The radius of the Earth is 6370km, the atmospheric pressure at sea level is 1 bar and the density at sea level is 1.2 kg/m^3.

Estimate the mass of the atmosphere assuming the height of the atmosphere is 11km.

Answer:

The cost to rent a trailer for 2.9 ​hours is $27.2.

The cost to rent a trailer for 3 ​hours is $28.

The cost to rent a trailer for 8.5 ​hours is $72.

The cost to rent a trailer for 9 ​hours is $76.

Step-by-step explanation:

It is given that the charge to rent a trailer is $20 for up to 2 hours plus $8 per additional hour or portion of an hour.

Let x be the number of hours.

The cost to rent a trailer for x ​hours is defined as

[tex]C(x)=\begin{cases}20 & \text{ if } x\leq 2 \\ 20+8(x-2) & \text{ if } x>2 \end{cases}[/tex]

For x>2,

[tex]C(x)=20+8(x-2)[/tex]

Substitute x=2.9 in the cost function.

[tex]C(x)=20+8(2.9-2)=27.2[/tex]

The cost to rent a trailer for 2.9 ​hours is $27.2.

Substitute x=3 in the cost function.

[tex]C(x)=20+8(3-2)=28[/tex]

The cost to rent a trailer for 3 ​hours is $28.

Substitute x=8.5 in the cost function.

[tex]C(x)=20+8(8.5-2)=72[/tex]

The cost to rent a trailer for 8.5 ​hours is $72.

Substitute x=9 in the cost function.

[tex]C(x)=20+8(9-2)=76[/tex]

The cost to rent a trailer for 9 ​hours is $76.

All ordered​ pairs, in the form of (hours,​ cost) are (2.9, 27.2), (3,28), (8.5, 72) and (9,76).

The graph of all ordered​ pairs is shown below.

Final answer:

The costs to rent a trailer for 2.9 and 3 hours are both $28, for 8.5 and 9 hours are both $76. To plot this function, one must mark these costs against the rental hours, showing a flat rate for the first two hours and additional charges thereafter.

Explanation:

The question asks for the cost of renting a trailer for varying numbers of hours and then requires plotting a graph with these costs against the hours. The rental system has a flat rate of $20 for the first 2 hours, and an additional charge of $8 for each hour or part of an hour thereafter.

To graph all ordered pairs, plot points for each of the time periods mentioned with their corresponding costs. Note how the graph demonstrates incremental jumps after the first 2 hours, reflecting the additional $8 charge per hour or part thereof.

Answer:

a. [tex]\bar X[/tex] is distributed [tex]N(84;4)[/tex]

b. [tex]P(\bar X \geq 87.6) = 0.03593[/tex]

c. [tex]P(\bar X \leq 79.2) = 0.00820[/tex]

d. [tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]

Step-by-step explanation:

a.

The central limit theorem states that, for large n, the sampling distribution of the sample mean is approximately normal with mean [tex]\µ[/tex] and variance [tex]\frac{\sigma^2}{n}[/tex], then, the sample mean is distributed as a normal random variable with means [tex]\mu_{\bar X}=\mu=84[/tex] and variance [tex]\sigma^2_{\bar X}=\frac{\sigma^2}{n}=\frac{16^2}{64}=4[/tex].

b.

[tex]P(\bar X \geq 87.6) = 0.03593[/tex]

c.

[tex]P(\bar X \leq 79.2) = 0.00820[/tex]

d.

[tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]

Answer:

11.70%

Step-by-step explanation:

Given;

Interest paid = $44

Principle amount = $1,153

Time = 119 days = [tex]\frac{\textup{119}}{\textup{365}}\textup{days}[/tex]  = 0.326 years

Now,

the interest is calculated as:

interest = Principle × Rate of interest × Time

thus,

$44 = $1,153 × Rate of interest × 0.326

or

Rate of interest = 0.1170

or

in percentage = Rate × 100 = 0.1170 × 100 = 11.70%

Final answer:

The qualitative statements are A. The flower is red. and C. The candy was sour. The quantitative statements are B. The bug is 5cm long. and D. You have three sisters.

Explanation:

The qualitative statement refers to descriptions that do not involve numerical values or measurements. On the other hand, quantitative statements involve numerical values or measurements. Based on these definitions, the qualitative statements in the given options are:

A. The flower is red.

C. The candy was sour.

The quantitative statements in the given options are:

B. The bug is 5cm long.

D. You have three sisters.

Answer:

83%

Step-by-step explanation:

100% - 17% = 83%

Cutting the number of pages by 17% leaves 83% of the original number of pages.

After reducing the number of pages by 17%, the final version of the book contains 83% of the original number of pages.

If the author of a book was told to cut the number of pages by 17%, we want to determine what percent of the original pages remained in the final version. To do this, we subtract the percentage of the pages cut from 100% (which represents the original number of pages).

100% - 17% = 83%.

Therefore, after a 17% cut, 83% percent of the pages are left in the final version of the book.

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Answer:

Step-by-step explanation:

Michael Perez deposited a total of $2,000 with two savings institutions.

One pays interest at a rate of 5% per year whereas the other pays interest at a rate of 8% per year.

Let x denote the amount of money invested at 5%

y = 2000 - x ---(1)

Let y denote the amount of money invested at 8%

so 5x/100 + 8(2000-x)/100 = $130 ----(2)

Michael earned $130 in interest during a single year.

Answer:

The proof will be as follows

      Statement                                  Reason

1. -2(x+1) = 8                                      Given

2.  -2(x+1)/ -2 = 8/-2                          Division Property of equality

3. x+1 = -4                                          Simplifying

4. x+1-1 = -4-1                                      Subtraction property of Equality

5. x = -5                                             Simplifying

The option of Distributive property of equality will not be used ..

Final answer:

There are 108 students in the class, calculated using the data from the class survey with the principle of inclusion-exclusion.

Explanation:

To find out how many students are in the class, we should analyze the survey data provided. We know that:

29 students watched TV on Monday.

24 students watched TV on Tuesday.

25 students watched TV on Wednesday.

13 students watched TV on only Monday.

9 students watched TV on only Tuesday.

10 students watched TV on only Wednesday.

12 students watched TV on all three days.

14 students watched TV on both Monday and Tuesday.

Since 14 watched on both Monday and Tuesday, and 12 of those also watched on Wednesday, there are 14 - 12 = 2 who watched on Monday and Tuesday but not Wednesday.

We can now calculate the total number of students using the principle of inclusion-exclusion:

Start with the total number of students who watched TV each day: 29 + 24 + 25 = 78.

Subtract the students who were counted twice because they watched on two different days: 78 - 2 (from Monday and Tuesday) = 76.

Subtract the students who watched on all three days, as they have been counted three times: 76 - 12 = 64.

Add back the number of students who watched TV on all three days to account for their previous subtraction: 64 + 12 = 76.

Add the students who watched TV on only one specific day to avoid double-counting those who were included in the two and three days' viewership: 76 + 13 (Monday only) + 9 (Tuesday only) + 10 (Wednesday only) = 108.

Therefore, there are 108 students in the class.

Answer:

  80

Step-by-step explanation:

The three highest exam scores have an average of 80. To maintain an average of 80, the final exam score must be at least 80.

  (83 +67 +90)/3 = 240/3 = 80

_____

Essentially, the final exam score counts as two tests, and the lowest test score is thrown out.

The lowest score the student can obtain is 80

The student would take 5 tests. The total marks obtainable is 500( 100 x 5). The student wants to score at least 80 in the course. So she needs to have at least a total score of 400(80 x 5).

In this calculation 52, which is the lowest score would be excluded. This is because the professor would replace this score with her exam score. The student has the scores for 3 tests already.

Total scores of the student for the 3 tests = 83 + 67 + 90 = 240

Sum of scores on the remaining two tests = 400 - 240 = 160

Average score on the remaining two tests = 160/ 2 =  80

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Answer:

[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Step-by-step explanation:

Given : [tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]

To find : ℒ{f(t)} by first using a trigonometric identity ?

Solution :

First we solve the function,

[tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]

Applying trigonometric identity, [tex]\cos (A-B)=\cos A\cos B+\sin A\sin B[/tex]

[tex]f(t)=16(\cos t\cos (\frac{\pi}{6})+\sin t\sin(\frac{\pi}{6})[/tex]

[tex]f(t)=16(\frac{\sqrt3}{2}\cos t+\frac{1}{2}\sin t)[/tex]

[tex]f(t)=\frac{16}{2}(\sqrt3\cos t+\sin t)[/tex]

[tex]f(t)=8(\sqrt3\cos t+\sin t)[/tex]

We know, [tex]L(\cos at)=\frac{s}{s^2+a^2}[/tex] and [tex]L(\sin at)=\frac{a}{s^2+a^2}[/tex]

Applying Laplace in function,

[tex]L\{f(t)\}=8\sqrt3L(\cos t)+8L(\sin t)[/tex]

[tex]L\{f(t)\}=8\sqrt3(\frac{s}{s^2+1})+8(\frac{1}{s^2+1})[/tex]

[tex]L\{f(t)\}=\frac{8\sqrt3s+8}{s^2+1}[/tex]

[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Therefore, The Laplace transformation is [tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Final answer:

In this college-level mathematics question, the task is to find ℒ{f(t)} by employing a trigonometric identity. By rewriting the function f(t) = 16cos(t−π/6) in terms of sine and utilizing a trigonometric identity, we find ℒ{f(t)} = 16s / (s^2 + 1).

Explanation:

To find ℒ{f(t)} by utilizing a trigonometric identity, first rewrite the function f(t) = 16cos(t−π/6) in terms of sine. Use the trigonometric identity cos(a) = sin(a + π/2) to rewrite cos(t−π/6) as sin(t−π/6 + π/2). This simplifies to sin(t−π/3). Thus, ℒ{f(t)} = 16 * ℒ{cos(t−π/6)}

= 16 * ℒ{sin(t−π/3)} = 16 * (s / (s^2 + 1)).

Therefore, the answer is 16s / (s^2 + 1).

Answer:

The required factors are: x, (x + 6) and (x - 3).

Step-by-step explanation:

As per the question,

The given polynomial is:

[tex]x^{3}+3x^{2}-18x[/tex]

Now,

BY factorization, we get

[tex]x^{3}+3x^{2}-18x[/tex]

[tex]=x(x^{2}+3x-18)[/tex]

By splitting the mid-term, that is split 3x like:

3x = 6x - 3x

Therefore,

[tex]x(x^{2}+6x-3x-18)[/tex]

Now on further solving by taking common factor out, we get

[tex]=x[x(x+6)-3(x+6)][/tex]

[tex]=x(x+6)(x-3)[/tex]

Therefore, the given second polynomial (x - 4), is not a factor of given polynomial [tex]x^{3}+3x^{2}-18x[/tex].

Hence, the given polynomial has three factor x, (x + 6) and (x - 3).

Answer:

Let V a vector space. And B a subset of elements in V.

B is a basis for V if satisfies the following conditions:

1. V= span(B). It means that every element of V can be written as a finite linear combination of elements of B.

2. B is a linear independent subset.

Answer:

(a) [tex]p \wedge -q[/tex]

(b) [tex]\neg p \Rightarrow \neg q[/tex]

(c) [tex]q \wedge -p[/tex]

Step-by-step explanation:

(a) You drive over 65 miles per hour, but you do not get a speeding ticket, it can be represented by: [tex]p \wedge -q[/tex]

(b) If you do not drive over 65 miles per hour, then you will not get a speeding ticket, it can be represented by: [tex]\neg p \Rightarrow \neg q[/tex]

(c) You get a speeding ticket, but you did not drive over 65 miles per hour, can be represented by: [tex]q \wedge -p[/tex]

Final answer:

The expected time between successive mosquito bites is calculated using the probabilities of landing and biting, resulting in an average time of 10 seconds between bites.

Explanation:

The expected time between successive mosquito bites can be calculated by considering the probabilities of two independent events: a mosquito landing on your neck and a mosquito bite, given that it has landed. As per the data provided, a mosquito lands with a probability of 0.5 per second, and out of those, it will bite with a probability of 0.2. The probability of getting bitten by a mosquito that has landed is then 0.5 (probability of landing) × 0.2 (probability of biting) = 0.1 per second.

To find the expected time between successive bites, we need to consider the inverse of this probability, which tells us that on average, you can expect to get bitten every 1/0.1 or 10 seconds.

The expected time between bites is 10 seconds.

To find the expected time between successive mosquito bites, we need to consider the probability of a mosquito biting you at any given second.

Given the probabilities:

The combined probability that a mosquito both lands and bites in any given second is the product of the two probabilities:
[tex]P(bite) = P(land) * P(bite\ |\ land) = 0.5 * 0.2 = 0.1.[/tex]

To find the expected time between successive bites, we take the reciprocal of the probability of being bitten in a given second:
[tex]Expected time = \frac{1}{P(bite)} = \frac{1}{0.1} = 10\ seconds[/tex].

The expected time between successive mosquito bites is approximately 10 seconds.

Answer:

a)

Reduced Row Echelon:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]

Solution to the system:

[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]

b)

Reduced Row Echelon:

[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]

Solution to the system:  

[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]

x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.

Step-by-step explanation:

To find the reduced row echelon form of the matrices, let's use the Gaussian-Jordan elimination process, which consists of taking the matrix and performing a series of row operations. For notation, R_i will be the transformed column, and r_i the unchanged one.

a) [tex]\left[\begin{array}{cccc}0&4&7&0\\2&1&0&0\\0&3&1&-4\end{array}\right][/tex]

Step by step operations:

1. Reorder the rows, interchange Row 1 with Row 2, then apply the next operations on the new rows:

[tex]R_1=\frac{1}{2}r_1\\R_2=\frac{1}{4}r_2[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&3&1&-4\end{array}\right][/tex]

2. Set the first row to 1

[tex]R_3=-3r_2+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]

3. Write the system of equations:

[tex]x_1+\frac{1}{2}x_2=0\\x_2+\frac{7}{4}x_3=0\\x_3=-4[/tex]

Now you have the  reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:

[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]

b)

[tex]\left[\begin{array}{cccc}4&3&0&7\\8&6&2&-3\\4&3&2&-10\end{array}\right][/tex]

1. [tex]R_2=-2r_1+r_2\\R_3=-r_1+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]

2. Write the system of equations:

[tex]4x_1+3x_2=7\\2x_3=-17[/tex]

Now you have the reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:

[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]

x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.

Answer:

(x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])

Step-by-step explanation:

We have to find point of intersection of two lines.

the given equations of line are:

10x - 4y = 43 - (1)

-3x - 3y = -15 - (2)

Multiplying the first equation by 3 we have:

(10x - 4y = 43)×3 = 30x - 12 y = 129 - (3)

Multiplying second equation by 10 we have :

(-3x - 3y = -15)×10 = -30x -30y = -150 - (4)

Now, adding equation (3) and (4)  we have:

-42y = -21

⇒ y = [tex]\frac{1}{2}[/tex]

Now, putting this value of y in equation (1), we have

10x - 2 = 43

⇒ 10x = 45

⇒x = [tex]\frac{9}{2}[/tex]

Hence, the intersection of given two lines is (x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])

By DeMoivre's theorem,

[tex]\cos(4\theta)+i\sin(4\theta)=(\cos\theta+i\sin\theta)^4[/tex]

Expanding the right side gives

[tex]\cos^4\theta+4i\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta-4i\cos\theta\sin^3\theta+\sin^4\theta[/tex]

Equating imaginary parts tells us

[tex]\sin(4\theta)=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta[/tex]

(Not sure what you mean by sin(e) and cos(e)...)

Answer:

[tex]x\ =\ \dfrac{209}{30}[/tex]

[tex]y\ =\ \dfrac{29}{18}[/tex]

[tex]z\ =\ \dfrac{64}{15}[/tex]

Step-by-step explanation:

Given equations are

15x + 15y + 10z = 106

5x + 15y + 25z = 135

15x + 10y - 5z = 42

The augmented matrix by using above equations can be written as

[tex]\left[\begin{array}{ccc}15&15&10\ \ |106\\5&15&25\ \ |135\\15&10&-5|42\end{array}\right][/tex]

[tex]R_1\ \rightarrow\ \dfrac{R_1}{15}[/tex]

[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\5&15&25|135\\15&15&-5|42\end{array}\right][/tex]

[tex]R_1\rightarrowR_2-5R1\ and\ R_3\rightarrow\ R_3-15R_1[/tex]

[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&10&\dfrac{65}{3}|\dfrac{299}{3}\\\\0&0&-15|-64\end{array}\right][/tex]

[tex]R_2\rightarrow\ \dfrac{R_2}{10}[/tex]

[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&-15|-64\end{array}\right][/tex]

[tex]R_3\rightarrow\ \dfrac{R_3}{-15}[/tex]

[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]

[tex]R_1\rightarrow\ R_1-R_2[/tex]

[tex]=\ \left[\begin{array}{ccc}1&0&\dfrac{-3}{2}|\dfrac{17}{30}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]

[tex]R_1\rightarrow\ R_1+\dfrac{3}{2}R_3[/tex]

[tex]=\ \left[\begin{array}{ccc}1&0&0|\dfrac{209}{30}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]

[tex]R_2\rightarrow\ R_2-\dfrac{65}{30}R_3[/tex]

[tex]=\ \left[\begin{array}{ccc}1&0&\0|\dfrac{209}{30}\\\\0&1&0|\dfrac{29}{18}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]

Hence, we can write from augmented matrix,

[tex]x\ =\ \dfrac{209}{30}[/tex]

[tex]y\ =\ \dfrac{29}{18}[/tex]

[tex]z\ =\ \dfrac{64}{15}[/tex]

Answer:

It took 14 hours to get out of the well.

Step-by-step explanation:

Consider the provided information.

The well was 70-feet deep, the frog climbs 7 feet per hour, and it slips back 2 feet while resting.

In first hr frog climbs 7 feet and slips back 2 feet, that means it climbs 5 feet in an hour.

In 2 hour it climbs 2 × 5 = 10 feet, in 3 hours it climbs 3 × 5 = 15 feet and so on.

Now we need to find the time taken by the frog to get out of the well.

By observing the above pattern we can say that after 13 hours it can climb 65 feet.

13 × 5 = 65 feet

In next hour frog will climb 65 feet + 7 feet = 72 feet.

That means after 14 hours frog can get out of the well.

Hence it took 14 hours to get out of the well.

Step-by-step explanation:

When you divide an integer number by n, you get a remainder of either 0, 1, 2, ..., n-1 (for example 5 divided by 2 leaves a remainder of 1, or 13 divided by 5 leaves a remainder of 3, or 16 divided by 2 leaves a remainder of 0, and so on).

So there are n different remainders we could get when dividing an integer number by n. If we are given n+1 numbers, they each leave a certain remainder when divided by n. Since there are only n possible remainders, and we have n+1 numbers, by the pigeonhole principle we know there must be at least 2 numbers that leave the same remainder when divided by n. Call them numbers a and b, and let's call r the remainder they leave when divided by n. So both a and b are of the form:

[tex] a=kn+r[/tex] (for some integer k)

[tex] b=ln+r[/tex] (for some integer l)

(this is exactly what it means to leave a remainder of r when divided by n)

And so their difference is

[tex] a-b=kn+r-(ln+r)=kn-ln=(k-l)n[/tex]

Which is divisible by n by definition of being divisible (or think of it as a-b being a multiple of n, so it's divisible by n).

Answer:

The patient would receive 1.05mg of the drug weekly.

Step-by-step explanation:

First step: How many mcg of the drug would the patient receive daily?

The problem states that he takes three doses of 50-mcg a day. So

1 dose - 50mcg

3 doses - x mcg

x = 50*3

x = 150 mcg.

He takes 150mcg of the drug a day.

Second step: How many mcg of the drug would the patient receive weekly?

A week has 7 days. He takes 150mcg of the drug a day. So:

1 day - 150mcg

7 days - x mcg

x = 150*7

x = 1050mcg

He takes 1050mcg of the drug a week.

Final step: Conversion of 1050 mcg to mg

Each mg has 1000 mcg. How many mg are there in 1050 mcg? So

1mg - 1000 mcg

xmg - 1050mcg

1000x = 1050

[tex]x = \frac{1050}{1000}[/tex]

x = 1.05mg

The patient would receive 1.05mg of the drug weekly.

The patient receives 0.15 milligrams of pergolide mesylate daily and thus 1.05 milligrams weekly.

To calculate the weekly dosage of pergolide mesylate in milligrams, we first need to understand the daily dosage. The patient takes three 50-mcg tablets daily, which is 150 mcg daily. Knowing that 1 mg (milligram) is equal to 1000 mcg (micrograms), we can convert the daily dose to milligrams by dividing by 1000 which is 0.15 mg daily. To find the weekly dosage, we multiply this daily total by 7 (since there are 7 days in a week), which gives us 1.05 milligrams weekly.

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Answer:

Step-by-step explanation:

Sin t . Sin 3t . Sin 5t = 1/4 [ - Sin t + Sin 3t + Sin 7t - Sin 9t ]

Take Right hand side and use the following formula

[tex]sin C - sin D = 2 Cos\left ( \frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )[/tex]

[tex]Cos C - Cos D = 2 Sin\left ( \frac{C+D}{2} \right )Sin\left ( \frac{D-C}{2} \right )[/tex]

Take right hand side

[tex]\frac{1}{4}\left (Sin 3t - Sin t + Sin 7t - Sin 9t  \right )[/tex]

[tex]\frac{1}{4}\left (2 Cos 2t Sin t +2 Sin (-t)Cos 8t \right )[/tex]

[tex]\frac{1}{4}\times 2 Sin t\left (Cos 2t-Cos8t \right )[/tex]

[tex]\frac{1}{4}\times 2 Sin t\ \times 2 \times Sin 5t\times 3t[/tex]

Sin t . Sin 3t . Sin 5t

So, LHS = RHS

Answer:

"The Hulk is not green AND the Iron Man is not red"

Step-by-step explanation:

DeMorgan's laws state that the negation of an statement whose structure is "p OR q" is "not p AND not q", and similarly, that the negation of an statement whose structure is "p AND q" is "not p OR not q". The statement we want to negate in our case is "The Hulk is green OR the Iron Man is red". This is an statement whose structure is of the type "p OR q", where p would be "The Hulk is green", and q would be "the Iron Man is red". So according to DeMorgan's laws, its negation should be the statement "not p AND not q". To put them in common english, not p would be "The Hulk is NOT green", and not q would be "The Iron Man is NOT red". So the statement "not p AND not q" is simply "The Hulk is not green AND the Iron Man is not red".

To negate the statement 'the Hulk is green or Iron Man is red' using DeMorgan's Laws, you rephrase it as 'the Hulk is not green and Iron Man is not red'. The formal representation switches the disjunction to a conjunction and applies negation to each individual proposition.

To write the negation of the statement 'the Hulk is green or Iron Man is red' using DeMorgan's Laws, we first need to understand what these laws state. DeMorgan's Laws tell us how to move a negation across a conjunction (and) or a disjunction (or). According to DeMorgan's Laws, the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations.

In formal logic, the original statement can be represented as (G \/ R), where G represents 'the Hulk is green' and R represents 'Iron Man is red'. Applying DeMorgan's Law to negate this statement would involve negating the entire proposition and then switching the 'or' to 'and'. Thus, the negation of the statement would be \u00AC(G \/ R) which translates to \u00ACG \u2227 \u00ACR using DeMorgan's Laws. This means 'the Hulk is not green and Iron Man is not red'.

Using DeMorgan's Laws is a way to express logical equivalence between propositions. By utilizing these transformation rules, one can simplify complex logical expressions or restate them in a different form without changing their meaning, making them powerful tools in formal logic and mathematics.

Answer:  [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]

Step-by-step explanation:

We know that the sum of measure of all the angles of a triangle is 180°.

Given : The measure of the angles of a triangle are x, 2x, and 3x.

Then, the sum of all the angle will be given by :-

[tex]x+2x+3x=180^{\circ}\\\\\Rightarrow\ 6x=180^{\circ}\\\\\Rightarrow\ x=\dfrac{180^{\circ}}{6}=30^{\circ}[/tex]

Then, the measures of angles of the triangle will be : [tex]30^{\circ},\ 2(30^{\circ}),\ 3(30^{\circ})[/tex]

i.e. [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]

Answer:

The maximum profit is $1600 at x=30 and y=10.

Step-by-step explanation:

Let x be the number of units of product X.

y be the number of units of product Y.

The profit for X is $35 and the profit for Y is $55.

Maximize [tex]Z = 35x + 55y[/tex]               ..... (1)

It requires 4 pounds of material and 2 hours of labor to produce one unit of X. It requires 4 pounds of material and 6 units of labor to produce one unit of Y.

Total material = 4x+4y

Total labor = 2x+6y

There are 160 pounds of material and 120 hours of labor available.

[tex]4x+4y\leq 160[/tex]             .... (2)

[tex]2x+6y\leq 120[/tex]            ..... (3)

[tex]x\geq 0,y\geq 0[/tex]

The related line of inequality (2) and (3) are solid line because the sign of equality "≤" contains all the point on line in the solution set.

Check the inequalities by (0,0).

[tex]4(0)+4(0)\leq 160[/tex]

[tex]0\leq 160[/tex]

This statement is true.

[tex]2x+6y\leq 120[/tex]  

[tex]2(0)+6(0)\leq 120[/tex]  

[tex]0\leq 120[/tex]  

It means shaded region of both inequalities contain (0,0).

The extreme points of common shaded region are (0,0), (0,20), (40,0) and (30,10).

At (0,0),

[tex]Z = 35(0) + 55(0)=0[/tex]

At (0,20),

[tex]Z = 35(0) + 55(20)=110[/tex]

At (40,0),

[tex]Z = 35(40) + 55(0)=140[/tex]

At (30,10),

[tex]Z = 35(30) + 55(10)=1600[/tex]

Therefore the maximum profit is $1600 at x=30 and y=10.

Answer:

46 minutes.

Step-by-step explanation:

You have a gift certificate worth $20 for one long-distance phone call.

The charge for the first minute = $1.10

Let the other additional minutes that you can talk = x

The charges for the x minutes = 0.42 per minute

the equation will be : 1.10 + 0.42x = 20

0.42x = 20 - 1.10

0.42x = 18.90

x = [tex]\frac{18.90}{0.42}[/tex]

x = 45

1 minute for $1.10 + 45 minutes for $0.42/min.

you can talk for 46 minutes.

The longest phone call you can make with a $20 gift certificate, given the cost structure of $1.10 for the first minute and $0.42 for each additional minute, is approximately 46 minutes.

The subject of this question is Mathematics, specifically related to linear equations and budget constraints in the context of phone call charges. To find the longest time you can talk on the phone using your gift certificate, you'll need to understand the cost structure. The charge is $1.10 for the first minute - that leaves you with $18.90 from the gift certificate for the remaining time ($20 - $1.10). Each additional minute costs $0.42. Now, divide the remaining amount in your gift certificate by the cost per additional minute: $18.90 ÷ $0.42 ≈ 45 minutes. Adding back the first minute, the longest call you can make with the gift certificate is approximately 46 minutes

.

Learn more about Linear Equations here:

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Answer:

a) The value of the Annual Payment is A=$17,258.80

b) Is the picture in the attachment file

c) As you can see it in the picture with each payment, balance comes down, due it is the interest base, Interest portion comes down too.

Step-by-step explanation:

Hi

a) First of all, we are going to list the Knowns: [tex]VP=47000[/tex], [tex]i=5[/tex]% and [tex]n=3[/tex], Then we can use [tex]A=\frac{VP}{\frac{1-(1+i)^{-n} }{i} } =\frac{47000}{\frac{1-(1+0.05)^{-3} }{0.03} }=17258.80[/tex]. So this is the value of the Annual Payment

Joan Messines's annual payment on her $47,000 loan at 5% interest over 3 years is $17,158.11. The interest portion of each payment declines over time due to the decreasing loan balance, leading to a smaller interest calculation base in each subsequent year.

Joan Messines borrowed $47,000 at a 5% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal, annual, end-of-year payments.

Calculation of the Annual Loan Payment

To calculate the annual payment, we use the formula for an annuity:

PV = PMT [(1 - (1 + r)^-n) / r]

Where:

PV is the present value of the loan (initial loan amount).

PMT is the annual payment.

r is the annual interest rate (expressed as a decimal).

n is the number of years.

Rearranging the formula to solve for PMT yields:

PMT = PV / [(1 - (1 + r)^-n) / r]

Substitute PV = $47,000, r = 0.05 (5%), and n = 3:

PMT = $47,000 / [(1 - (1 + 0.05)^-3) / 0.05]

PMT = $17,158.11 (rounded to the nearest cent).

Loan Amortization Schedule

Year 1: Interest = $47,000 * 5% = $2,350; Principal = $17,158.11 - $2,350 = $14,808.11; Remaining Balance = $47,000 - $14,808.11 = $32,191.89

Year 2: Interest = $32,191.89 * 5% = $1,609.59; Principal = $17,158.11 - $1,609.59 = $15,548.52; Remaining Balance = $32,191.89 - $15,548.52 = $16,643.37

Year 3: Interest = $16,643.37 * 5% = $832.17; Principal = $17,158.11 - $832.17 = $16,325.94; Remaining Balance = $16,643.37 - $16,325.94 = $317.43

Why the Interest Portion Declines Over Time

The interest portion of each payment declines with the passage of time because as the loan principal is paid down, there is a smaller balance on which interest is calculated. This results in a decreasing interest payment and an increasing principal payment with each subsequent payment until the loan is paid off.

Answer:

12 units

Step-by-step explanation:

The given function is

[tex]S(q) = 5q^2 + 1000q + 100[/tex]

where, S(q) is the price in dollars at which q units are supplied.

We need to find the quantity supplied in a month when the company sets the price of its olive oil press at $12,820.

Substitute S(q)=12820 in the given function.

[tex]12820 = 5q^2 + 1000q + 100[/tex]

Subtract both sides by 12820.

[tex]0= 5q^2 + 1000q - 12720[/tex]

Taking out GCF.

[tex]0= 5(q^2 + 200q - 2544)[/tex]

Now solve the equation for q by splitting the middle term.

[tex]0= 5(q^2 + 212q-12q - 2544)[/tex]

[tex]0= 5(q(q + 212)-12(q + 212))[/tex]

[tex]0= 5(q + 212)(q-12)[/tex]

Using zero product property we get

[tex]q + 212=0\Rightarrow q=-212[/tex]

[tex]q-12=0\Rightarrow q=12[/tex]

q is number of units. So the value of q can not be negative.

Therefore the quantity supplied in a month when the company sets the price of its olive oil press at $12,820 is 12 units.

Answer: There are 133 chickens and 20 pigs.

Step-by-step explanation:

Let x be the number of chickens and y be the number of pigs.

Given : Number of heads of pigs and chicken = 153

Number of feet = 346

Since one chicken has 2 legs and one pig has 4 legs.

By considering the given information, we have the following system of equations:-

[tex]x+y=153------(1)\\\\ 2x+4y=346---------(2)[/tex]

Multiply 2 on both sides of (1), we get

[tex]2x+2y=306-----(3)[/tex]

Subtract (3) from (2), we get

[tex]2y=40\\\\\Rightarrow\ y=\dfrac{40}{2}=20[/tex]

Put value of y in (1), we get

[tex]x+20=153\\\\\Rightarrow\ x=153-20=133[/tex]

Hence, there are 133 chickens and 20 pigs.

Final answer:

The problem is a system of linear equations in Mathematics, where we find that there are 133 chickens and 20 pigs in the barnyard after setting up and solving the equations based on the given number of heads and feet.

Explanation:

Let's denote the number of chickens as C and the number of pigs as P. Therefore, we have two equations based on the given information:

C + P = 153 (since each animal has one head)

2C + 4P = 346 (since chickens have 2 feet and pigs have 4 feet)

By solving these equations, we can find the values for C and P. Multiplying the first equation by -2 and adding it to the second equation, we eliminate C and get:

-2C - 2P = -306
2C + 4P = 346
-----------------
2P = 40

Dividing both sides by 2, we find that P = 20. Substituting this value back into the first equation, we get C + 20 = 153, which means C = 133.

Therefore, there are 133 chickens and 20 pigs in the barnyard.