Directions Evaluate each limit or indicate that the limit does not exist. Show all steps needed to find your solution. Only work the problems that are assigned iress 10. lim x→1
​
x
​
−1
x 2
+2x+1
​
= ? 11. lim x→1
​
x 2
−2x+1
x 2
+2x+1
​
= ?

The length of another diagonal will be 3 inches.

The formula for a parallelogram relationship between its sides and diagonals is

(D1)² +  (D2)² = 2A² + 2B²

were

D1 represents one diagonal,

D2 represents the second diagonal,

A stand for one side and B stands for the adjacent side.

Putting the mentioned values in this formula will give -

= 7² +(D2)²  = 2*2² + 2*5²

= 49 + (D2)² = 2*4 + 2*25

= 49 + (D2)² = 8 + 50

= 49 + (D2)² = 58

= D2 = 3 inch

So finally, the length of the other diagonal will be 3 inches.

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This means that the total cost for drilling and maintaining the wells for 5 years would be $37,500.

The expression representing the cost (in dollars) to drill and maintain the wells for n years is given by:

34,500 + 540n

In the given expression, the constant term 34,500 represents the initial cost of drilling the wells, which includes expenses such as equipment, labor, and permits. The term 540n represents the cost of maintaining the wells for n years, with 540 being the annual maintenance cost per well.

Interpreting the expression:

The expression allows us to calculate the total cost of drilling and maintaining the wells for a given number of years, n. As the value of n increases, the cost will increase proportionally, reflecting the additional expenses incurred for maintenance over time.

For example, if we plug in n = 5 into the expression, we can calculate the cost of drilling and maintaining the wells for 5 years:

[tex]\(34,500 + 540 \times 5 = 37,500\).[/tex]

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The empirical probability that a person  will suffer both a loss of appetite and a loss of sleep is 5%.

First step is to find the Number of subjects who reported both adverse reactions

Number of subjects who reported both adverse reactions = 1,000 - (60 + 90 + 800)

Number of subjects who reported both adverse reactions = 50

Now let find the Empirical Probability

Empirical Probability = Number of subjects who reported both adverse reactions / Total number of test subjects

Empirical Probability = 50 / 1,000

Empirical Probability = 0.05 or 5%

Therefore the empirical probability is 5%.

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The function g(x) is discontinuous at x = -2 and x = 7. The correct choice is B) g(x) has two discontinuities. The lesser discontinuity can be removed by defining g to beat that value. The greater discontinuity cannot be removed.

The function g(x) is discontinuous at x = -2 and x = 7.

x = -2

The denominator of g(x) is equal to 0 at x = -2. This means that g(x) is undefined at x = -2. The discontinuity at x = -2 cannot be removed.

x = 7

The numerator of g(x) is equal to 0 at x = 7. This means that g(x) approaches ∞ as x approaches 7. The discontinuity at x = 7 can be removed by defining g(7) to be 3.

Choice

The correct choice is B. The lesser discontinuity can be removed by defining g(-2) to be 3. The greater discontinuity cannot be removed.

Explanation

The function g(x) is defined as follows:

g(x) = x + 2 / ([tex]x^2[/tex] - 5x - 14) = x + 2 / ((x - 7)(x + 2))

The denominator of g(x) is equal to 0 at x = -2 and x = 7. This means that g(x) is undefined at x = -2 and x = 7.

The discontinuity at x = -2 cannot be removed because the denominator of g(x) is equal to 0 at x = -2. However, the discontinuity at x = 7 can be removed by defining g(7) to be 3. This is because the two branches of g(x) approach the same value, 3, as x approaches 7.

The following table summarizes the discontinuities of g(x) and how they can be removed:

x Value of g(x) Can the discontinuity be removed?

-2 undefined No

7       3         Yes

Therefore, the correct choice is B.

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The answer is P(x < 85) = 0.1587

Given that the random variable x is normally distributed with mean μ=90 and standard deviation σ=5. We need to find the probability P(x < 85).

Normal Distribution

The normal distribution refers to a continuous probability distribution that has a bell-shaped probability density curve. It is the most important probability distribution, particularly in the field of statistics, because it describes many natural phenomena.

P(x < 85)Using z-score:

When a dataset follows a normal distribution, we can transform the data using z-scores so that it follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1, as shown below:z = (x - μ) / σ = (85 - 90) / 5 = -1P(x < 85) = P(z < -1)

We can find the area under the standard normal curve to the left of -1 using a z-table or a calculator.

Using a calculator, we can use the normalcdf function on the TI-84 calculator to find P(z < -1). The function takes in the lower bound, upper bound, mean, and standard deviation, and returns the probability of the z-score being between those bounds, as shown below:

normalcdf(-10, -1, 0, 1) = 0.1587

Therefore, P(x < 85) = P(z < -1) ≈ 0.1587 (to four decimal places).Hence, the answer is P(x < 85) = 0.1587 (rounded to four decimal places).

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It will take 12 seconds for Hudson and Knox to have run the same number of feet.

Let's first write the equation to represent the situation described in the problem.

Let's assume it takes t seconds for Hudson and Knox to run the same number of feet. In that time, Hudson will have run a distance of 8.8t feet, and Knox will have run a distance of 30 + 6.3t feet. Since they are running the same distance, we can set these two expressions equal to each other:

8.8t = 30 + 6.3t

Now we can solve for t:

8.8t - 6.3t = 30

2.5t = 30

t = 12

Therefore, it will take 12 seconds for Hudson and Knox to have run the same number of feet.

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The probability that a particular book is free from misprints is 0.2231. option D is correct.

The average number of misprints per page (λ) is given as 1.5.

The probability of having no misprints (k = 0) can be calculated using the Poisson probability mass function:

[tex]P(X = 0) = (e^{-\lambda}\times \lambda^k) / k![/tex]

Substituting the values:

P(X = 0) = [tex](e^{-1.5} \times 1.5^0) / 0![/tex]

Since 0! (zero factorial) is equal to 1, we have:

P(X = 0) = [tex]e^{-1.5}[/tex]

Calculating this value, we find:

P(X = 0) = 0.2231

Therefore, the probability that a particular book is free from misprints is approximately 0.2231.

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Question 13: The average number of misprints per page of a book is 1.5.Assuming the distribution of number of misprints to be Poisson. The probability that a particular book is free from misprints,is B. 0.435 D. 0.2231 A. 0.329 C. 0.549​

Answer:

y = -4x

Step-by-step explanation:

We can find the equation of the line in slope-intercept form, whose general equation is given by:

y = mx + b, where

Finding the slope (m):

m = (y2 - y1) / (x2 - x1), where

Thus, we can plug in (0, 0) for (x1, y1) and (2, -8) for (x2, y2) to find m, the slope of the line:

m = (-8 - 0) / (2 - 0)

m = -8/2

m = -4

Thus, the slope of the line is-4.

Finding the y-intercept (b):

Thus, the equation of the line is y = -4x.

The marginal revenue when 12,000 video games have been produced and sold is 105 dollars.

Given function, p=420/(x-6)+4000

To find the marginal revenue function, R′(x)

As we know, Revenue, R = price x quantity

R = p * x (price, p and quantity, x are given in the function)

R = (420/(x-6) + 4000) x

Revenue function, R(x) = (420/(x-6) + 4000) x

Differentiating R(x) w.r.t x,

R′(x) = d(R(x))/dx

R′(x) = [d/dx] [(420/(x-6) + 4000) x]

On expanding and simplifying,

R′(x) = 420/(x-6)²

Now, to approximate the marginal revenue when 12,000 video games have been produced and sold, we need to put the value of x = 12

R′(12) = 420/(12-6)²

R′(12) = 105 dollars

Therefore, the marginal revenue when 12,000 video games have been produced and sold is 105 dollars.

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The Brady family received 12 letters on December 25th.

They received 9 magazines.

They received 3 bills.

They received 3 ads.

To solve this problem, we can use algebra. Let x be the number of bills the Brady family received. We know that they received three more magazines than bills, so the number of magazines they received is x + 3.

We also know that they received a total of 27 pieces of mail, so we can set up an equation:

x + (x + 3) + 12 + 3 = 27

Simplifying this equation, we get:

2x + 18 = 27

Subtracting 18 from both sides, we get:

2x = 9

Dividing by 2, we get:

x = 3

So the Brady family received 3 bills. Using x + 3, we know that they received 3 + 3 = 6 magazines. We also know that they received 12 letters and 3 ads. Therefore, the Brady family received 12 letters on December 25th.

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Answer:

Since the graph of a certain quadratic function has no x-intercepts, the discriminant has to be negative, so A and D are possible values for the discriminant.

When multiplied, 5 1/6 and -2/5 equals -31/15.

To multiply 5 1/6 by -2/5, we first need to convert the mixed number to an improper fraction:

5 1/6 = (6 x 5 + 1) / 6 = 31/6

Now we can multiply the fractions:

(31/6) x (-2/5) = -(62/30)

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (2):

-(62/30) = -31/15

Therefore, when multiplied, 5 1/6 and -2/5 equals -31/15.

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Solution to initial value problem is u = (125/19)e^(20t) + (53/19)e^(-18t)

Given differential equation is

2d²y/dt² - 2dy/dt + y = 0;

y(0) = 4; y'(0) = 1.

And another differential equation is

y'' - 2y' + y = 343e^(8t);

u(0) = 8,

u'(0) = 6.

For the first differential equation,Let us find the characteristic equation by assuming

y = e^(mt).d²y/dt²

= m²e^(mt),

dy/dt = me^(mt)

Substituting these values in the given differential equation, we get

2m²e^(mt) - 2me^(mt) + e^(mt) = 0

Factorizing, we get

e^(mt)(2m - 1)² = 0

The characteristic equation is 2m - 1 = 0 or m = 1/2

Taking the first case 2m - 1 = 0

m = 1/2

Since this root is repeated twice, the general solution is

y = (c1 + c2t)e^(1/2t)

Differentiating the above equation, we get

dy/dt = c2e^(1/2t) + (c1/2 + c2/2)te^(1/2t)

Applying the initial conditions,

y(0) = 4c1 = 4c2 = 4

The solution is y = (4 + 4t)e^(1/2t)

For the second differential equation,

Let us find the characteristic equation by assuming

u = e^(mt).

u'' = m²e^(mt);

u' = me^(mt)

Substituting these values in the given differential equation, we get

m²e^(mt) - 2me^(mt) + e^(mt) = 343e^(8t)

We have e^(mt) commonm² - 2m + 1 = 343e^(8t - mt)

Dividing throughout by e^(8t), we get

m²e^(-8t) - 2me^(-8t) + e^(-8t) = 343e^(mt - 8t)

Setting t = 0, we get

m² - 2m + 1 = 343

Taking square roots, we get

(m - 1) = ±19

Taking first case m - 1 = 19 or m = 20

Taking the second case m - 1 = -19 or m = -18

Substituting the roots in the characteristic equation, we get

u1 = e^(20t); u2 = e^(-18t)

The general solution is

u = c1e^(20t) + c2e^(-18t)

Differentiating the above equation, we get

u' = 20c1e^(20t) - 18c2e^(-18t)

Applying the initial conditions,

u(0) = c1 + c2 = 8u'(0) = 20c1 - 18c2 = 6

Solving the above equations, we get

c1 = 125/19 and c2 = 53/19

Hence, the solution is

u = (125/19)e^(20t) + (53/19)e^(-18t)

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The given information describes a parabola with vertex at (4,3), axis of symmetry with equation y=3, and a latus rectum length of 4. The value of 4p is positive.

1. The axis of symmetry is a horizontal line passing through the vertex, so the equation y=3 represents the axis of symmetry.

2. Since the latus rectum length is 4, we know that the distance between the focus and the directrix is also 4.

3. The focus is located on the axis of symmetry and is equidistant from the vertex and directrix, so it has coordinates (4+2, 3) = (6,3).

4. The directrix is also a horizontal line and is located 4 units below the vertex, so it has the equation y = 3-4 = -1.

5. The distance between the vertex and focus is p, so we can use the distance formula to find that p = 2.

6. Since 4p>0, we know that p is positive and thus the parabola opens to the right.

7. Finally, the equation of the parabola in standard form is (y-3)^2 = 8(x-4).

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When finding the difference between 74 and 112, a student might say, "First, I added 6 onto 74 to get a number that ends in 0, specifically 80, to get to 100 because that's another ten."

To find the difference between 74 and 112, the student is using a strategy of breaking down the numbers into smaller parts and manipulating them to simplify the subtraction process. In this case, the student starts by adding 6 onto 74, resulting in 80. By doing so, the student is aiming to create a number that ends in 0, which is closer to 100 and represents another ten. This approach allows for an easier mental calculation when subtracting 80 from 112 since it involves subtracting whole tens instead of dealing with more complex digit-by-digit subtraction.

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The required roots of the given expressions are:

(1) (1/√2 + i/√2), (-1/√2 + i/√2), (-1/√2 - i/√2), (1/√2 - i/√2).

(2)1

(3) [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].

Formula used:For finding roots of a complex number `a+bi`,where `a` and `b` are real numbers and `i` is an imaginary unit with property `i^2=-1`.

If `r(cosθ + isinθ)` is the polar form of the complex number `a+bi`, then its roots are given by:r^(1/n) [cos(θ+2kπ)/n + isin(θ+2kπ)/n],where `n` is a positive integer and `k = 0,1,2,...,n-1.

Calculations:

(1) (-16)^(1/4)

This expression (-16)^(1/4) can be written as [16 × (-1)]^(1/4).

Therefore (-16)^(1/4) = [16 × (-1)]^(1/4) = 2^(1/4) × [(−1)^(1/4)] = 2^(1/4) × [cos((π + 2kπ)/4) + isin((π + 2kπ)/4)],where k = 0,1,2,3.

Therefore (-16)^(1/4) = 2^(1/4) × [(1/√2) + i(1/√2)], 2^(1/4) × [(−1/√2) + i(1/√2)],2^(1/4) × [(−1/√2) − i(1/√2)], 2^(1/4) × [(1/√2) − i(1/√2)].

Hence, the roots of (-16)^(1/4) are (1/√2 + i/√2), (-1/√2 + i/√2), (-1/√2 - i/√2), (1/√2 - i/√2).

(2) 1^(1/5)

This expression 1^(1/5) can be written as 1^[1/(2×5)] = 1^(1/10).

Now, 1^(1/10) = 1 because any number raised to power 0 equals 1.

Hence, the only root of 1^(1/5) is 1.

(3) i^(1/4).

Now, i^(1/4) can be written as (cos(π/2) + isin(π/2))^(1/4).Now, the modulus of i is 1 and its argument is π/2.
Therefore, its polar form is: 1(cosπ/2 + isinπ/2).

Therefore i^(1/4) = 1^(1/4)[cos(π/2 + 2kπ)/4 + isin(π/2 + 2kπ)/4], where k = 0, 1,2,3.

Therefore i^(1/4) = [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].

Therefore, the roots of i^(1/4) are [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].

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According to the Empirical Rule, the percentage of values that fall within one standard deviation of the mean is approximately 68%.

The percentage of values that fall within two standard deviations of the mean is approximately 95%. The percentage of values that fall within three standard deviations of the mean is approximately 99.7%. The body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.21 °F and a standard deviation of 0.69 °F. Using the Empirical Rule, we need to determine the approximate percentage of healthy adults with body temperatures within 2 standard deviations of the mean, or between 96.3 °F and 99.59 °F, as well as the percentage of healthy adults with body temperatures between 96.14 °F and 100.28 °F. The Empirical Rule is based on the normal distribution of data, and it states that the percentage of values that fall within one, two, and three standard deviations of the mean is approximately 68%, 95%, and 99.7%, respectively. Thus, we can use the Empirical Rule to solve the problem. For part a, the range of body temperatures within two standard deviations of the mean is given by:

98.21 - 2(0.69) = 96.83 to 98.21 + 2(0.69) = 99.59.

Therefore, the percentage of healthy adults with body temperatures within this range is approximately 95%. For part b, the range of body temperatures between 96.14 and 100.28 is more than two standard deviations away from the mean. Therefore, we cannot use the Empirical Rule to determine the approximate percentage of healthy adults with body temperatures in this range. However, we can estimate the percentage by using Chebyshev's Theorem. Chebyshev's Theorem states that for any data set, the percentage of values that fall within k standard deviations of the mean is at least 1 - 1/k2, where k is any positive number greater than 1. Therefore, the percentage of healthy adults with body temperatures between 96.14 and 100.28 is at least 1 - 1/32 = 1 - 1/9 = 8/9 = 0.8889, or approximately 89%.

Approximately 95% of healthy adults in this group have body temperatures within 2 standard deviations of the mean, or between 96.83 °F and 99.59 °F. The percentage of healthy adults with body temperatures between 96.14 °F and 100.28 °F cannot be determined exactly using the Empirical Rule, but it is at least 89% according to Chebyshev's Theorem.

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Mean absolute deviation (MAD) is a measure of variability that indicates the average distance between each observation and the mean of the data set.

The MAD is calculated by adding the absolute values of the deviations from the mean and dividing by the number of observations. The MAD is always a non-negative value

In general, when data has more dispersion, the MAD will come out to a larger number. This is because the larger the dispersion of data, the greater the differences between the data points and the mean, which leads to a larger sum of deviations when calculating MAD. Hence, it can be concluded that data with more dispersion will result in a larger MAD. The range, on the other hand, is simply the difference between the largest and smallest data points in the data set. This means that the range is only dependent on two observations and is therefore sensitive to extreme values. The MAD, on the other hand, considers all of the observations in the data set, so it is more resistant to outliers and extreme values. This means that it is possible for the range to come out to a large number and for the MAD to come out to a much smaller number with the same set of data. If the data set has a few extreme values that increase the range, but the other values are relatively close to each other and the mean, then the MAD will come out to a much smaller number.

MAD is a more robust measure of variability than range, as it takes into account all the observations in the data set, making it more resistant to extreme values. Additionally, a larger dispersion of data will result in a larger MAD, while the range is more sensitive to extreme values. Hence, it is possible for the range to come out to a large number and for the MAD to come out to a much smaller number with the same set of data.

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xn+1 = (18xn + 53) mod 4913; x1 = 4600 We are given that the value of x1 is 4600 and we are to find the value of x0.Let's substitute the given value of x1 in the LCG equation and solve for x0. Thus,x2 = (18 * 4600 + 53) mod 4913x2 = 82853 mod 4913x2 = 1427... and so on.

Substituting x2 in the equation,

x3 = (18 * 1427 + 53) mod 4913x3 = 25751 mod 4913x3 = 2368...

and so on.Substituting x3 in the equation,

x4 = (18 * 2368 + 53) mod 4913x4 = 42657 mod 4913x4 = 1504...

and so on.This is a process of backward iteration of LCG. Since it is a backward iteration, x0 is the last generated random number before x1. So x0 is the random number generated after x4. Hence, x0 = 4600. We have been provided with a linear congruential generator (LCG), which is defined by the equation:xn+1 = (a xn + c) mod m...where xn is the nth random number, xn+1 is the (n+1)th random number, and a, c, and m are constants.Let's substitute the given values in the above equation,

xn+1 = (18 xn + 53) mod 4913; x1 = 4600

We can use backward iteration to solve for x0. In backward iteration, we start with the given value of xn and move backward in the sequence until we find the value of x0.Let's use the backward iteration to find the value of x0. Thus,

x2 = (18 * 4600 + 53) mod 4913x2 = 82853 mod 4913x2 = 1427...

and so on.Substituting x2 in the equation,

x3 = (18 * 1427 + 53) mod 4913x3 = 25751 mod 4913x3 = 2368...

and so on.Substituting x3 in the equation,

x4 = (18 * 2368 + 53) mod 4913x4 = 42657 mod 4913x4 = 1504...

and so on.The last generated random number before x1 is x0. Hence, x0 = 4600.Therefore, the value of x0 is 4600. This is the solution.

Thus, we can conclude that the value of x0 is 4600. We have solved this by backward iteration of LCG. This method involves moving backward in the sequence of random numbers until we find the value of x0.

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The polynomial solution for deg p = 3 is p(t, x) = At³ + Bx³ + Ct² + Dx² - 3At² - 2Ct - 3Bx² - 2Dx, where A, B, C, and D are constants.

(a) Case: deg p ≤ 2

Let's assume p(t, x) = At² + Bx² + Ct + Dx + E, where A, B, C, D, and E are constants.

Substituting p(t, x) into the wave equation, we have:

(p_tt) = 2A,

(p_zz) = 2B,

(p_t) = 2At + C,

(p_z) = 2Bx + D.

Therefore, the wave equation becomes:

2A = 2B.

This implies that A = B.

Next, we consider the terms involving t and x:

2At + C = 0,

2Bx + D = 0.

From the first equation, we get C = -2At. Substituting this into the second equation, we have D = -4Bx.

Finally, we have the constant term:

E = 0.

So, the polynomial solution for deg p ≤ 2 is p(t, x) = At² + Bx² - 2At - 4Bx, where A and B are constants.

(b) Case: deg p = 3

Let's assume p(t, x) = At³ + Bx³ + Ct² + Dx² + Et + Fx + G, where A, B, C, D, E, F, and G are constants.

Substituting p(t, x) into the wave equation, we have:

(p_tt) = 6At,

(p_zz) = 6Bx,

(p_t) = 3At² + 2Ct + E,

(p_z) = 3Bx² + 2Dx + F.

Therefore, the wave equation becomes:

6At = 6Bx.

This implies that A = Bx.

Next, we consider the terms involving t and x:

3At² + 2Ct + E = 0,

3Bx² + 2Dx + F = 0.

From the first equation, we get E = -3At² - 2Ct. Substituting this into the second equation, we have F = -3Bx² - 2Dx.

Finally, we have the constant term:

G = 0.

So, the polynomial solution for deg p = 3 is p(t, x) = At³ + Bx³ + Ct² + Dx² - 3At² - 2Ct - 3Bx² - 2Dx, where A, B, C, and D are constants.

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Two possible solutions to the equation W = 250 + 132.5T are:

T = 2, W = 515

T = 5, W = 912.5

To find possible solutions to the equation W = 250 + 132.5T, we need to substitute values for T and calculate the corresponding value of W.

Let's consider two possible values for T:

Solution 1: T = 2 (indicating 2 T-shirts in the box)

W = 250 + 132.5 * 2

W = 250 + 265

W = 515

So, one possible solution is T = 2 and W = 515.

Solution 2: T = 5 (indicating 5 T-shirts in the box)

W = 250 + 132.5 * 5

W = 250 + 662.5

W = 912.5

Therefore, another possible solution is T = 5 and W = 912.5.

Hence, two possible solutions to the equation W = 250 + 132.5T are:

T = 2, W = 515

T = 5, W = 912.5

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Answer:

1/3

Step-by-step explanation:

I assume that 2/5 and -7/5 are exponents.

3^(2/5) × 3^(-7/5) = 3^(2/5 + (-7/5)) = 3^(-5/5) = 3^(-1) = 1/3

Answer: 136/5

Step-by-step explanation: First simplify the fraction

1) 3 2/5 = 17/5

3 multiply by 5 and add 5 into it.

2) 3(-7/5) = 8/5

3 multiply by 5 and add _7 in it.

By multiplication of 2 fractions,

17/5 multiply 8/5 = 136/5

=136/5

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The amount Kori spent on rent this month if she spent a total of $2600 this month and 26% of her total budget is spent on rent is $676

Total amount Kori spent this month = $2600

Percentage spent on rent = 26%

Amount spent on rent = Percentage spent on rent × Total amount Kori spent this month

= 26% × $2600

= 0.26 × $2,600

= $676

Hence, Kori spent $676 on rent.

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To predict the cost of driving 1800 miles per month, substitute 1800 in the given function C(d) = 0.2d + 280C(1800) = 0.2 (1800) + 280= $640 per month. Therefore, the cost of driving 1800 miles per month is $640.

(b) Graph is shown below:(c)The slope of the graph represents the rate of change of the cost of driving a car per mile. The slope is given by 0.2, which means that for every mile Lynn drives, the cost increases by $0.2.The y-intercept of the graph represents the fixed cost (amount she pays even if she does not drive).

The y-intercept is given by 280, which means that even if Lynn does not drive the car, she has to pay $280 per month.The linear function gives a suitable model in this situation because the monthly cost increases as the number of miles driven increases.

This is shown by the positive slope of the graph. The fixed cost is also included in the function, which is represented by the y-intercept. Therefore, a linear function is a suitable model in this situation.

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The percentage error between log n! and the approximation when n = 10 is approximately 100%. This means that the approximation n log(n) - n is not very accurate for calculating log n! when n = 10.

The given approximation for log n! can be derived by expanding the logarithm of each term in the n! product and then approximating the resulting sum by an integral.

When we take the logarithm of each term in n!, we have log(n!) = log(1) + log(2) + log(3) + ... + log(n).

Using the properties of logarithms, this can be simplified to log(n!) = log(1 * 2 * 3 * ... * n) = log(1) + log(2) + log(3) + ... + log(n).

Next, we approximate this sum by an integral. We can rewrite the sum as an integral by considering that log(x) is approximately equal to the area under the curve y = log(x) between x and x+1. So, we approximate log(n!) by integrating the function log(x) from 1 to n.

∫(1 to n) log(x) dx ≈ ∫(1 to n) log(n) dx = n log(n) - n.

Therefore, the approximation for log n! is given by log(n!) ≈ n log(n) - n.

To calculate the percentage error between log n! and the approximation n log(n) - n when n = 10, we need to compare the values of these expressions and determine the difference.

Exact value of log(10!):

Using a calculator or logarithmic tables, we can find that log(10!) is approximately equal to 15.1044.

Approximation n log(n) - n:

Substituting n = 10 into the approximation, we have:

10 log(10) - 10 = 10(1) - 10 = 0.

Difference:

The difference between the exact value and the approximation is given by:

15.1044 - 0 = 15.1044.

Percentage Error:

To calculate the percentage error, we divide the difference by the exact value and multiply by 100:

(15.1044 / 15.1044) * 100 ≈ 100%.

Therefore, the percentage error between log n! and the approximation when n = 10 is approximately 100%. This means that the approximation n log(n) - n is not very accurate for calculating log n! when n = 10.

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The time required for the account balance to reach $8000 is 26.187 months(using compund interest), which is approximately equal to 2.18 years, after rounding to two decimal places.

Given,

Homer invests $3000 in an account paying 10% interest compounded monthly.

The interest rate, r = 10% per annum = 10/12% per month = 0.1/12

The amount invested, P = $3000.

The final amount, A = $8000

We need to find the time required for the account balance to reach $8000.

Let n be the number of months required to reach the balance of $8000.

Using the formula for compound interest,

we can calculate the future value of the investment in n months.

It is given by:A = P(1 + r/n)^(n*t)

Where, P is the principal or investment,

r is the annual interest rate,

t is the number of years,

and n is the number of times the interest is compounded per year.

Substituting the given values in the above formula, we get:

8000 = 3000(1 + 0.1/12)^(n)t

Simplifying this equation, we get:

(1 + 0.1/12)^(n)t = 8/3

Taking the log of both sides, we get:

n*t * log(1 + 0.1/12) = log(8/3)

Dividing both sides by log(1 + 0.1/12), we get:

n*t = log(8/3) / log(1 + 0.1/12)

Solving for n, we get:

n = (log(8/3) / log(1 + 0.1/12)) / t

Let us assume t = 1 year, and then we can calculate n as:

n = (log(8/3) / log(1 + 0.1/12)) / t

    = (log(8/3) / log(1 + 0.1/12)) / 1

     = 26.187 (approx.)

Therefore, the time required for the account balance to reach $8000 is 26.187 months, which is approximately equal to 2.18 years, after rounding to two decimal places.

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The point is located at (6,8). In the coordinate plane, the point is defined by an ordered pair of numbers, one for the x-coordinate and one for the y-coordinate. The first number represents the x-coordinate, and it specifies the horizontal position of the point, while the second number represents the y-coordinate and it specifies the vertical position of the point.

In this particular case, the point is located six units to the right of the y-axis and 8 units above the x-axis. This means that the x-coordinate is 6, and the y-coordinate is 8. In other words, the point is 6 units to the right of the y-axis, which means that it is on the positive x-axis, and it is 8 units above the x-axis, which means that it is in the positive y-direction.

Therefore, the point is at (6,8) which means that it is six units to the right of the y-axis and 8 units above the x-axis. This point is in the first quadrant of the coordinate plane, which is where both the x- and y-coordinates are positive.The coordinate plane is an essential tool in algebra that helps graphically represent functions and equations. It is divided into four quadrants by two perpendicular lines, the x-axis, and the y-axis. These axes intersect at the origin, which has the coordinates (0,0).

The location of a point in the coordinate plane is determined by its ordered pair of x- and y-coordinates. By plotting these points on the coordinate plane, we can graph lines, functions, and other mathematical concepts. The coordinate plane is also helpful in finding solutions to equations by identifying the points that satisfy the equation or inequality.

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The system as an augmented matrix is given by;[2 -3 1 | 5][-1 -6 -2 | -6][3 0 -1 | 4], the reduced row echelon form is;[1 0 0 | 1][0 1 0 | -1/3][0 0 1 | 23/24]. The solution set of the given system of equations is{(x,y,z) : x = 1, y = -1/3, z = 23/24}.

a. The system as an augmented matrix is given by;[2 -3 1 | 5][-1 -6 -2 | -6][3 0 -1 | 4]

b. Using Gaussian elimination to reduce the matrix into row echelon form;[2 -3 1 | 5][-1 -6 -2 | -6][3 0 -1 | 4]R1 <- R1/2[1 -3/2 1/2 | 5/2][-1 -6 -2 | -6][3 0 -1 | 4]R2 <- R2 + R1[1 -3/2 1/2 | 5/2][0 -15/2 -3/2 | -7/2][3 0 -1 | 4]R3 <- R3 - 3R1[1 -3/2 1/2 | 5/2][0 -15/2 -3/2 | -7/2][0 9/2 -5/2 | -5/2]R2 <- R2/(-15/2)[1 -3/2 1/2 | 5/2][0 1 1/5 | 7/30][0 9/2 -5/2 | -5/2]R1 <- R1 + (3/2)R2[1 0 8/5 | 29/15][0 1 1/5 | 7/30][0 9/2 -5/2 | -5/2]R3 <- R3 - (9/2)R2[1 0 8/5 | 29/15][0 1 1/5 | 7/30][0 0 -8/5 | -23/30]R3 <- R3/(-8/5)[1 0 8/5 | 29/15][0 1 1/5 | 7/30][0 0 1 | 23/24]R1 <- R1 - (8/5)R3R2 <- R2 - (1/5)R3[1 0 0 | 1][0 1 0 | -1/3][0 0 1 | 23/24].Therefore, the reduced row echelon form is;[1 0 0 | 1][0 1 0 | -1/3][0 0 1 | 23/24]

c. The solution set of the given system of equations is{(x,y,z) : x = 1, y = -1/3, z = 23/24}.This can be explained as follows;The above matrix is already in reduced row echelon form, thus; x = 1, y = -1/3 and z = 23/24. Therefore, the solution set of the given system of equations is{(x,y,z) : x = 1, y = -1/3, z = 23/24}.

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The statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.

Given data:

7, 11, 12, 18, 20, 22, 43.

To find out whether the last observation is an outlier or not, let's use the three standard deviation criterion.

That is, if a data value is more than three standard deviations from the mean, then it is considered an outlier.

The formula to find standard deviation is:

S.D = \sqrt{\frac{\sum_{i=1}^{N}(x_i-\bar{x})^2}{N-1}}

Where, N = sample size,

             x = each value of the data set,

    \bar{x} = mean of the data set

To find the mean of the given data set, add all the numbers and divide the sum by the number of terms:

Mean = $\frac{7+11+12+18+20+22+43}{7}$

          = $\frac{133}{7}$

          = 19

Now, calculate the standard deviation:

$(7-19)^2 + (11-19)^2 + (12-19)^2 + (18-19)^2 + (20-19)^2 + (22-19)^2 + (43-19)^2$= 1442S.D

                                                                                                                               = $\sqrt{\frac{1442}{7-1}}$

                                                                                                                                ≈ 10.31

To determine whether the value of x = 43 is an outlier, we need to compare it with the mean and the standard deviation.

Therefore, compute the z-score for the last observation (x=43).Z-score = $\frac{x-\bar{x}}{S.D}$

                                                                                                                      = $\frac{43-19}{10.31}$

                                                                                                                      = 2.32

Since the absolute value of z-score > 3, the value of x = 43 is considered an outlier.

Therefore, the statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.

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