Ali has worked at a fashion magazine for the last 5 years. Her current annual salary is $64,000. When she was hired, she was told that she had four days of paid vacation time. For each year that she worked at the magazine, she would gain another three days of paid vacation time to a maximum of 26 days. How many paid vacation days does she now get at the end of 5 years of employment?

Answer:

(a) 0.9838 (b) 0.6553 (c) 13.05198

Step-by-step explanation:

We have that the daily viewing time is a random variable normally distributed with mean and standard deviation

[tex]\mu[/tex] = 8.35 hours  and

[tex]\sigma[/tex] = 2.5 hours

respectively. If we call the random variable X, the density function of this random variable is given by

f(x) = [tex]\frac{1}{\sqrt{2\pi}2.5}\exp[-\frac{(x-8.35)^{2}}{2(2.5)^{2}}][/tex], and we can calculate the next probabilities using a computer or a table from a book.

(a) P(X>3)=[tex]\int\limits^{\infty}_3 {f(x)} \, dx[/tex]=0.9838

in the R statistical programming language we use the instruction pnorm(3, mean = 8.35, sd = 2.5, lower.tail = FALSE)

(b) P([tex]5\leq X\leq 10[/tex]) = [tex]\int\limits^{10}_5 {f(x)} \, dx[/tex] = 0.6553

in the R statistical programming language we use the instruction

pnorm(10, mean = 8.35, sd = 2.5) - pnorm(5, mean = 8.35, sd = 2.5)

(c) You should find a value [tex]x_{0}[/tex] such that

[tex]P(X\geq x_{0}) = 0.03[/tex], this value is  [tex]x_{0}[/tex]=13.05198

The instruction qnorm(0.03, mean = 8.35, sd = 2.5, lower.tail = FALSE) give us 13.05198 in the R statistical programming language.

Answer:

a. P=0.98

b. P=0.66

c. The top 3% of all TV viewing households watch 12.95 hours or more.

Step-by-step explanation:

We have a normal distribution with these parameters:

[tex]\mu=8.35\\\\\sigma=2.50[/tex]

a. What is the probability that a household views television more than 3 hours a day?

To calculate this, first we calculate the z-value for X=3 and then calculate the probability according to the standard normal distribution.

[tex]z=(X-\mu)/\sigma=(3-8.25)/2.50=-2.1\\\\P(X>3)=P(z>-2.1)=0.98214[/tex]

b. What is the probability that a household spends 5 – 10 hours watching television more a day?

[tex]z_1=(X_1-\mu)/\sigma=(5-8.25)/2.50=-1.3\\\\z_2=(X_2-\mu)/\sigma=(10-8.25)/2.50=0.7\\\\P(5<X<10)=P(-1.3<z<0.7)\\\\P(-1.3<z<0.7)=P(z>-1.3)-P(z>0.7)=0.9032-0.2412=0.662[/tex]

c. How many hours of television viewing must a household have in order to be in the top 3% of all television viewing households?

To calculate this we have to know the z-value for [tex]P(z>z_1)=0.03[/tex].

This z-value, according to the standard normal distribution is z=1.88.

Then, we can calculate the number of hours X as:

[tex]X=\mu+z*\sigma=8.25+1.88*2.5=8.25+4.7=12.95[/tex]

The top 3% of all TV viewing households watch 12.95 hours or more.

Answer:

35

Step-by-step explanation:

Given that A,B, C are three non empty sets.

[tex]n(A) =n(B) =n(C) =17\\n(A \bigcap C) = 7\\n(A \bigcap B) = 5\\n(B \bigcap C) = 6\\n(A \bigcap B \bigcap C) = 7[/tex]

Use the addition theory for finding no of elements in union of two or more sets

We have addition theorem as

[tex]n(AUBUC) = n(A)+n(B)+n(C)-n(A \bigcap B)-n(B  \bigcap C)-n(A  \bigcap C)+n(A  \bigcap B \bigcap C)[/tex]

Now substitute for each entry from the given information

[tex]n(AUBUC) = 17+17+17-5-6-7+2\\= 53-18\\=35[/tex]

Answer:

Solution for the linear system:

a)  [tex]X_1=3, X_2=1, X_3=2[/tex]

b) [tex]x= w-1\\y=2z[/tex]

z and w are free, meaning that can have any value, for this reason, this system has infinite solutions.

Step-by-step explanation:

Gaussian-Jordan elimination consists of taking an augmented matrix, and transform it into its Row echelon form by means of row operation.  For notation, R_i will be the transform column, and r_i the actual one.

Linear System a)

First, you have to convert the system into matrix notation, in this case, column 1 corresponds to variable x_1, column 2 to x_2, column 3 to x_3 and column 4 to the system constants:

[tex]\left[\begin{array}{cccc}1&1&2&8\\-1&-2&3&1\\3&-7&4&10\end{array}\right][/tex]

Operations:

[tex]R_2=r_1+r_2\\R_3=-3r_1+r_3[/tex]

[tex]\left[\begin{array}{cccc}1&1&2&8\\0&-1&5&9\\0&-10&-2&-14\end{array}\right][/tex]

Operations:

[tex]R_2=-r_2[/tex]

[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&-10&-2&-14\end{array}\right][/tex]

Operations:

[tex]R_3=10r_2+r_3[/tex]

[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&-52&-104\end{array}\right][/tex]

Operations:

[tex]R_3=-\frac{1}{52}r_3[/tex]

[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&1&2\end{array}\right][/tex]

[tex]x_1+x_2+2x_3=8\\x_2-5x_3=-9\\x_3=2[/tex]

[tex]x_3=2\\x_2=-9+5*2=1\\x_1=8-1-4=3[/tex]

Linear System b)

For this system, the process is the same as the above.  

Convert the system into matrix form

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

Operations:

[tex]R_2=-2r_1+r_2\\R_3=r_1+r_3\\R_4=-3r_1+r_4[/tex]

[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&3&-6&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right][/tex]

Operations:

[tex]R_2=\frac{1}{3}r_2[/tex]

[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&1&-2&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right][/tex]

Operations:

[tex]R_3=-r_2+r_3\\R_4=-3r_2+r_4[/tex]

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

Now you can write the system as equations:

[tex]x-y+2z-w=-1\\y-2z=0[/tex]

For w and z there is no unique answer, so the system result is expressed in terms of those variables. This system has infinite solutions.

Solution:

[tex]x= w-1\\y=2z[/tex]

z and w are free values.

Answer:

Alice's weight is 100 lbs.

Step-by-step explanation:

Let's denote Jane's weight by J, and Alice's weight by A.

The exercise says that Jane is 20 lbs heavier than Alice. So that if you add 20 lbs to Alice's weight, you get Jane's weight. In equation form:

[tex]20+A=J[/tex]

It also mentions that Jane's weight is 120% that of Alice. So that if you multiply Alice's weight by 1.2, you get Jane's weight. In equation form:

[tex]1.2 \cdot A=J[/tex]

Plugging this second equation onto the first equation, we get:

[tex]20+A=1.2A[/tex]

And now solving for A:

[tex]20=1.2A-A[/tex]

[tex]20=0.2A[/tex]

[tex]\frac{20}{0.2}=A[/tex]

[tex]100=A[/tex]

Therefore Alice's weight is 100 lbs.

Answer:

1 light-minute ≈ 1.12×10⁷ miles, three significant figures.

Step-by-step explanation:

Light-second is equal to the distance traveled per second by light in space, which is equal to 299,792,458 metres. Other units used are light-day, light-hour and light-minute.

Significant figures are the figures or digits of a number that carry meaning and contribute to the precision of the given number.

1 light-minute = 1.118×10⁷ miles, has four significant figures.

To express this number in three significant figures, the given number is rounded.

1 light-minute = 1.118×10⁷ miles ≈ 1.12×10⁷ miles, has three significant figures, as the non-zero digits are significant.

Answer:

The negation will be : The shirt I am wearing to my interview is not orange.

Step-by-step explanation:

The negation of the statement means adding not, or nor to the statement.

The given statement is :

The shirt I’m wearing to my interview is orange.

The negation will be : The shirt I am wearing to my interview is not orange.

The particular solution to the differential equation with the initial condition P(0) = 3000 is given by P = 3000e^((ln(2)/6)t), where P represents the population at time t. The growth rate is proportional to the population, and the constant of proportionality is ln(2)/6. This equation can be derived by solving the differential equation and using the initial condition.

In this problem, we are given that the population of a suburb grows at a rate proportional to the population. We are also given that the population doubles in size from 3000 to 6000 in a 6-month period. We need to find the particular solution to the differential equation with the initial condition P(0)=3000.

Let's denote the population at time t as P(t). Since the growth rate is proportional to the population, we can write the differential equation as dP/dt = kP, where k is the proportionality constant.

Integrating both sides of the equation, we get: ∫dP/P = ∫kdt. This gives us ln|P| = kt + C, where C is the constant of integration.

Using the initial condition P(0) = 3000, we can substitute t = 0 and P = 3000 into the equation to get: ln|3000| = 0 + C. Solving for C, we find C = ln|3000|.

Substituting C = ln|3000| back into the equation, we have ln|P| = kt + ln|3000|. Simplifying, we get ln|P| - ln|3000| = kt.

Since ln|P| - ln|3000| = ln|(P/3000)|, we can write the equation as ln|(P/3000)| = kt.

Taking the exponential of both sides, we get |(P/3000)| = e^(kt).

Since the population cannot be negative, we can remove the absolute value sign and write the equation as (P/3000) = e^(kt).

Substituting the doubling in size from 3000 to 6000 in a 6-month period, we have (6000/3000) = e^(k(6)).

Simplifying, we get 2 = e^(6k).

Taking the natural logarithm of both sides, we have ln(2) = ln(e^(6k)).

Using the property ln(a^b) = bln(a), we can rewrite the equation as ln(2) = 6kln(e).

Since ln(e) = 1, we have ln(2) = 6k.

Solving for k, we get k = ln(2)/6.

Substituting k = ln(2)/6 back into the equation, we have (P/3000) = e^((ln(2)/6)t).

Multiplying both sides by 3000, we get P = 3000e^((ln(2)/6)t).

This is the particular solution to the differential equation with the initial condition P(0) = 3000.

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Answer:

[tex]\frac{(2m+3n)}{5mn}=\frac{2}{5n}+\frac{3}{5m}[/tex] is a rational number for any m and n; nonzero integers.

Step-by-step explanation:

We have been given that 'm' and 'n' are nonzero integers. We are asked to show that [tex]\frac{(2m+3n)}{5mn}[/tex] is a rational number.

We can rewrite our given number as:

[tex]\frac{2m}{5mn}+\frac{3n}{5mn}[/tex]

Cancelling out common terms:

[tex]\frac{2}{5n}+\frac{3}{5m}[/tex]

Since 'm' and 'n' are nonzero integers, so each part will be a rational number.

We know that sum of two rational numbers is always rational, therefore, our given number is a rational number.

Answer:

There are two unknow values, your 8% which is the decrease and the main value which is "the sales last year".

It is correct, the number you gave as an answer $9798,91. Let's get the explanation by a rule of three.

Step-by-step explanation:

The 100 % was "the sales last year", the 8% are what the sales decreased but you don't have that number, you have the result in the substraction, $9015. So this are the step by step, hope you understand!

Answer:

Conjecture: next term is 45

Step-by-step explanation:

One very common sequence is one where the difference between one term and the previous one follows a recognizable pattern. Let's inspect the difference from one term to the previous one in the sequence:

Difference from 9 to 5:      9 - 5 = 4

Difference from 15 to 9:    15 - 9 = 6

Difference from 23 to 15: 23 -15 = 8

Difference from 33 to 23: 33-23= 10

At this point the pattern is clear, the differences are just even numbers increasing 2 by 2. We would expect next difference to be 12, and so the next term on the sequence should be 33 + 12 = 45.

Answer: The purchase price of the property is $112,500.

Step-by-step explanation:

Let the original value of property be 'x'.

Discount rate = 3%

Amount of discount = $2700

According to question, it becomes,

[tex]\dfrac{3}{100}\times x=2700\\\\x=\dfrac{2700\times 100}{3}\\\\x=\$90000[/tex]

Rate of down payment = 20%

So, Remaining rate of payment = 100-20 = 80%

So, Purchase price of the property would be

[tex]\dfrac{80}{100}\times x= 90000\\\\0.8\times x=90000\\\\x=\dfrac{90000}{0.8}\\\\x=\$112,500[/tex]

Hence, the purchase price of the property is $112,500.

Answer:

See definitions and relation below

Step-by-step explanation:

Given points x and y of a certain set S in a metric space, a path from x to y is a continuous map  f:[a,b]-->S of some closed interval [a,b] in the real line into S, such that

f(a)=x and f(b)=y

In this case, we can also say that the points x and y are joined by a path or arc.

A set S in metric space is said to be path connected or arcwise connected if every pair of points x, y of S  can be joined by a path.

The relation between arcwise connectedness and connectedness of a set is that every arcwise connected set is also connected, but the converse does not hold; not every connected space is also path connected.

As an example, consider the unit square [0,1]X[0,1] with the dictionary order topology.

It can be proved that this space is connected but not path connected.

Arcwise connectedness is defined as the presence of a continuous path between any two points in a set in a metric space. There is a relation between arcwise connectedness and connectedness, where any path connected set is also connected, but the converse is not necessarily true.

A set in a metric space is said to be arcwise connected or path connected if there exists a continuous curve or path that connects any two points in the set.

The relation between arcwise connectedness and connectedness of a set is that any arcwise connected set is also connected, but the converse is not necessarily true. In other words, every path connected set is connected, but not every connected set is path connected.

For example, consider a set consisting of two separate points in a metric space. This set is connected because we cannot find two disjoint open sets that cover the set, but it is not arcwise connected because there is no continuous path connecting the two points.

Answer:

The confidence interval is -5.3444 to 6.453 .

Step-by-step explanation:

We are given that  In a survey of 458 likely voters, 254 said that they would vote "yes" on the referendum.

So, n = 458

x = 254

We will use sample proportion over here

[tex]\widehat{p}=\frac{x}{n}[/tex]

[tex]\widehat{p}=\frac{254}{458}[/tex]

[tex]\widehat{p}=0.5545[/tex]

Confidence level = 95% = 0.95

Level of significance = 1-0.95 = 0.05

z value at 0.05 significance level = 1.96

Formula of confidence interval : [tex]\widehat{p}-x\times \sqrt{\frac{\widehat{p} \times (1-\widehat{p})}{n}[/tex] to [tex]\widehat{p}+x\times \sqrt{\frac{\widehat{p} \times (1-\widehat{p})}{n}[/tex]

Confidence interval : [tex]0.5545-254\times \sqrt{\frac{0.5545\times (1-0.5545)}{458}}[/tex] to [tex]0.5545+254\times \sqrt{\frac{0.5545\times (1-0.5545)}{458}}[/tex]

Confidence interval : [tex]-5.3444[/tex] to [tex]6.453[/tex]

Hence The confidence interval is -5.3444 to 6.453 .

Answer:

Step-by-step explanation:

We have given,              

x=254          

n=458          

Estimate for sample proportion

Level of significance is =1-0.95=0.05      

Z critical value(using Z table)=1.96      

Confidence interval formula is              

 =(0.5091,0.6001)              

Lower limit for confidence interval=0.5091

Upper limit for confidence interval=0.6001

Answer:

[tex]y(x)=x^2+5x[/tex]

Step-by-step explanation:

Given: [tex]y'=\sqrt{4y+25}[/tex]

Initial value: y(1)=6

Let [tex]y'=\dfrac{dy}{dx}[/tex]

[tex]\dfrac{dy}{dx}=\sqrt{4y+25}[/tex]

Variable separable

[tex]\dfrac{dy}{\sqrt{4y+25}}=dx[/tex]

Integrate both sides

[tex]\int \dfrac{dy}{\sqrt{4y+25}}=\int dx[/tex]

[tex]\sqrt{4y+25}=2x+C[/tex]

Initial condition, y(1)=6

[tex]\sqrt{4\cdot 6+25}=2\cdot 1+C[/tex]

[tex]C=5[/tex]

Put C into equation

Solution:

[tex]\sqrt{4y+25}=2x+5[/tex]

or

[tex]4y+25=(2x+5)^2[/tex]

[tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex]

[tex]y(x)=x^2+5x[/tex]

Hence, The solution is [tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex] or [tex]y(x)=x^2+5x[/tex]

Answer:

[tex]12.0\%=0.12[/tex]

Step-by-step explanation:

We have been given that your friend borrows $100 from you and promises to pay you back $109 in 8 months.

We will use simple interest formula to solve our given problem.

[tex]A=P(1+rt)[/tex], where,

A = Amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form,

t = Time in years.

Convert 8 months to year:

[tex]\frac{8}{12}\text{ year}=\frac{2}{3}\text{ year}[/tex]

[tex]108=100(1+r*\frac{2}{3})[/tex]

[tex]108=100+r*\frac{2}{3}\times 100[/tex]

[tex]108-100+r*\frac{200}{3}[/tex]

[tex]108-100=100-100+r*\frac{200}{3}[/tex]

[tex]8=r*\frac{200}{3}[/tex]

[tex]8\times \frac{3}{200}=r*\frac{200}{3}\times \frac{3}{200}[/tex]

[tex]\frac{24}{200}=r[/tex]

[tex]r=\frac{24}{200}[/tex]

[tex]r=0.12[/tex]

Convert to percent:

[tex]0.12\times 100\%=12\%[/tex]

Therefore, you are charging 12% APR to you friend.

Answer:

the primes are all the even numbers in the equation

Step-by-step explanation:

Final answer:

The negation, contrapositive, converse, and inverse of a statement relating the divisibility of a number by 6, 2, and 3 are constructed by logically altering the original condition and consequent. These reflect different ways to express the relationship between the divisibility properties.

Explanation:

The original statement is: "If n is divisible by 6, then n is divisible by 2 and n is divisible by 3." Let's define the following propositions:

P: n is divisible by 6.

Q: n is divisible by 2.

R: n is divisible by 3.

The original statement can be written in logical form as P → (Q ∧ R).

Negation

The negation of the original statement is: "It is not the case that if n is divisible by 6, then n is divisible by 2 and n is divisible by 3." In logical form: ¬(P → (Q ∧ R)).

Contrapositive

The contrapositive of the original statement is: "If n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6." In logical form: (¬Q ∨ ¬R) → ¬P.

Converse

The converse of the original statement is: "If n is divisible by 2 and n is divisible by 3, then n is divisible by 6." In logical form: (Q ∧ R) → P.

Inverse

The inverse of the original statement is: "If n is not divisible by 6, then n is not divisible by 2 or n is not divisible by 3." In logical form: ¬P → (¬Q ∨ ¬R).

Answer:

The absolute and relative error of 355/113 compared with π is less than when π is compared with 22/7, that's why 355/113 is a better approximation for the actual value of π.

Step-by-step explanation:

The absolute error is the difference between a value measured and the real value.  

abs = π - approximation of π

The relative error indicates how large the absolute error is when compared with the actual value of π.

Now, let's calculate the absolute an relative error for each approximation of π, for simplicity the calculations will be rounded to 4 decimal digits.

rel = abs / π

abs = π - 22/7

abs = -0.0013

rel = (π - 22/7) / π

rel = -0.0402 %

abs = π - 355/113

abs = -2.6676 x10-7

rel =  (π - 355/113) / π

rel = -8.4914 x10-6 %

You can see that both the value of the absolute and relative error for the 355/113 approximation are smaller numbers, in conclusion, 355/113 is a better approximation for π.

Answer:

The business should reject the supplier's claim as mean length is not equal to claimed value of 26.70 inches.      

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 26.70 inches

Sample mean, [tex]\bar{x}[/tex] = 26.77 inches

Sample size, n = 48

Alpha, α = 0.05

Population standard deviation, σ = 0.20 inches

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 26.70\text{ inches}\\H_A: \mu \neq 26.70\text{ inches}[/tex]

We use Two-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{26.77 - 26.70}{\frac{0.20}{\sqrt{48}} } = 2.425[/tex]

Now, [tex]z_{critical} \text{ at 0.05 level of significance } = 1.96[/tex]

Since,

[tex]z_{stat} > z_{critical}[/tex]

We reject the null hypothesis and accept the alternate hypothesis. Thus, the business should reject the supplier's claim as mean length is not equal to claimed value of 26.70 inches.

Answer:

Step-by-step explanation:

Since your population are the students in math class, you can use the z-score formula [tex]z=(x-\mu)/\sigma[/tex] in order to comparing the two math test scores. Where [tex]\mu [/tex] is the mean for the class, [tex]\sigma [/tex] is the standars deviation and x is Mary score.

For the first test [tex]\mu=.71 , \sigma=.18,x=.87[/tex] , so ,[tex]z_{1} = (.87-.71)/(.18)=.88[/tex].

For the second test [tex]\mu=.53 , \sigma=.14,x=.66[/tex] , so ,[tex]z_{1} = (.66-.53)/(.14)=.93[/tex]

Mary do better in the second test, relative to the rest of the class (because  [tex].88 \leq .93[/tex], it means the second score is nearer to the mean score of the class than the first one )  

Answer:

The 5.5 would be called a parameter.

Step-by-step explanation:

A parameter in statistics is a data/number/quantity that gives you information about an entire population. Given that Dr. Johnson asked ALL of her students to respond the question and the median of 5.5 was based on ALL of her students, we can say that this number would be a parameter for the population (in this case, Dr. Johnson's class)

Answer:

equilibrium point (10,340)

Step-by-step explanation:

To find the equilibrium point, equal the demand and the supply:

[tex]D(q)=S(q)\\\\-6.10q^2-5q+1000=3.2q^2+10q-80[/tex]

Reorganize the terms in one side and reduce similar terms:

[tex]3.2q^2+6.1q^2+5q+10q-80-1000=0\\\\9.3q^2+15q-1080=0[/tex]

that's a cuadratic equation, solve with the general formula when:

a=9.3, b=15, c=-1080

[tex]q_{1}=\frac{-b+\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{1}=\frac{-15+\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{1}=\frac{-15+201}{18.6}\\\\q_{1}=\frac{186}{18.6}\\\\q_1=10[/tex]

q can't be negative because it is the quantity of bike frames, so:

[tex]q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-15-\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{2}=\frac{-15-201}{18.6}\\\\q_{2}=\frac{-216}{18.6}\\\\[/tex]

This value of q can't be considered.

Then substitute the value of q in D(q) to find the price p:

[tex]D(10) = -6.10(10)^2-5(10) + 1000\\\\D(10)=340=p[/tex]

The equilibrium point (q,p) is (10,340).

To find the equilibrium point in supply and demand equations, calculate where the two equations intersect to determine the equilibrium quantity and price.

Equilibrium Point Calculation:

Answer:

If you purchase it, you are going to save 0.43 = 43% of the original price

Step-by-step explanation:

This problem can be solved by a rule of three.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

In this problem, we have the following measures:

-The prices

-The percentage that the price represents.

As the value of one measure increases, so do the value of the other. It means that we have a direct rule of three.

The problem states that the model you really like happens to be on sale for $400. It's original price is $700.

You saved $700-$400 = $300.

What percent of the original price will you save if you purchase it?

How much is $300 of $700.

$700 - 1

$300 - x

700x = 300

[tex]x = \frac{300}{700}[/tex]

[tex]x = 0.43[/tex]

If you purchase it, you are going to save 0.43 = 43% of the original price

Step-by-step explanation:

To prove it we just use the definition of similar matrices and properties of determinants:

If [tex] A,B[/tex] are similar matrices, then there is an invertible matrix [tex]C[/tex], such that [tex] A=C^{-1}BC}[/tex] (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:

[tex]det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)[/tex]

[tex]=\frac{1}{det(C)}det(B)det(C)=det(B)[/tex]

(Determinant of a product of matrices is the product of their determinants, and the determinant of [tex]C^{-1}[/tex] is just [tex]\frac{1}{det(C)}[/tex])

Answer:

  67.73625

Step-by-step explanation:

The dot (scalar) product is also the sum of products of corresponding vector components.

~a • ~b = 9·2.97 +6.75·6.075 +0·0 = 27.73 +41.00625 = 67.73625

The scalar product or dot product of two vectors, ~a and ~b, in Cartesian form is calculated by multiplying the matching components of the two vectors and then adding them. Performing these steps will give us 67.73625, which represents the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.

In physics, the scalar product, also known as dot product, of two vectors like ~a = h9, 6.75, 0i and ~b = h2.97, 6.075, 0i can be determined using their magnitudes and the cosine of the angle between them. However, in the given question, the vectors are in Cartesian form (i,j,k coordinate system), and we can calculate their dot product directly. The dot product is calculated by multiplying the respective i, j, and k components of the two vectors and then adding them. Let's do this step by step:

This scalar product gives the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.

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Answer:

No , any two quadrilaterals may not be similar if  their corresponding angles are congruent.

Step-by-step explanation:

We need to check that whether two quadrilaterals are similar if

their corresponding angles are congruent.

A  quadrilateral is a polygon having two sides .

Two figures are said to be similar if they have same shape .

Two angles are said to be congruent if they have same measure .

Consider two quadrilaterals :  rectangle and square

Each angle of square and rectangle is equal to [tex]90^{\circ}[/tex] . So, their corresponding angles are congruent .

But square and rectangle are not similar as they have different shape .

Answer:

a) 1 unit per hour will be infused.

b) 5mL per hour will be infused.

Step-by-step explanation:

Each question can be solved as a rule of three with direct measures, that means we have a cross multiplication.

a. How many units per hour will be infused?

20 units are going to be infused in 20 hours. How many units are going to be infused each hour?

1 hour - x units

20 hours - 20 units

[tex]20x = 20[/tex]

[tex]x = \frac{20}{20}[/tex]

[tex]x = 1[/tex]

1 unit per hour will be infused.

b. How many milliliters per hour will be infused?

100mL are going to be infused in 20 hours. How many mL are going to be infused each hour?

1 hour - x mL

20 hours - 100 mL

[tex]20x = 100[/tex]

[tex]x = \frac{100}{20}[/tex]

[tex]x = 5[/tex]

5mL per hour will be infused.

Answer:

11/4

Step-by-step explanation:

Step 1: Simplify both sides of the equation.

12x+3−1=35

12x+3+−1=35

(12x)+(3+−1)=35(Combine Like Terms)

12x+2=35

12x+2=35

Step 2: Subtract 2 from both sides.

12x+2−2=35−2

12x=33

Step 3: Divide both sides by 12.

12x / 12 = 33 / 12x =

11 / 4

Answer: There are 1,816,214,400 ways for arrangements.

Step-by-step explanation:

Since we have given that

"REPRESENTATION"

Here, number of letters = 14

There are 2 R's, 3 E's, 2 T's, 2 N's

So, number of permutations would be

[tex]\dfrac{14!}{2!\times 3!\times 2!\times 2!}\\\\=1,816,214,400[/tex]

Hence, there are 1,816,214,400 ways for arrangements.

Answer:

  (a)  C(x) = 5550 +8.25x

  (b)  R(x) = 36x

  (c)  P(x) = 27.75x -5550; 200 hours to break even

Step-by-step explanation:

(a) Howen's costs include fixed costs and a cost per hour. Then her total cost will be the sum of the fixed cost (5550) and the product of hours (x) and the cost per hour (8.25):

  C(x) = 5550 +8.25x

__

(b) Howen plans to charge a given amount (36) per hour, so her revenue will be the product of that amount and the number of hours she works:

  R(x) = 36x

__

(c) Her profit function is the difference between revenue and cost:

  P(x) = R(x) -C(x)

  P(x) = 36x -(5550 +8.25x)

  P(x) = 27.75x -5550

Howen's break-even point is the number of hours required to make profit be zero:

  0 = 27.75x -5550

  0 = x - 200 . . . . . . . . . divide by 27.75

  200 = x . . . . . . . . . . . . add 200

She needs to work 200 hours to break even.

Bea's cost function for operating her snow plow is C(x) = 5550.00 + 8.25x. Her revenue function for the amount she earns is R(x) = 36.00x. The profit function, which is the revenue minus the cost, simplifies to P(x) = 27.75x - 5550. To break even, she needs to work approximately 200 hours.

The cost function C(x) for operating the snow plow for x hours includes the initial cost of the snow plow plus the hourly operating cost. This can be written as C(x) = 5550.00 + 8.25x.

The revenue function R(x), representing the amount of revenue gained from x hours of use, can be given as R(x) = 36.00x as she charges $36 for each hour.

The profit function P(x), representing the amount of profit, is the revenue function minus the cost function.so, P(x) = R(x) - C(x) which simplifies to P(x) = 36x - (5550 + 8.25x). Simplify that to get P(x) = 27.75x - 5550.

To find when she breaks even, we set the profit function equal to zero and solve for x:
0 = 27.75x - 5550
Adding 5550 to both sides gives: 27.75x = 5550
Dividing both sides by 27.75 gives: x ≈ 200. Therefore, she needs to work approximately 200 hours to break even.

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