Simplify sin 2
(t)
sin 2
(t)+cos 2
(t)
​
to an expression involving a single trig function with no fractions. If needed, enter squared trigonometric expressions using the following notation. Example: Enter sin 2
(t) as (sin(t)) 2
.
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9.  the number of ways to arrange k men and k women in a group is (2k)!.

a) To determine if the relation R is reflexive, we need to check if (a, a) ∈ R for all elements a ∈ A.

In this case, the relation R does not contain any pairs of the form (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), or (6, 6). Therefore, (a, a) ∈ R is not true for all elements a ∈ A, and thus the relation R is not reflexive.

b) To determine if the relation R is symmetric, we need to check if whenever (a, b) ∈ R, then (b, a) ∈ R.

In this case, we have (1, 2) and (2, 1) ∈ R, but we don't have (2, 1) ∈ R. Therefore, the relation R is not symmetric.

c) To determine if the relation R is transitive, we need to check if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

In this case, we have (1, 2) and (2, 1) ∈ R, but we don't have (1, 1) ∈ R. Therefore, the relation R is not transitive.

To summarize:

a) The relation R is not reflexive.

b) The relation R is not symmetric.

c) The relation R is not transitive.

8. a) To choose 12 individuals from a group of 19 firefighters and 16 police officers, we can use the combination formula. The number of ways to choose 12 individuals from a group of 35 individuals is given by:

C(35, 12) = 35! / (12!(35-12)!)

Simplifying the expression, we find:

C(35, 12) = 35! / (12!23!)

b) To choose a president, vice president, and secretary from the group of 16 police officers, we can use the permutation formula. The number of ways to choose these three positions is given by:

P(16, 3) = 16! / (16-3)!

Simplifying the expression, we find:

P(16, 3) = 16! / 13!

9. To arrange k men and k women in a group, we can consider them as separate entities. The total number of people is 2k.

The number of ways to arrange 2k people is given by the factorial of 2k:

(2k)!

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Approximately 68.27% of the scores are between 485 and 515.

Since the distribution of scores is not known to be normal, we can use the empirical rule, also known as the 68-95-99.7 rule, to estimate the percentage of scores between 485 and 515.

According to the empirical rule, for a normal distribution:

Approximately 68.27% of the data falls within one standard deviation of the mean.

Approximately 95.45% of the data falls within two standard deviations of the mean.

Approximately 99.73% of the data falls within three standard deviations of the mean.

Given that the sample mean is 500 and the standard deviation is 15, we can consider the interval of one standard deviation on either side of the mean.

Lower bound: 500 - 15 = 485

Upper bound: 500 + 15 = 515

Therefore, approximately 68.27% of the scores are between 485 and 515.

Approximately 68.27% of the scores fall between 485 and 515 based on the assumption that the distribution is approximately normal using the empirical rule.

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The product is zw = 15 ∠ 45° and the quotient is c = 3(cos 15° + i sin 15°).

Given that z = 5(cos 30° + i sin 30°) and w = 3(cos 15° + i sin 15°)

We need to find zw:

zw = z * wzw = 5(cos 30° + i sin 30°) * 3(cos 15° + i sin 15°)

zw = 15(cos 30° + i sin 30°)(cos 15° + i sin 15°)

zw = 15(cos 30°cos 15° − sin 30°sin 15° + i(sin 30°cos 15° + cos 30°sin 15°))

zw = 15(cos (30° + 15°) + i sin(30° + 15°))

zw = 15(cos 45° + i sin 45°)

zw = 15(cos 45° + i sin 45°)

zw = 15 ∠ 45°

Now, we need to find the quotient:

zw = 15 ∠ 45°c

= zw / zc

= 15 ∠ 45° / 5(cos 30° + i sin 30°)c

= 3(cos 15° + i sin 15°)

Hence, the product is zw = 15 ∠ 45° and the quotient is c = 3(cos 15° + i sin 15°).

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The present value of $41,230.00 due in nine months with an interest rate of 11.1% is approximately $37,725.66.

To calculate the present value of an amount due in the future, we need to discount it by considering the interest rate and the time period. The present value formula is:

Present Value = Future Value / (1 + interest rate)^time

Let's calculate the present value for the given scenario:

Future Value (FV): $41,230.00 (amount due in nine months)

Interest Rate (r): 11.1% (convert to decimal by dividing by 100, so r = 0.111)

Time (t): 9 months (expressed in years, so t = 9/12 = 0.75)

Using the formula, we can substitute the values:

Present Value = $41,230.00 / (1 + 0.111)^0.75

Calculating the value inside the parentheses:

(1 + 0.111)^0.75 ≈ 1.09337

Substituting this value back into the formula:

Present Value ≈ $41,230.00 / 1.09337

Calculating the present value:

Present Value ≈ $37,725.66

Therefore, the present value of $41,230.00 due in nine months with an interest rate of 11.1% is approximately $37,725.66.

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a. The value of n is 114.

b. The value of n is 2/9.

c. The given expression can be written as a single logarithm, which is In(84).

d. The given expression can be rewritten as log(24/5).

a. If logs (6) + loge (19) = log, (n) then n = log(6×19)=log(114)= log(n)

Thus, the value of n is 114.

b. If log(36) + log(n) = log(8) then n = (8/36)= (2/9)

Hence, the value of n is 2/9.

c. We know that In(12) + In(7) = In(12×7)

Therefore, the given expression can be written as a single logarithm, which is In(84).

d. We know that log(6) - log(5) + log(4) = log(24/5)

Therefore, the given expression can be rewritten as log(24/5).

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The expression can be simplified to In(84).

a. If logs(6) + loge(19) = log(n) then

n = logs(6) + loge(19) = loge(6) / loge(10) + loge(19) / loge(e) = loge(6.19) ≈ 2.119

b. If log(36) + log(n) = log(8) then

n =log(36) + log(n) - log(8) = log[36n/8] = log(4.5n)

c. Rewrite the following expression as a single logarithm.

In(12) + In(7) = In(12 x 7) = In(84)

d. Rewrite the following expression

3ln(2x-1) - 5ln(x+3) = ln[(2x-1)³/(x+3)⁵]

(a) If logs(6) + loge(19) = log(n), then n = 6 * 19 = 114.

(b) If log(36) + log(n) = log(8), we can use the logarithmic property of addition to combine the logarithms:

log(36 * n) = log(8)

This implies that 36 * n = 8, so n = 8/36 = 2/9.

(c) To rewrite In(12) + In(7) as a single logarithm, we can use the logarithmic property of addition:

In(12) + In(7) = In(12 * 7) = In(84) =

d. Therefore, the expression can be simplified to In(84).

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Naruto paid an interest of $55.25 to his credit card company for the purchase of his TV.

The interest Naruto paid for the purchase of his TV can be calculated using the simple interest formula:

Interest = Principal × Rate × Time

In this case, the principal is $850, the rate is 13% (or 0.13 as a decimal), and the time is 6 months (or 0.5 years). Plugging these values into the formula, we get:

Interest = $850 × 0.13 × 0.5 = $55.25

Therefore, Naruto paid an interest of $55.25 to his credit card company for the purchase of his TV.

The correct answer is option d. Naruto paid an interest of $55.25.

It's important to note that in this scenario, Naruto paid off the total cost of the TV after six months. Since the interest rate is annual, the interest is calculated based on the principal amount for the duration of six months. If Naruto had taken longer to pay off the TV or had not paid it off within a year, the interest amount would have been higher. However, in this case, Naruto paid off the TV before a year's time, so the interest amount is relatively low.

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To find  [tex]\( a_{1} \)[/tex] , given that [tex]\( S_{14}=168 \)[/tex]  and [tex]\( a_{14}=25 \)[/tex] we can use the formula for the sum of an arithmetic series. By substituting the known values into the formula, we can solve for [tex]a_{1}[/tex].

To find the value of [tex]a_{1}[/tex] we need to determine the formula for the sum of an arithmetic series and then use the given information to solve for [tex]a_{1}[/tex]

The sum of an arithmetic series can be calculated using the formula

[tex]S_{n}[/tex] = [tex]\frac{n}{2} (a_{1} + a_{n} )[/tex] ,  

where [tex]s_{n}[/tex] represents the sum of the first n terms [tex]a_{1}[/tex]  is the first term, and [tex]a_{n}[/tex] is the nth term.

Given that [tex]\( S_{14}=168 \) and \( a_{14}=25 \)[/tex]  we can substitute these values into the formula:

168= (14/2)([tex]a_{1}[/tex] + 25)

Simplifying the equation, we have:

168 = 7([tex]a_{1}[/tex] +25)

Dividing both sides of the equation by 7  

24 = [tex]a_{1}[/tex] + 25

Finally, subtracting 25 from both sides of the equation

[tex]a_{1}[/tex] = -1

Therefore, the first term of the arithmetic series is -1.

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Given,Face value (FV) of the note = $1000Issued date = April 21, 2005Rate of interest (r) = 5.5%Time period (t) = 90 daysNow, we have to find the maturity value of the note.To compute the maturity value, we have to find the interest and then add it to the face value (FV) of the note.

To find the interest, we use the formula,Interest (I) = (FV x r x t) / (100 x 365)where t is in days.Putting the given values in the above formula, we get,I = (1000 x 5.5 x 90) / (100 x 365)= 150.14So, the interest on the note is $150.14.Now, the maturity value (MV) of the note is given by,MV = FV + I= $1000 + $150.14= $1150.14Therefore, the maturity value of the note is $1150.14.

On computing the maturity value of a 90-day note with a face value of $1000 issued on April 21, 2005, at an interest rate of 5.5%, it is found that the maturity value of the note is $1150.14.

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We need to consider the general problem: \[-(ku')' + cu' + bu = f\]If we discretize by the finite element method with four elements.

On the first and last elements, we use linear shape functions, and on the middle two elements, we use quadratic shape functions. The resulting basis functions are given by:The basis functions ϕ1 and ϕ4 are linear while ϕ2 and ϕ3 are quadratic in nature. These basis functions are such that they follow the property of linearity and quadratic nature on each of the elements.

For the structure of the stiffness matrix K, we need to consider the discrete problem given by \[KU=F\]where U is the vector of nodal values of u, K is the stiffness matrix and F is the load vector. Considering the above equation and assuming constant values of k and c on each of the element we can write\[k_{1}\begin{bmatrix}1 & -1\\-1 & 1\end{bmatrix}+k_{2}\begin{bmatrix}2 & -2 & 1\\-2 & 4 & -2\\1 & -2 & 2\end{bmatrix}+k_{3}\begin{bmatrix}2 & -1\\-1 & 1\end{bmatrix}\]Here, the subscripts denote the element number. As we can observe, the resulting stiffness matrix K is symmetric and has a banded structure.

The element [1 1] and [2 2] are common to two elements while all the other elements are present on a single element only. Hence, we have four elements with five degrees of freedom. Thus, the stiffness matrix will be a 5 x 5 matrix and the structure of K is as follows:

$$\begin{bmatrix}k_{1}+2k_{2}& -k_{2}& & &\\-k_{2}&k_{2}+2k_{3} & -k_{3} & & \\ & -k_{3} & k_{1}+2k_{2}&-k_{2}& \\ & &-k_{2}& k_{2}+2k_{3}&-k_{3}\\ & & & -k_{3} & k_{3}+k_{2}\end{bmatrix}$$Conclusion:In this question, we considered the general problem given by -(ku')' + cu' + bu = f. We discretized it by the finite element method with four elements. On the first and last elements, we used linear shape functions, and on the middle two elements, we used quadratic shape functions. We sketched the resulting basis functions. The structure of the stiffness matrix K was then determined by ignoring boundary conditions. We observed that it is symmetric and has a banded structure.

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The probability of a child being a boy is 2  1, and a family plans to have 5 children, the odds against having all boys in this case are 31 to 1.

To calculate the odds against having all boys, we need to determine the probability of not having all boys and then calculate the odds based on that probability.

The probability of having all boys is given by the product of the individual probabilities for each child being a boy. In this case, the probability of a child being a boy is 1/2.

So, the probability of having all boys is (1/2) × (1/2) × (1/2) × (1/2)× (1/2) = 1/32.

The probability of not having all boys is 1 - (1/32) = 31/32.

The odds against having all boys can be calculated as the ratio of the probability of not having all boys to the probability of having all boys.

Odds against having all boys = (31/32) / (1/32) = 31.

Therefore, the odds against having all boys in this case are 31 to 1.

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Answer: x = 8

Step-by-step explanation:

The two lines are of the same length. We can write the equation 11 + 7x = 67 to represent this. We can simplify (solve) this equation by isolating our variable.

11 + 7x = 67 becomes:

7x = 56

We've subtracted 11 from both sides.

We can then isolate x again. By dividing both sides by 7, we get:

x = 8.

Therefore, x = 8.

The average woman burns 8.2 calories per minute while riding a bicycle. If she rides for 35 minutes, she will burn a total of 287 calories (option b).

To calculate the total number of calories burned, we multiply the number of minutes by the rate of calorie burn per minute. In this case, the woman burns 8.2 calories per minute, and she rides for 35 minutes. So, the total calories burned can be calculated as:

Total calories burned = Rate of calorie burn per minute × Number of minutes

                    = 8.2 calories/minute × 35 minutes

                    = 287 calories

Therefore, the correct answer is option b, 287 calories. This calculation assumes a constant rate of calorie burn throughout the duration of the ride.

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The coefficient "c" in the quadratic equation 2Q(x) = ax² + bx + c determines the vertical shift of the graph of y = Q(x).

The coefficient value "c" in the general formula 2Q(x) = ax² + bx + c affects the shape of the graphs of y = Q(x) by determining the vertical shift or displacement of the graph.

To understand the impact of the coefficient "c" on the graph, let's consider different scenarios:

When c > 0: If the value of "c" is positive, it will shift the graph of y = Q(x) vertically upward by c units. The graph will be higher compared to the graph of y = ax² + bx, but the overall shape of the parabola remains the same.

When c < 0: If the value of "c" is negative, it will shift the graph of y = Q(x) vertically downward by |c| units. The graph will be lower compared to the graph of y = ax² + bx, but the shape of the parabola remains unchanged.

When c = 0: If the value of "c" is zero, the graph of y = Q(x) will coincide with the graph of y = ax² + bx. The parabola will pass through the origin (0, 0), and there will be no vertical shift.

A positive value of "c" shifts the graph upward, a negative value shifts it downward, and when "c" is zero, there is no vertical shift. The coefficient "c" does not alter the shape of the parabola, but it affects its position on the y-axis.

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The roots of the quadratic equation obtained using the quadratic formula method are [tex]$\frac{4}{3}$ and $\frac{5}{3}$.[/tex]

The method used to factorize the expression -3x² + 8x-5 is completing the square method.

That coefficient is half of the coefficient of the x term squared; in this case, it is (8/(-6))^2 = (4/3)^2 = 16/9.

So, we have -3x² + 8x - 5= -3(x^2 - 8x/3 + 16/9 - 5 - 16/9)= -3[(x - 4/3)^2 - 49/9]

By simplifying the above expression, we get the final answer which is: -3(x - 4/3 + 7/3)(x - 4/3 - 7/3)

Now, we can use another method of factorization to check the answer is correct.

Let's use the quadratic formula.

The quadratic formula is given by:

                    [tex]$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$[/tex]

Comparing with our expression, we get a=-3, b=8, c=-5

Putting these values in the quadratic formula and solving it, we get

        [tex]$x=\frac{-8\pm \sqrt{8^2 - 4(-3)(-5)}}{2(-3)}$[/tex]

which simplifies to:

              [tex]$x=\frac{4}{3} \text{ or } x=\frac{5}{3}$[/tex]

Hence, the factors of the given expression are [tex]$(x - 4/3 + 7/3)(x - 4/3 - 7/3)$.[/tex]

The roots of the quadratic equation obtained using the quadratic formula method are [tex]$\frac{4}{3}$ and $\frac{5}{3}$.[/tex]

As we can see, both methods of factorisation gave the same factors, which proves that the answer is correct.

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the correct option is (d) {2, 3, 4, 5, 6, 7, 8, 9}.

The given universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1}. We are to find the complement of A.

The complement of A, A' is the set of elements that are not in A but are in the universal set. It is denoted by A'.

Therefore,

A' = {2, 3, 4, 5, 6, 7, 8, 9}

The complement of A is the set of all elements in U that do not belong to A. Since A contains only the element 1, we simply remove this element from U to obtain the complement.

Hence, A' = {2, 3, 4, 5, 6, 7, 8, 9}.

The complement of the set A = {1} is the set of all the remaining elements in the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

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The solution is: sin(A - B) = -0.7071. We can use the following formula to find sin(A - B): sin(A - B) = sin A cos B - cos A sin B

We are given that cos(A) = -7/25 and cos(B) = -12/13. Since A is in Quadrant III, we know that sin(A) is positive. Since B is in Quadrant II, we know that sin(B) is negative.

Plugging in the values, we get:

```

sin(A - B) = (-7/25) * (-12/13) - (-7/25) * (-13/13)

= 84/325 - 91/325

= -0.7071

```

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To determine the number of minutes it takes for the coffee to cool from 95°C to 77°C at a rate of 1°C per minute, we can set up an equation and solve for the unknown variable.

Let's proceed with the calculation:

Step 1: Determine the temperature difference:

The temperature of the coffee decreases from 95°C to 77°C, resulting in a temperature difference of 95°C - 77°C = 18°C.

Step 2: Calculate the time taken:

Since the coffee is cooling at a rate of 1°C per minute, the time taken for a temperature difference of 18°C is simply 18 minutes.

The coffee will take approximately 18 minutes to cool from 95°C to 77°C at a rate of 1°C per minute using equation

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Looking at the graph, the correct answer is in option B; An employee would earn $730 if they work for 27 hours on a project.

A scatterplot is a type of graphical representation that displays the relationship between two numerical variables. It is particularly useful for visualizing the correlation or pattern between two sets of data points.

We can see that we can trace the statement that is correct when we try to match each of the points on the graph. When we do that, we can see that 27 hours can be matched with $730 earnings.

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Answer:

Step-by-step explanation:

\begin{align*}

T_{1,1} &= \frac{1}{2} (f(0) + f(1)) \\

&= \frac{1}{2} (1 + \frac{1}{2}) \\

&= \frac{3}{4}

\end{align*}

Now, for two subintervals:

\begin{align*}

T_{2,1} &= \frac{1}{4} (f(0) + 2f(1/2) + f(1)) \\

&= \frac{1}{4} \left(1 + 2 \left(\frac{1}{1 + \left(\frac{1}{2}\right)^2}\right) + \frac{1}{1^2}\right) \\

&= \frac{1}{4} \left(1 + 2 \left(\frac{1}{1 + \frac{1}{4}}\right) + 1\right) \\

&= \frac{1}{4} \left(1 + 2 \left(\frac{1}{\frac{5}{4}}\right) + 1\right) \\

&= \frac{1}{4} \left(1 + 2 \cdot \frac{4}{5} + 1\right) \\

&= \frac{1}{4} \left(1 + \frac{8}{5} + 1\right) \\

&= \frac{1}{4} \left(\frac{5}{5} + \frac{8}{5} + \frac{5}{5}\right)

\end{align*}

Thus, the approximate value of the integral using Romberg's method is T_2,1, and this can also be used to obtain an approximate value of π.

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Architectural elements and their corresponding numbers are given below:

1. Nave (4): The main part of a church building where the congregation usually stands, as distinct from the chancel, choir, and sanctuary.

2 Side Aisles (2): The two narrow passages that run parallel to the nave on either side.

3 Narthex (1): A porch or vestibule leading to the nave of a church.

4 Apse (5): A semicircular recess, usually in the sanctuary of a church, used as a place for the altar.

5 Transept (3): A transverse part of any building, which lies across the main body of the building.

(5)The nave is the main body of the church where the congregation sits. Side Aisles are the two narrow passages that run parallel to the nave on either side. The transept is a transverse part of any building that lies across the main body of the building. The narthex is a porch or vestibule leading to the nave of a chruch. An apse is a semicircular recess, usually in the sanctuary of a church, used as a place for the altar.

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The probability that all four numbers generated by the computer program are greater than 10 is 1/16. This is obtained by multiplying the individual probabilities of each number being greater than 10, which is 1/2. The probability of randomly selecting four balls, one at a time, from a bag containing 20 balls numbered 1 to 20, and having all four of them be numbers greater than 10 is 168/517.

a) For each number generated by the computer program, the probability of it being greater than 10 is 10/20 = 1/2, since there are 10 numbers out of the total 20 that are greater than 10. Since the numbers are generated independently, the probability of all four numbers being greater than 10 is (1/2)^4 = 1/16.

b) When taking out the balls from the bag, the probability of the first ball being greater than 10 is 10/20 = 1/2. After removing one ball, there are 19 balls left in the bag, and the probability of the second ball being greater than 10 is 9/19.

Similarly, the probability of the third ball being greater than 10 is 8/18, and the probability of the fourth ball being greater than 10 is 7/17. Since the events are dependent, we multiply the probabilities together: (1/2) * (9/19) * (8/18) * (7/17) = 168/517.

Note: The probability in part b) assumes sampling without replacement, meaning once a ball is selected, it is not put back into the bag before the next selection.

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The question asks to estimate the rainfall at station D in the year 1980 based on the rainfall data of other stations. The answer is that station D's rainfall in 1980 can be estimated by comparing the rainfall patterns of all stations in the basin.

To estimate the rainfall at station D in 1980, we can analyze the relationship between the rainfall at different stations. In this case, we can observe that there is a correlation between the annual rainfall at stations A, B, C, and D. By comparing the rainfall patterns of these stations, we can make an estimation for station D.

In the given data, station D is inoperative in 1980, but the rainfall data for stations A, B, and C are available. By analyzing the rainfall patterns in previous years and considering the correlation between the stations, we can estimate the rainfall at station D in 1980 based on the rainfall data of the other stations in the basin.

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The price that will maximize your profits is $10 per dog, and you will walk 6 dogs. Increasing the price to $14 would result in a loss of 2 dogs per hour and potentially lower your overall profits.

To determine the price that maximizes your profits, you need to consider the relationship between price, quantity, and revenue. Increasing the price per dog to $14 would result in a loss of 2 dogs per hour, meaning you would only walk 4 dogs. At $10 per dog, you can walk 6 dogs.

Let's calculate the profits at each price point. At $14 per dog, with 4 dogs walked, your revenue would be 4 * $14 = $56. However, your costs would still be the same as when you walked 6 dogs, resulting in a higher cost per dog. This would likely lead to lower profits.

At $10 per dog, with 6 dogs walked, your revenue would be 6 * $10 = $60. Although the price is lower, the increased quantity of dogs walked allows you to generate higher overall revenue. Assuming your costs remain constant, this would likely result in higher profits.

Therefore, to maximize your profits, it would be advisable to keep the price at $10 per dog and continue walking 6 dogs per hour.

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The equation in slope-intercept form is y = x - 3

The points are drawn on the graph

The line is drawn

The solution area is shaded

The point (0, 0) is a point in the solution area

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

y > x - 3

The above expression is an inequality

And it is already in a slope-intercept form

Express as equation

y = x - 3

So, y = x - 3 is the slope-intercept form

We have

x = -1, 0 and 1

So, we have

y = -1 - 3 = -4

y = 0 - 3 = -3

y = 1 - 3 = -2

So, the table of values is

x   -1     0     1

y   -4     -3   -2

From the table, we have the points

(-1, -4), (0, -3) and (1, -2)

The points are on the graph and the line is also drawn

Also, the solution area is shaded

We have

y > x - 3

Set x = 0 and y = 0

So, we have

0 > 0 - 3

Evaluate

0 > -3

This is true

So, (0, 0) is a point in the solution area

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The composition (f∘g)(x) is equal to x^(1/39), and (g∘f)(x) is equal to (x^3)^(1/3), which simplifies to x. Therefore, (f∘g)(x) is not equal to (g∘f)(x).

The composition (f∘g)(x) means applying the function g(x) first and then applying f(x) to the result.

In this case, g(x) = 13√x, so substituting this into f(x) gives (f∘g)(x) = f(g(x)) = (13√x)³ = 13³ * (√x)³ = 13³ * x^(3/2) = x^(1/39).

On the other hand, the composition (g∘f)(x) means applying the function f(x) first and then applying g(x) to the result.

Using f(x) = x³, we have (g∘f)(x) = g(f(x)) = g(x³) = 13√(x³) = 13√(x^(3/2)) = (x^(3/2))^(1/2) = x.

Comparing (f∘g)(x) and (g∘f)(x), we can see that they are not equal. (f∘g)(x) = x^(1/39) and (g∘f)(x) = x.

Therefore, the compositions of the functions f(x) and g(x) do not yield the same result.

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The surface area of a sphere with a radius of 2 units is approximately 50.24 square units, and the surface area of a sphere with a radius of 4 units is approximately 200.96 square units.

To find the surface area of a sphere with radius rr, we can use the formula S(r)=4πr2S(r)=4πr2.

Let's substitute the given values into the formula:

For r=2r=2, we have:

S(2)=4π⋅22S(2)=4π⋅22

S(2)=4π⋅4S(2)=4π⋅4

S(2)=16πS(2)=16π

For r=4r=4, we have:

S(4)=4π⋅42S(4)=4π⋅42

S(4)=4π⋅16S(4)=4π⋅16

S(4)=64πS(4)=64π

Now, let's approximate the values to two decimal places using a calculator:

S(2)≈16⋅3.14≈50.24S(2)≈16⋅3.14≈50.24

S(4)≈64⋅3.14≈200.96S(4)≈64⋅3.14≈200.96

Therefore, S(2)≈50.24S(2)≈50.24 and S(4)≈200.96S(4)≈200.96 (rounded to two decimal places).

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Answer:

24 meters

Step-by-step explanation:

To find the span of the pitched roof, we can use the given ratio of rise to span. The ratio states that for every 3 units of rise, there are 8 units of span.

Given that the rise of the roof is 9 meters, we can set up a proportion to solve for the span:

(3 units of rise) / (8 units of span) = (9 meters) / (x meters)

Cross-multiplying, we get:

3 * x = 8 * 9

3x = 72

Dividing both sides by 3, we find:

x = 24

Therefore, the span of the pitched roof is 24 meters.

As per Sharia, any stock that is involved in the following activities is considered haram or non-permissible:Speculative and High-risk businesses; businesses that deal with any sort of prohibited substances like alcohol, tobacco, drugs, and more.

Mohammed wishes to buy some stocks in a reputable company with a 4% tobacco activity, a total debt of $30,000, total cash of $40,000, and a total asset of $100,000. Determine whether this stock is Sharia compliant so Mohammed can invest.According to the information given, the company has 4% tobacco activity. Thus, this stock is considered haram or non-permissible as per Sharia law because it involves activities related to tobacco.So, Mohammed cannot invest in this stock as it is not Sharia compliant.

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In order to determine if the stock is Sharia-compliant or not, we must first determine if the company's primary business activities are halal (permissible) or haram (impermissible).

In this case, the company's primary business activity is tobacco, which is considered haram (impermissible) according to Islamic principles. As a result, the stock is not considered Sharia-compliant, and Mohammed should not invest in it.

Islamic finance refers to financial activities that are consistent with Islamic law (Sharia). The primary goal of Islamic finance is to promote social welfare and economic development while adhering to the principles of fairness, justice, and transparency.

To achieve these goals, Islamic finance prohibits certain activities that are considered haram (impermissible), such as charging or paying interest (riba), engaging in speculative transactions (gharar), and investing in businesses that are involved in haram activities such as gambling or the production of alcohol or tobacco.

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Here are the steps to show that (S, *) is a group where a * b = a + b + ab for all a, b ∈ S. Let us take S as the set of all real numbers except -1.

Proof of Group Axioms for (S, *):Closure: Let a, b ∈ S, then a + b + ab ∈ S, because S is closed under multiplication and addition. So, S is closed under *.

Associativity: Let a, b, c ∈ S, then:  a * (b * c) = a * (b + c + bc) = a + (b + c + bc) + a(b + c + bc) = a + b + c + ab + ac + bc + abc = (a + b + ab) + c + (a + b + ab)c = (a * b) * c. So, * is associative on S.

Identity: Let e = 0 be the identity element of (S, *). Then, a * e = a + e + ae = a for all a ∈ S, because a + 0 + 0a = a. Therefore, e is an identity element of S.Inverse:

Let a ∈ S, then -1 ∈ S. Let b = -1 - a, then b ∈ S because S is closed under addition and -a is in S. Then a * b = a + b + ab = a + (-1 - a) + a(-1 - a) = -1, which is the additive inverse of -1.

Therefore, every element of S has an inverse under *.

So, (S, *) is a group where a * b = a + b + ab for all a, b ∈ S.  

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(a) The volume using the Riemann sum:V ≈ Σ[[tex](x_i * y_i)[/tex] * (Δx * Δy)] for i = 1 to m, j = 1 to n

(b) V ≈ Σ[[tex](x_m * y_m)[/tex] * (Δx * Δy)] for i = 1 to m, j = 1 to n

To estimate the volume of the solid that lies below the surface z = xy and above the given rectangle R = (x, y) | 10 ≤ x ≤ 16, 6 ≤ y ≤ 10, we can use the provided methods: (a) Riemann sum with m = 3, n = 2 using the upper right corner of each square, and (b) Midpoint Rule.

(a) Riemann Sum with Upper Right Corners:

First, let's divide the rectangle R into smaller squares. With m = 3 and n = 2, we have 3 squares in the x-direction and 2 squares in the y-direction.

The width of each x-square is Δx = (16 - 10) / 3 = 2/3.

The height of each y-square is Δy = (10 - 6) / 2 = 2.

Next, we'll evaluate the volume of each square by using the upper right corner as the sample point. The volume of each square is given by the height (Δz) multiplied by the area of the square (Δx * Δy).

For the upper right corner of each square, the coordinates will be [tex](x_i, y_i),[/tex] where:

[tex]x_1[/tex] = 10 + Δx = 10 + (2/3) = 10 2/3

x₂ = 10 + 2Δx = 10 + (2/3) * 2 = 10 4/3

x₃ = 10 + 3Δx = 10 + (2/3) * 3 = 12

y₁ = 6 + Δy = 6 + 2 = 8

y₂ = 6 + 2Δy = 6 + 2 * 2 = 10

Using these coordinates, we can calculate the volume for each square and sum them up to estimate the total volume.

V = Σ[Δz * (Δx * Δy)] for i = 1 to m, j = 1 to n

To calculate Δz, substitute the coordinates [tex](x_i, y_i)[/tex] into the equation z = xy:

Δz = [tex]x_i * y_i[/tex]

Now we can estimate the volume using the Riemann sum:

V ≈ Σ[[tex](x_i * y_i)[/tex] * (Δx * Δy)] for i = 1 to m, j = 1 to n

(b) Midpoint Rule:

The Midpoint Rule estimates the volume by using the midpoint of each square as the sample point. The volume of each square is calculated similarly to the Riemann sum, but with the coordinates of the midpoint of the square.

For the midpoint of each square, the coordinates will be [tex](x_m, y_m)[/tex], where:

[tex]x_m[/tex] = 10 + (i - 1/2)Δx

[tex]y_m[/tex] = 6 + (j - 1/2)Δy

V ≈ Σ[[tex](x_m * y_m)[/tex] * (Δx * Δy)] for i = 1 to m, j = 1 to n

Now that we have the formulas, we can calculate the estimates for both methods.

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