You can afford monthly deposits of $270 into an account that pays 4.8% compounded monthly. How long will it be until you have $8,200 to buy a​ boat?

Answer:  Yes, the given set of vectors is a linearly independent subset of R³.

Step-by-step explanation:  We are given to show that the following set of three vectors is a linearly independent subset of R³ :

B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} .

Since the given set contains three vectors which is equal to the dimension of R³, so it is a subset of R³.

To check the linear independence, we will find the determinant formed by theses three vectors as rows.

If the value of the determinant is non zero, then the set of vectors is linearly independent. Otherwise, it is dependent.

The value of the determinant can be found as follows :

[tex]D\\\\\\=\begin{vmatrix}1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1\end{vmatrix}\\\\\\=1(0\times1-1\times1)+1(1\times0-1\times1)+0(1\times1-0\times0)\\\\=1\times(-1)+1\times(1)+0\\\\=-1-1\\\\=-2\neq0.[/tex]

Since the determinant is not equal to 0, so the given set of vectors is a linearly independent subset of R³.

Thus, the given set is a linearly independent subset of R³.

Answer:

The total amount to be repaid is equal to $949.66

Step-by-step explanation:

Simple interest is a type of interest which is usually applied on short term loans, where when a payment is made towards this kind of interest the payment first goes towards monthly interest and then the remainder is reverted towards the principal.

FORMULA FOR CALCULATING SIMPLE INTEREST =

[tex]\frac{PRINCIPAL \times RATE OF INTEREST \times TIME PERIOD}{100}[/tex]

Here principal = $922

         interest rate = 6%

         time period = 6 months (when made per annum it will be 6/12)

[tex]\frac{\$ 922 \times 6 \times 1}{100\times 2}[/tex]

SIMPLE INTEREST IS EQUAL TO $27.66

The total amount that is to be repaid is equal to

   PRINCIPAL + SIMPLE INTEREST

= $922 + $27.66

= $949.66

Answer: a)  0.6141

b) 0.9772

Step-by-step explanation:

Given : Mean : [tex]\mu= 62.3\text{ in}[/tex]

Standard deviation : [tex]\sigma = \text{2.4 in}[/tex]

The formula for z -score :

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

a) Sample size = 1

For x= 63 in. ,

[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{1}}}=0.29[/tex]

The p-value = [tex]P(z<0.29)=[/tex]

[tex]0.6140918\approx0.6141[/tex]

Thus, the probability is approximately = 0.6141

b)  Sample size = 47

For x= 63 ,

[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{47}}}\approx2.0[/tex]

The p-value = [tex]P(z<2.0)[/tex]

[tex]=0.9772498\approx0.9772[/tex]

Thus , the probability that they have a mean height less than 63 in =0.9772.

Answer:

Domain : [-4, ∞)

Range : [ -2, ∞)

Step-by-step explanation:

The given function is y = [tex]\sqrt{(x+4)}-2[/tex]

Domain of the given function will be

(x + 4) ≥ 0

[ Since square root of numbers less than 0 is not possible ]

Domain : x ≥ (-4)

Or [-4, ∞) will be the domain

Now range of the function will be x ≥ -2

[ -2, ∞) will be the range of the given function.

Answer:

p(F)=0.73

Step-by-step explanation:

we have by identity

[tex]p(A\cup B)=p(A)+p(B)-p(A\cap B)[/tex]

Thus for given events E and F

we have[tex]p(E\cup F)=p(E)+p(F)-p(E\cap F)[/tex]

Applying values we get

[tex]p(F)=p(E\cup F)+p(E\cap F)-p(E)[/tex]

Thus

p(F) = 0.82+0.09-0.18

p(F) = 0.73

Answer: The proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.

Step-by-step explanation:

Since we have given that

Number of tankers is drawn = 474

Number of tankers did not have spills = 318

Number of tankers have spills = 474 - 318 = 156

Proportion of oil tankers that had spills is given by

[tex]\dfrac{Containing\ spill}{Total}=\dfrac{156}{474}=0.329[/tex]

Hence, the proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.

Answer: 0.1587

Step-by-step explanation:

Given : The foreman of a bottling plant has observed that the amount of soda in each 16-ounce bottle is actually a normally distributed random variable, with

[tex]\mu=15.9\text{ ounces}[/tex]

Standard deviation : [tex]\sigma=0.1\text{ ounce}[/tex]

Let x be the amount of soda in a randomly selected bottle.

Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{16-15.9}{0.1}=1[/tex]

The probability that the bottle will contain more than 16 ounces using standardized normal distribution table  :

[tex]P(x>16)=P(z>1)=1-P(z<1)\\\\=1-0.8413447=0.1586553\approx0.1587[/tex]    

Hence, the probability that the bottle will contain more than 16 = 0.1587

Answer:

The Hindu-Arabic numeral form of the given expanded numeral is 842.

Step-by-step explanation:

The given expanded numeral is

[tex](8\times 10^2)+(4\times 10)+(2\times 1)[/tex]

We need to express the given expanded numeral as a Hindu-Arabic numeral.

According to Hindu-Arabic numeral the given expanded numeral is written as

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(8\times 100)+(4\times 10)+(2\times 1)[/tex]

On simplification we get,

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(800)+(40)+(2)[/tex]

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=842[/tex]

Therefore the Hindu-Arabic numeral form of the given expanded numeral is 842.

Answer:

The correct option is 4.

Step-by-step explanation:

It is given that the rate of recipt of income from the sales of vases from 1988 to 1993 can be approximated by

[tex]R(t)=\frac{100}{(t+0.87)^2}[/tex]

billion dollars per year, where t is time in years since January 1988.

We need to estimate the total change in income from January 1988 to January 1993.

[tex]I=\int_{0}^{5}R(t)dt[/tex]

[tex]I=\int_{0}^{5}\frac{100}{(t+0.87)^2}dt[/tex]

[tex]I=100\int_{0}^{5}\frac{1}{(t+0.87)^2}dt[/tex]

On integration we get

[tex]I=-100[\frac{1}{(t+0.87)}]_{0}^{5}[/tex]

[tex]I=-100(\frac{1}{5+0.87}-\frac{1}{0+0.87})[/tex]

[tex]I=-100(-0.979)[/tex]

[tex]I=97.9[/tex]

[tex]I\approx 98[/tex]

The total change in income from January 1988 to January 1993 is $98. Therefore the correct option is 4.

Answer:

There are  ways for quickly multiply out a binomial that's being raised by an exponent. Like

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3

and so on and so on

but there was this mathematician named Blaise Pascal and he found a numerical pattern, called Pascal's Triangle, for quickly expanding a binomial like the ones from earlier. It looks like this

1                           1      

2                   1     2     1

3                1     3     3   1

4           1     4     6     4     1

5       1     5     10     10     5     1

Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle.

Hope this helps!

The Binomial Theorem uses Pascal's triangle for expanding binomials raised to positive integer powers. Pascal's triangle provides the coefficients for each term in the binomial expansion, simplifying the expansion process.

The Binomial Theorem uses Pascal's triangle to expand binomials that are raised to positive integer power. Pascal's triangle is a triangular array of binomial coefficients. Each line of the triangle represents the coefficients of the terms of a binomial expansion. Let's take for instance binomial expansion of (a + b)n = an+nan-1b+…from Pascal's triangle, the coefficients are 1,n, and so on. The role of Pascal's triangle in this case is pivotal in knowing the coefficients of each term in the binomial expansion and thus facilitates the expansion process.

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Answer:  C. 25

Step-by-step explanation:

Given : The average rat (of this strain) can learn to run this type of maze in a box without any special coloring : [tex]\mu=25[/tex]

The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper= [tex]M= 11[/tex]

We know that the sampling distribution D is given by :-

[tex]\mu_D=\mu[/tex]

Similarly the mean of the distribution M in the given situation is given by :_

[tex]\mu_M=\mu=25[/tex]

The mean of the distribution M in the given situation is 25. Then the correct option is C.

It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.

An experimental psychologist is interested in whether the color of an animal's surroundings affects the learning rate.

He tests 16 rats in a box with colorful wallpaper.

The average rate (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution.

The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.

We know that the sampling distribution D is given by

μD = μ

Similarly, the mean of the distribution M in the given situation is given by

μD = μ = 25

More about the normal distribution link is given below.

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Answer:

The amount invested at 4.5% was [tex]\$3,000[/tex]

The amount invested at 5% was [tex]\$6,000[/tex]

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Final Interest Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

Let

x -----> the amount invested at 4.5%

9,000-x -----> the amount invested at 5%

in this problem we have

[tex]t=1\ year\\ P1=\$x\\ P2=\$(9,000-x)\\I=\$435\\r1=0.045\\r2=0.05[/tex]

substitute

[tex]435=x(0.045*1)+(9,000-x)(0.05*1)[/tex]

[tex]435=0.045x+450-0.05x[/tex]

[tex]0.05x-0.045x=450-435[/tex]

[tex]0.005x=15[/tex]

[tex]x=\$3,000[/tex]

so

[tex]9,000-x=\$6,000[/tex]

therefore

The amount invested at 4.5% was [tex]\$3,000[/tex]

The amount invested at 5% was [tex]\$6,000[/tex]

The generating function is [tex]f(x)[/tex] where

[tex]f(x)=\displaystyle\sum_{k=0}^\infty a_kx^k[/tex]

with [tex]a_k=k^3[/tex] for [tex]k\ge0[/tex].

Recall that for [tex]|x|<1[/tex], we have

[tex]g(x)=\dfrac1{1-x}=\displaystyle\sum_{k=0}^\infty x^k[/tex]

Taking the derivative gives

[tex]g'(x)=\dfrac1{(1-x)^2}=\displaystyle\sum_{k=1}^\infty kx^{k-1}=\sum_{k=0}^\infty(k+1)x^k[/tex]

[tex]\implies g'(x)-g(x)=\dfrac x{(1-x)^2}=\displaystyle\sum_{k=0}^\infty kx^k[/tex]

Taking the derivative again, we get

[tex]g''(x)=\dfrac2{(1-x)^3}=\displaystyle\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k^2+3k+2)x^k[/tex]

[tex]\implies g''(x)-3g'(x)+g(x)=\dfrac{x^2+x}{(1-x)^3}=\displaystyle\sum_{k=0}^\infty k^2x^k[/tex]

Take the derivative one last time to get

[tex]g'''(x)=\dfrac6{(1-x)^4}=\displaystyle\sum_{k=3}^\infty k(k-1)(k-2)x^{k-3}=\sum_{k=0}^\infty(k^3+6k^2+11k+6)x^k[/tex]

[tex]\implies g'''(x)-6g''(x)+7g'(x)-g(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}=\displaystyle\sum_{k=0}^\infty k^3x^k[/tex]

So the generating function is

[tex]\boxed{f(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}}[/tex]

Answer:  

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

Step-by-step explanation:

For each question we have 3 choices. So,total choices will be :

[tex]3\times3\times3\times3\times3\times3=729[/tex]

Getting 4 correct means, 4 corrects and two wrongs

Now, as there are 3 answer choices, out of which only one will be correct, so 2/3 is the probability if a question is answered wrong.

And 1/3 is the probability if a question is answered correctly.

Hence, we can consider this probability :

[tex]P=(2/3)*(2/3)*(1/3)*(1/3)*(1/3)*(1/3)[/tex] = 4/729

=> P = 0.00548696

We can select any combination of 2 from 6 for being wrong, so we will multiply P by (6,2)=6!/(2!*4!) = 15

So the answer is P*15 =[tex]0.00548696*15=0.0823044[/tex]

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

With 3 choices per question, the probability of getting at least 4 out of 6 questions correct is approximately 0.0823044

1: Total Choices

Each question has 3 possible answers.

So, the total choices for 6 questions would be 3 raised to the power of 6 (3^6).

2: Probability of Getting 4 Correct and 2 Wrong

Getting 4 correct and 2 wrong means selecting 4 correct answers out of 6 questions.

The probability of a question being answered correctly is 1/3, and the probability of being answered incorrectly is 2/3.

So, the probability of getting 4 correct and 2 wrong is calculated using combinations (6 choose 4) multiplied by (1/3)^4 multiplied by (2/3)^2.

3: Calculate Probability

(6 choose 4) is the number of ways to choose 4 correct answers out of 6 questions, which is 15.

The probability (P) is then calculated as 15 multiplied by (1/3)^4 multiplied by (2/3)^2.

4: Multiply by Number of Combinations

Since there are 15 ways to choose 4 correct answers out of 6 questions, multiply the probability by 15.

So, the probability that the student will get at least 4 of the questions right is approximately 0.0823044.

To learn more about probability

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Answer:

Step-by-step explanation:

We know that X-Y = X∩Y'

Using it ,we get

A ∩(B∩C') which can be written as (A∩B)∩C' or (A∩B) - C

And right hand side is

(A∩B)-(B∩C) =B ∩(A-C) = B∩(A∩C') = A∩B∩C'

Since both left and right side both leads to same expression A∩B∩C'

Therefore both are equal.

Step-by-step explanation:

I just found the answer and I hope that this helps :)!!

The rate at which the distance between the ships is changing at 4 PM depends on their velocities.

To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. Let's consider ship B as the reference point. Ship A is moving west at 20 knots (which is equivalent to 20 nautical miles per hour), and ship B is moving north at 21 knots. The distance between the ships can be considered as the hypotenuse of a right triangle, with the velocities of the ships representing the triangle's sides.

Using the Pythagorean theorem, we can write the equation: d^2 = x^2 + y^2, where d is the distance between the ships, x is the velocity of ship A, and y is the velocity of ship B. We need to find the rate of change of d with respect to time (dt).

Taking the derivative on both sides of the equation with respect to time, we get: 2d * (dd/dt) = 2x * (dx/dt) + 2y * (dy/dt).

Substituting the given values, x = -20 knots (negative because ship A is moving west), y = 21 knots, and dx/dt = dy/dt = 0 (since ship B is not changing its velocity), we can solve for dd/dt, which represents the rate at which the distance between the ships is changing.

Therefore, dd/dt = 2x * (dx/dt) + 2y * (dy/dt) = 2 * -20 knots * 0 + 2 * 21 knots * 0 = 0.

Thus, the distance between the ships is not changing at 4 PM.

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There are 36 total possible outcomes.

Rolling a sum of 11 or higher, there are 3 possible rolls, to make a 3/36 = 1/12 probability.

Rolling a sum of 4 there are also 3 possibilities, so the chance would be the same.

Rolling a sum of 9, there are 4 possibilities, which is a better chance.

Rolling a sum less than 5, there is 6 possibilities, which is a better chance.

Rolling greater than 5 but less than 7 means rolling a sum of 6, there are 5 chances, which is a better chance.

Rolling greater than 9 but less than 11, means rolling a 10, there are 3 possibilities, which is the same.

Rolling greater than 2 and less than 4 means rolling a 3, there are 2 possibilities, which is less.

The answers would be:

Rolling a sum of 9,

Rolling a sum less than 5

Rolling greater than 5 but less than 7

Answer:

B, C, D

Step-by-step explanation:

I got it right on Edg

Answer:1,2,3

Step-by-step explanation:

F(x)=[tex]x^{3}[/tex]+2[tex]x^2[/tex]-[tex]5x[/tex]-[tex]6[/tex]=0

disintegrating 2[tex]x^2[/tex] to [tex] x^2[/tex] + [tex]x^2[/tex]

[tex]x^{3}[/tex]+[tex]x^2[/tex]+[tex]x^2[/tex]-[tex]5x-6[/tex]=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-5x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-6x+x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]\left (x-6 \right )[/tex][tex]\left ( x+1\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+3x-2x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x+3\right )[/tex][tex]\left ( x-2\right )[/tex]=0

Step-by-step explanation:

1.Assuming the same sample size and considering the same value for the errors ( not taking into consideration the type of music, the volume of the sound and cow familiar the runner is with that type of stimuli, age group, time of the day/ number of days, running conditions like wether and equipment, distance) one can state:

A. Music has no influence over the running speed ( when jogging) in Washington DC

B.When listening to music, people ( in Washington DC) run faster while jogging

The mean running speed is a simple, ponderate or other type of mean ( that takes into consideration the variations of speed at the beginning and by the end of the race?

[tex]y''+2y'-3y=0 [/tex]

Second order linear homogeneous differential equation with constant coefficients, ODE has a form of,

[tex]ay''+by'+cy=0[/tex]

From here we assume that for any equation of that form has a solution of the form, [tex]e^{yt}[/tex]

Now the equation looks like this,

[tex]((e^{yt}))''+2((e^{yt}))'-3e^{yt}=0[/tex]

Now simplify to,

[tex]e^{yt}(y^2+2y-3)=0[/tex]

You can solve the simplified equation using quadratic equation since,

[tex]e^{yt}(y^2+2y-3)=0\Longleftrightarrow y^2+2y-3=0[/tex]

Using the QE we result with,

[tex]\underline{y_1=1}, \underline{y_2=-3}[/tex]

So,

For two real roots [tex]y_1\neq y_2[/tex] the general solution takes the form of,

[tex]y=c_1e^{y_1t}+c_2e^{y_2t}[/tex]

Or simply,

[tex]\boxed{y=c_1e^t+c_2e^{-3t}}[/tex]

Hope this helps.

r3t40

The statement is given by:

∀ x ,  [tex]x^4>x[/tex]

This statement is false

Since, if we consider,

[tex]x=\dfrac{1}{2}[/tex]

then we have:

[tex]x^4=(\dfrac{1}{2})^4\\\\i.e.\\\\x^4=\dfrac{1}{2^4}\\\\i.e.\\\\x^4=\dfrac{1}{16}[/tex]

Also, we know that:

[tex]\dfrac{1}{16}<\dfrac{1}{2}[/tex]

( Since, two number with same numerator; the number with greater denominator is smaller than the number with the smaller denominator )

Hence, we get:

[tex]x^4<x[/tex]

when [tex]x=\dfrac{1}{2}[/tex]

Hence, the result :

[tex]x^4>x[/tex] is not true for all x belonging to real numbers.

Hence, the given statement is a FALSE statement.

Final answer:

The statement (∀x)(x⁴ > x) is false, as it does not hold true for all real numbers. For instance, when x is a negative number like -1, the inequality x⁴ > x is false.

Explanation:

False

Explanation:

Given statement: (∀x)(x⁴ > x)

This statement asserts that for all real numbers x, x⁴ will be greater than x. However, this statement is false because it doesn't hold for all real numbers. For instance, when x is a negative number such as -1, (-1)4 is greater than -1, which means the inequality x⁴ > x is false.

Answer:

Step-by-step explanation: manej

Final answer:

In algebra, a tenth of a number is algebraically represented by multiplying the number by [tex]10^{-1}[/tex], which is equivalent to dividing the number by 10. This application of negative exponents simplifies expressions, especially in scientific notation, making it easier to work with large and small quantities.

Explanation:

In algebra, when we refer to a tenth of a number, we are usually dealing with fractions or exponential notation. A tenth of a number can be represented algebraically as the number divided by 10, which is the same as multiplying the number by [tex]10^{-1}[/tex]. This is because negative exponents indicate the reciprocal of a number; in other words, 10-1 equals 1/10 or 0.1.

This concept relates to the powers of ten and how each power of 10 affects the size of a number. For instance, 102 is 100, and 101 is 10, which is ten times smaller than 100. Conversely, 100 is 1, which is ten times smaller than 10, and thus, logically, [tex]10^{-1}[/tex] is 0.1, which is ten times smaller still. In expressing measurements in scientific work, especially for very small numbers, we frequently use this exponential form.

Thus, a tenth of an algebraic expression would mean multiplying the expression by [tex]10^{-1}[/tex] or dividing the expression by 10. This process is a form of simplification and re-scaling of numbers that are commonly used in scientific notation, which includes both positive and negative exponents. By understanding these principles, one can efficiently work with both large and small quantities in scientific and mathematical contexts.

Answer:

The answer is - a) 800

Step-by-step explanation:

In a periodic review system, we calculate the quantity of an item, a company has on hand at specified and fixed interval of time.

Given is : The target inventory level is 1000 units and the on-hand inventory level is 200 units.

So, the quantity will be =[tex]1000-200=800[/tex] units.

The answer is option A.

Answer:

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Step-by-step explanation:

You need to set up a two equation problem

[tex]\left \{ {{258,000=6,000y+3,000x} \atop {60=y+x}} \right.[/tex]

Now you have to clear "x" or "y" from the second equation:

[tex]y = 60 - x[/tex]

And replace on the first equation:

[tex]258,000 = 6,000 (60 - x) + 3,000x\\258,000 = 360,000 - 6,000x + 3,000x\\3,000x = 360,000 - 258,000\\x = 102,000/3,000\\x = 34[/tex]

And now you use this "x" vale on the second equation

[tex]60 = y + x\\60 = y + 34\\60 - 34 = y\\y = 26[/tex]

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Answer:

2.8

Step-by-step explanation:

The weighted average is found by dividing the total number of points by the total number of credits.

GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)

GPA = 56 / 20

GPA = 2.8

The GPA of the student will be 2.8.

Algebra is the study of mathematical symbols, and the rule is the manipulation of those symbols.

One common system for computing a grade point average​ (GPA) assigns 4 points to an​ A, 3 points to a​ B, 2 points to a​ C, 1 point to a​ D, and 0 points to an F.

Then the GPA of a student who gets an A in a 3​-credit ​course, a B in each of three 4​-credit ​courses, a C in a 3​-credit ​course, and a D in a 2​-credit ​course will be

GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)

GPA = 56 / 20

GPA = 2.8

More about the Algebra link is given below.

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Answer:Given below

Step-by-step explanation:

A function is said to be odd if

[tex]F\left ( x\right )=F\left ( -x\right )[/tex]

(a)[tex]F\left ( x\right )=x^3+3x[/tex]

[tex]F\left ( -x\right )=-x^3-3x=-\left ( x^3+3x\right )[/tex]

odd function

(b)[tex]F\left ( x\right )=4sin2x[/tex]

[tex]F\left ( -x\right )=-4sin2x[/tex]

odd function

(c)[tex]F\left ( x\right )=x^2+|x|[/tex]

[tex]F\left ( -x\right )=\left ( -x^2\right )+|-x|=x^2+|x|[/tex]

even function

(d)[tex]F\left ( x\right )=e^x[/tex]

[tex]F\left ( -x\right )=e^{-x}[/tex]

neither odd nor even

(e)[tex]F\left ( x\right )=\frac{1}{x}[/tex]

[tex]F\left ( -x\right )=-\frac{1}{x}[/tex]

odd

(f)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x+e^{-x}\right )[/tex]

[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]

even function

(g)[tex]F\left ( x\right )=xcosx(h)[/tex]

[tex]F\left ( -x\right )=-xcosx(h)[/tex]

odd function

(h)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x-e^{-x}\right )[/tex]

[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]

odd function

Step-by-step explanation:

If p is the number of physics books and s is the number of sociology books, then:

p + s = 440

s = p - 54

Substituting:

p + (p - 54) = 440

2p - 54 = 440

2p = 494

p = 247

Solving for s:

s = p - 54

s = 247 - 54

s = 193

The store sold 247 physics books and 193 sociology books.

Answer:

A) (-17+5k,17-4k)

B)  (-4+3k,4-2k)

C) No integer pairs.

Step-by-step explanation:

To do this, I'm going to use Euclidean's Algorithm.

4x+5y=17

5=4(1)+1

4=1(4)

So going backwards through those equations:

5-4(1)=1

-4(1)+5(1)=1

Multiply both sides by 17:

4(-17)+5(17)=17

So one integer pair satisfying 4x+5y=17 is (-17,17).

What is the slope for this equation?

Let's put it in slope-intercept form:

4x+5y=17

Subtract 4x on both sides:

     5y=-4x+17

Divide both sides by 5:

      y=(-4/5)x+(17/5)

The slope is down 4 and right 5.

So let's show more solutions other than (-17,17) by using the slope.

All integer pairs satisfying this equation is (-17+5k,17-4k).

Let's check:

4(-17+5k)+5(17-4k)

-68+20k+85-20k

-68+85

17

That was exactly what we wanted since we were looking for integer pairs that satisfy 4x+5y=17.

Onward to the next problem.

6x+9y=12

9=6(1)+3

6=3(2)

Now backwards through the equations:

9-6(1)=3

9(1)-6(1)=3

Multiply both sides by 4:

9(4)-6(4)=12

-6(4)+9(4)=12

6(-4)+9(4)=12

So one integer pair satisfying 6x+9y=12 is (-4,4).

Let's find the slope of 6x+9y=12.

6x+9y=12

Subtract 6x on both sides:

      9y=-6x+12

Divide both sides by 9:

       y=(-6/9)x+(12/9)

Reduce:

       y=(-2/3)x+(4/3)

The slope is down 2 right 3.

So all the integer pairs are (-4+3k,4-2k).

Let's check:

6(-4+3k)+9(4-2k)

-24+18k+36-18k

-24+36

12

That checks out since we wanted integer pairs that made 6x+9y=12.

Onward to the last problem.

4x+10y=9

10=4(2)+2

4=2(2)

So the gcd(4,10)=2 which means this one doesn't have any solutions because there is no integer k such that 2k=9.

Answer:

303.6 feet

Step-by-step explanation:

Given,

The average Ferris wheel rotates at 6.9 miles per hour.

So, the speed of the wheel = 6.9 miles per hour,

We know that,

Distance = Speed × Time

So, the distance covered by the wheel in 5 minutes ( or 1/12 hours because 1 hour = 60 minutes ) ride = [tex]6.9\times \frac{1}{12}[/tex]

[tex]=\frac{6.9}{12}[/tex]

[tex]=0.575\text{ miles}[/tex]

Since, 1 mile = 5280 feet,

Hence, the distance covered by the wheel in 5 minutes = 0.575 × 528 = 303.6 feet.

The average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.

To calculate the circular distance covered by the average Ferris wheel in a 5-minute ride, we need to convert the speed from miles per hour to feet per minute. There are 5,280 feet in a mile and 60 minutes in an hour, so we can convert 6.9 miles per hour to feet per minute using the formula:

6.9 miles/hour x 5,280 feet/mile x 1 hour/60 minutes = 604.8 feet/minute

Now that we know the Ferris wheel covers 604.8 feet in 1 minute, we can calculate the circular distance covered in 5 minutes by multiplying the feet per minute by the number of minutes:

604.8 feet/minute x 5 minutes = 3024 feet

Therefore, the average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.

Answer: There are 17576000 ways to generate different license plates.

Step-by-step explanation:

Since we have given that

Numbers are given = 0 to 9 = 10 numbers

Number of letters = 26

We need to generate the license plate numbers.

Since there are repetition allowed.

We would use "Fundamental theorem of counting".

So, the number of different license numbers may be created as given as

[tex]26\times 26\times 26\times 10\times 10\times 10\\\\=26^3\times 10^3\\\\=17576\times 1000\\\\=17576000[/tex]

Hence, there are 17576000 ways to generate different license plates.