which are solutions to the given quadratic equation as required to be determined are; x = (1 ± i√11) / 2.

It follows from the task content that the given quadratic equation is to be solved for variable, x.

3x² + 4x - 5 = 5x² + 2x + 1;

By collect like terms and evaluating; we have that;

2x² - 2x + 6 = 0

By solving the equation by means of the formula method; we find that;

x = (1 ± i√11) / 2

Ultimately, the values of x which holds True are; x = (1 ± i√11) / 2.

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The curve is given by: y(t) = -4t + 4[tex]t^2[/tex] And the derivative is: y'(t) = -4 + 8t

(a) Given that (3, 12) is a local minimum, we know that the derivative of y(t) at t = 3 is zero. So, y'(3) = 0. This eliminates options (i) and (iv) for the first question.

Since y(3) = 12, the correct answer to the first question is (iii) y(3) = 12.

(b) To find y'(t), we take the derivative of y(t) with respect to t:

y'(t) = a + b

(c) We know that y(3) = 12, so we can substitute t = 3 and get:

y(3) = a(3) + b(3) = 12

We also know that y'(3) = 0, so we can substitute t = 3 into y'(t) and get:

y'(3) = a + b = 0

We now have two equations with two unknowns:

a(3) + b(3) = 12

a + b = 0

Solving for a and b, we get:

a = -4

b = 4

Therefore, the curve is given by:

y(t) = -4t + 4[tex]t^2[/tex]

And the derivative is:

y'(t) = -4 + 8t

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The set of parametric equations for the line is:

x = -13 - t

y = 1 - √(3)t

z = 3 + 2t

To find the set of parametric equations for the line tangent to the space curve r(t) = (-2 sin t, 2 cos t, 4 sin t) at the point P(-13, 1, 3), we need to find the derivative of r(t) and evaluate it at t = t₀, where t₀ is the value of t that corresponds to the point P.

The derivative of r(t) is:

r'(t) = (-2 cos t, -2 sin t, 4 cos t)

To find t₀, we need to solve the equation r(t₀) = P:

(-2 sin t₀, 2 cos t₀, 4 sin t₀) = (-13, 1, 3)

From the second component, we can see that cos t₀ = 1/2, which means t₀ = π/3 or t₀ = -π/3.

Substituting t₀ = π/3 into r'(t), we get:

r'(π/3) = (-2 cos(π/3), -2 sin(π/3), 4 cos(π/3)) = (-1, -sqrt(3), 2)

So the line tangent to the space curve at P has direction vector (-1, -√(3), 2), which means the set of parametric equations for the line is:

x = -13 - t

y = 1 - √(3)t

z = 3 + 2t

where t is a parameter that varies along the line.

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We can calculate the experimental probability of selling a can of tomato soup, and then use that probability to predict the number of tomato soup cans sold in the next 20 cans.

Step 1: Calculate the experimental probability of selling a can of tomato soup.
Probability = (Number of tomato soup cans sold) / (Total number of cans sold)
Probability = 6 / 12 = 0.5

Step 2: Use the probability to predict the number of tomato soup cans sold in the next 20 cans.
Expected number of tomato soup cans = Probability × Total number of cans
Expected number of tomato soup cans = 0.5 × 20 = 10

Based on the experimental probability, you should expect 10 of the next 20 cans sold to be tomato soup.

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