Write an equation of the circle with center (6, 2) and radius 4.

In a true/false test where answers are guessed, the mean number of correct answers out of 18 would be 9 (50%) and the standard deviation would be 3. Passing by guessing with a minimum of 14 correct answers would be unusual as it is significantly above the mean and more than a single standard deviation away.

This question is about applying the binary distribution, specifically dealing with a true/false test. Before diving into the question, let's understand what we're working with. In a true/false test where students are guessing, the probability of a correct answer (p) is 0.5, as the expected value (mean) is np (number of trials times the probability of success) and the standard deviation is sqrt(np(1-p)).

In this scenario, for 18 questions, the mean (expected value) would be 18*0.5 = 9 and the standard deviation would be sqrt(18*0.5*0.5) = 3. Therefore, the mean is 9 and the standard deviation is 3.

For a student to 'pass' by guessing, they would need at least 14 correct answers. This number of successes is much greater than our mean expectation of 9, and it's more than a single standard deviation away. Therefore, it would be unusual for a student to pass the test by simply guessing.

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Answer:

Solution --> [tex] y(x) = x + \frac{2}{x^{2}} [/tex]

when x --> infinity the y goes to infinity too.

Step-by-step explanation:

We have the eq.

[tex] x y'+2 y = 3 x [/tex]

with y(1) = 3. So a reconfiguration o this last one equation can be like:

[tex] y' + \frac{2}{x} y = 3 [/tex]

so is of the form y'+p(x) y = f(x), where the integral factor is given by:

[tex] \mu = \exp[ \int p(x) dx] [/tex]

is,

[tex] \mu = \exp[ \int \frac{2}{x} dx] = x^{2} [/tex]

Multiplying the whole equation with this integral factor we can write the expression:

[tex] \int \frac{d}{dx}[y x^{2}] dx = \int 3x^{2} dx [/tex]

and from this we obtain,

[tex] y x^{2} = x^{3} + c,[/tex]

So when y(x=1) = 3, c = 2 and the complete solution can be writen as:

[tex] y(x) = x - \frac{2}{x^{2}} [/tex].

And we can see that when x --> infinity the second term of the solution is practically zero and the first is infinity so y also goes to infinity when x does.

The first-order linear ODE xy' + 2y = 3x is  y = x + 2/x² and the end behavior of y is approximately as x.

To solve the first-order linear ordinary differential equation (ODE) of the form xy' + 2y = 3x using an integrating factor, follow these steps:

1. Rewrite the Equation

2. Identify the Integrating Factor

3. Multiply by the Integrating Factor

4. Simplify and Integrate

5. Solve for y

6. Use Initial Condition

7. Find the End Behavior

Therefore, as x → ∞, y behaves approximately as x.

Complete Question:

Use an integrating factor to solve the following first order linear ODE. xy' + 2y = 3x, y(1) = 3 Find the end behavior of y as x rightarrow infinity.

Answer:

242.

Step-by-step explanation:

C = 0.03 x 2 − 8 x + 700

When the marginal cost = $522 we have:

522 = 0.03 x 2 − 8 x + 700

0.03x^2 - 8x + 700 - 522 = 0

0.03x^2 - 8x + 178 = 0

Using the quadratic formula:

x =  [ -(-8) +/- sqrt( 64 - 4 * 0.03 * 178)] / 0.06

= 8 +/- sqrt42.64 / 0.06

= 242  players.

242 DVD/Blu-ray players were produced

Marginal cost is the change in cost as a result of producing additional number of units.

Given that the marginal cost is:

C = 0.03x² - 8x + 700

Therefore for a marginal cost of $522, the number of DVD players is:

522 = 0.03x² - 8x + 700

0.03x² - 8x + 178 = 0

x = 24 and x = 242

Therefore 242 DVD/Blu-ray players were produced

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I guess the series is

[tex]\displaystyle\sum_{n=3}^\infty\frac8{(n-1)(n+1)}[/tex]

Splitting into partial fractions gives

[tex]\dfrac8{(n-1)(n+1)}=\dfrac4{n-1}-\dfrac4{n+1}[/tex]

The series telescopes, which we can see by considering the [tex]k[/tex]-th partial sum:

[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=(2-1)+\left(\frac43-\frac45\right)+\cdots+\left(\frac4{k-2}-\frac4k\right)+\left(\frac4{k-1}-\frac4{k+1}\right)[/tex]

(feel free to write out more terms to convince yourself that the following is true)

[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=2+\frac43-\frac4k-\frac4{k+1}[/tex]

Then as [tex]k\to\infty[/tex], we end up with

[tex]\displaystyle\lim_{k\to\infty}\sum_{n=3}^k\frac8{(n-1)(n+1)}=\boxed{\frac{10}3}[/tex]

The series is convergent with a sum of -1.143

The given series is:
∞ 8 (n-1)(n+1) / n = 3

First, we can use partial fraction decomposition to simplify the series. Factoring (n-1)(n+1) gives us [tex]n^2[/tex] - 1. So, the series becomes:

∞ 8 ([tex]n^2[/tex] - 1) / n = 3

Next, we can rewrite the series as two separate series:

∞ 8 [tex]n^2[/tex] / n - 8 / n = 3

Simplifying each series:

∞ 8n - 8 / n = 8∞ n³ - 1∞

The first series is a geometric series with a common ratio of 8 and a first term of 8. Using the formula for the sum of a geometric series, we can find its sum:

S = a / (1-r)

Substituting the values:

S = 8 / (1-8) = 8 / -7 = -1.143

Therefore, the series is convergent, and its sum is -1.143.

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When you have a sample of twelve people and 40 percent of them prefer chocolate ice cream, on average you would expect about 4.8 people to pick chocolate as their favorite flavor. Depending on how you interpret 'expect', this could mean you expect 4 or you 'expect' 5 people.

To calculate the expected number of people who prefer chocolate ice cream, we would multiply the total sample size by the percentage that indicated chocolate as their favorite flavor. In this case, the sample size is twelve people, and the percentage that prefers chocolate is 40 percent (or 0.4 when expressed as a decimal).

So, we calculate: 12 * 0.4 = 4.8

However, since we cannot have a fraction of a person, we would either round down to 4 or round up to 5. It depends on what you understand by 'expect'. If you expect at least some amount, then you expect 5 people (because you can't get 4.8 people, you need at least 5 to cover the 4.8). But if you understand 'expect' as an average, then you expect 4 people because fractions are permissible in averages.

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The correct answer is:

Expected number of people who would name chocolate as their favorite flavor is 5.

To solve this problem, we will calculate the expected number of people who would name chocolate as their favorite flavor out of a sample of twelve people, given that 40 percent indicated chocolate was their favorite flavor.

First, we need to determine what 40 percent of twelve people is. To do this, we convert the percentage to a decimal by dividing by 100, which gives us 0.40. Then we multiply this decimal by the number of people in the sample:

[tex]\[ \text{Expected number of people} = 0.40 \times 12 \] \[ \text{Expected number of people} = 4.8 \][/tex]

Since we cannot have a fraction of a person, the expected number is 4.8 people. Rounding off the answer we get 5 as expected number of people.

Answer:

12.1%

Step-by-step explanation:

First calculate the z-score:

z = (x − μ) / σ

z = (55000 − 48000) / 6000

z = 1.17

Look up in a z-score table.

P(z>1.17) = 1 − 0.8790

P(z>1.17) = 0.1210

There's a 12.1% probability that a randomly selected individual with a master's in statistics will have a starting salary of at least $55,000.

The probability that a randomly selected individual with a Master's in statistics will get a starting salary of at least $55,000 is 12.17%.

The problem you're dealing with involves the normal distribution which is a type of continuous probability distribution for a real-valued random variable. Given that we have a mean (μ) of $48,000, and a standard deviation (σ) of $6,000, we are asked to find the probability that a randomly selected individual with a master's in statistics will get a starting salary of at least $55,000.

This involves calculating a 'Z-score', which tells us how many standard deviations an element is from the mean. To find it, you would subtract the mean from the value of interest and divide by the standard deviation.

So, in this case, the Z-score would be: Z = (55000 - 48000) / 6000 = 1.167. Using standard Z-score tables, this corresponds to a probability of 0.8783.

However, as we are interested in the probability of a salary being at least $55,000, we need to subtract this value from 1 to get the final probability. So, the answer is: 1 - 0.8783 = 0.1217 or 12.17%.

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Answer:

-18

Step-by-step explanation:

Simplify the following:

abs(3 - 10) - (12/4 + 2)^2

The gcd of 12 and 4 is 4, so 12/4 = (4×3)/(4×1) = 4/4×3 = 3:

abs(3 - 10) - (3 + 2)^2

3 + 2 = 5:

abs(3 - 10) - 5^2

3 - 10 = -7:

abs(-7) - 5^2

5^2 = 25:

abs(-7) - 25

Since -7<=0, then abs(-7) = 7:

7 - 25

7 - 25 = -18:

Answer:  -18

For this case we must resolve the expression:

[tex]| 3-10 | - (\frac {12} {4}+2) ^ 2[/tex]

We have the following definitions:

Different signs are subtracted and the sign of the major is placed.

Absolute value definition:[tex]| -a | = a[/tex]

Resolving we have:

[tex]| 3-10 | = | -7 | = 7[/tex]

[tex](\frac {12} {4}+2) ^ 2 = (3+2) ^ 2 = 5 ^ 2 = 25[/tex]

Then, rewriting the expression:

[tex]7-25 = -18[/tex]

Answer:

-18

Answer: There are 52 men we can expect to have been involved in a minor traffic accident.

Step-by-step explanation:

Since we have given that

Percentage of men has been involved in a minor traffic accident = 13%

Number of men surveyed = 400

So, Number of sample men we can expect to have been involved in a minor traffic accident is given by

[tex]\dfrac{13}{100}\times 400\\\\=0.13\times 400\\\\=52[/tex]

Hence, there are 52 men we can expect to have been involved in a minor traffic accident.

[tex]\bf \begin{cases} t=\textit{tv ads}\\ r=\textit{radio ads}\\ n=\textit{newspaper ads} \end{cases}~\hspace{7em}t+r+n=88.9 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{tv, radio ads 3.1 more than newspaper}}{t+r=n+3.1}~\hfill \stackrel{\textit{newspaper ads 5.4 more than tv's}}{\boxed{n} = t+5.4}[/tex]

[tex]\bf \stackrel{\textit{substituting on the 2nd equation}}{t+r=\boxed{t+5.4}+3.1}\implies ~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+r=~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+8.5\implies \blacktriangleright r=8.5 \blacktriangleleft \\\\\\ \stackrel{\textit{we know that}}{t+r+n=88.9}\implies t+8.5+n=88.9\implies t+n=80.4[/tex]

[tex]\bf \boxed{n}=80.4-t~\hspace{7em}\stackrel{\textit{substituting on the 3rd equation}}{n=t+5.4\implies \boxed{80.4-t}=t+5.4} \\\\\\ 75-t=t\implies 75=2t\implies \cfrac{75}{2}=t\implies \blacktriangleright 37.5=t \blacktriangleleft \\\\\\ \stackrel{\textit{since we know that}}{n = t+5.4}\implies n=37.5+5.4\implies \blacktriangleright n=42.9 \blacktriangleleft[/tex]

The amounts spent on Newspaper, Television and Radio ads are $40.3 billion, $34.9 billion, and $13.7 billion, respectively.

Let us represent the three advertising channels - newspaper, television, and radio - as N, T, and R respectively. Given that total spending on advertising is $88.9 billion, we can write this formally as N + T + R = 88.9 (equation 1). The problem also mentions that total spending on television and radio ads is $3.1 billion more than spending on newspaper ads. This can be written formally as T + R = N + 3.1 (equation 2). Lastly, it is given that spending on newspaper ads is $5.4 billion more than what was spent on television ads. Writing this we get N = T + 5.4 (equation 3). Substituting equation 3 into equation 2, we get (T + 5.4) + R = T + R + 3.1. This simplifies to T - R = 2.3. Solving these equations, we find that $40.3 billion is spent on Newspaper ads, $34.9 billion is spent on Television ads, and $13.7 billion is spent on Radio ads.

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[tex](x-a)^2+(y-b)^2=r^2[/tex]

center - [tex](a,b)[/tex]

[tex]x^2 + y^2 + 6x -8y +21 = 0 \\x^2+6x+9+y^2-8y+16-4=0\\(x+3)^2+(y-4)^2=4[/tex]

center - [tex](-3,4)[/tex]

[tex]r=2[/tex]

Answer:

The center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]

Step-by-step explanation:

The general equation of a circle has the following formula:

[tex](x-h)^2 + (y-k)^2 = r^2\\[/tex]

Where r is the radius of the circle and (h, k) is the center of the circle

In this case we have the following equation

[tex]x^2 + y^2 + 6x -8y +21 = 0[/tex]

To find the radius and the center of this cicunference we must rewrite the equation in the general form of a circumference completing the Square

[tex](x^2 + 6x)+ (y^2 -8y) +21 = 0\\\\(x^2 + 6x+9)+ (y^2 -8y+16) +21 = 9+16\\\\(x^2 + 6x+9)+ (y^2 -8y+16) = 9+16-21\\\\(x+3)^2+ (y-4)^2 = 4\\\\(x+3)^2+ (y-4)^2 = 2^2[/tex]

Then the center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]

Answer:

The symbol used for the sample standard deviation is the 's', while the symbol used for the population standard deviation is a "sigma" or σ.

On the other hand, the sample variance comes to be the square of the sample standard deviation: s². Likewise, the the population variance comes to be the square of the population standard deviation which is: σ²

The symbols used for sample standard deviation, population standard deviation, sample variance, and population variance are s, σ (sigma), s², and σ² (sigma squared), respectively.

When discussing statistics, there are specific symbols used for different concepts. For the (a) sample standard deviation, the most commonly used symbol is s. In terms of (b) population standard deviation, the symbol commonly used for this is a Greek letter, σ (lowercase sigma).

Then, when shifting the focus from standard deviation to variance, there are different symbols. For (c) sample variance, the symbol used is s², which is the sample standard deviation squared. As for the (d) population variance, this is represented by the square of the population standard deviation, σ² (sigma squared).

The symbols used for each of the following are:

(a) Sample standard deviation: s

(b) Population standard deviation: σ (lowercase sigma)

(c) Sample variance: s²

(d) Population variance: σ² (lowercase sigma squared)

To calculate the standard deviation, you first need to calculate the variance. The variance is the average of the squares of the deviations. The sample standard deviation is represented by s, while the population standard deviation is represented by lowercase Greek letter σ (sigma).

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Complete question

Identify the symbols used for each of the​ following:

(a) sample standard​ deviation;

(b) population standard​ deviation;

(c) sample​ variance;

(d) population variance.

a. The symbol for sample standard deviation is▼ss squaredsigmasigma squared.

b. The symbol for population standard deviation is▼ss squaredsigmasigma squared.

c. The symbol for sample variance is▼ss squaredsigmasigma squared.

d. The symbol for population variance is▼ss squaredsigmasigma squared.

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The correct option is B.

Any prism volume is V = BH where B is an area of the base and H is the height of the prism, so find the area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.

How to Find the volume of the prism?

The triangles on the two ends are congruent.

So, to find the prism area, it is enough to take the area of one of the triangles and multiply it by the prism depth, which is 20 units.

Use the Pythagorean theorem to find the height of the triangle.

5^2 - 3^2 = 4^2, so the height is 4.

Multiply half times height times base: 1/2 x 4 x 3 = 6.

Multiply that by the depth, 20.

6x20=120 units^3.

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Answer: 0.0325

Step-by-step explanation:

Binomial probability distribution formula for x successes in n trials:-

[tex]P(X=x)=^nC_x\ p^x\ (1-p)^{n-x}[/tex], where n is the number of trials , p is the probability of success.

Given : The probability that taxpayers who filed their tax return are electronically​ self-prepared their taxes : [tex]p= 0.319[/tex]

If three tax returns submitted electronically are randomly​ selected, then the probability that all three were​ self-prepared is given by :-

[tex]P(X=3)=^3C_3\ (0.319)^3\ (1-0.319)^{3-3}\\\\=(1) (0.319)^3(1)\\=0.032461759\approx0.0325[/tex]

Hence, the probability that all three were​ self-prepared =0.0325

Final answer:

The probability that all three tax returns submitted electronically are self-prepared, when each has a 31.9% chance of being self-prepared, is the product of their individual probabilities, resulting in approximately 3.24%.

Explanation:

To compute this, we will use the property that the probability of independent events all occurring is the product of their individual probabilities.

The probability of one taxpayer self-preparing is 0.319. Since the question implies that the taxpayers are chosen independently, the probability that all three returns are self-prepared is:

P(First self-prepared) = 0.319,

P(Second self-prepared) = 0.319,

P(Third self-prepared) = 0.319.

To find the combined probability, we multiply these probabilities together:

P(All three self-prepared) = 0.319 × 0.319 × 0.319,

This calculation results in approximately 0.0324, or 3.24%.

A) The retail price of the suit would be $320.

B) The wholesale price of the pair of jeans would be $50.

Given that;

Retail prices in a department store are obtained by marking up the wholesale price by 60 %. That​ is, the retail price is obtained by adding 60 % of the wholesale price to the wholesale price.

(A) Now for the retail price of the suit, we need to add 60% of the wholesale price to the wholesale price itself.

So, if the wholesale price is $200, calculate the retail price as follows:

Retail price = Wholesale price + (60% of Wholesale price)

Retail price = $200 + (0.60 × $200)

Retail price = $200 + $120

Retail price = $320

Therefore, the retail price of the suit would be $320.

(B) Now, for the wholesale price of the pair of jeans, reverse the calculation.

If the retail price is $80 and the markup is 60%, calculate the wholesale price as follows:

Wholesale price = Retail price / (1 + Markup percentage)

Wholesale price = $80 / (1 + 0.60)

Wholesale price = $80 / 1.60

Wholesale price = $50

Hence, the wholesale price of the pair of jeans would be $50.

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The retail price of a suit with a wholesale price of $200 would be $320. To find the wholesale price of a pair of jeans that retails for $80, you calculate the original wholesale price to be $50.

To answer question (A), we calculate the retail price by adding 60% of the wholesale price to the wholesale price itself. For a wholesale price of $200, this would be:

Retail Price = Wholesale Price + (0.60 × Wholesale Price)
Retail Price = $200 + (0.60 × $200)
Retail Price = $200 + $120
Retail Price = $320

For question (B), we find the wholesale price of a pair of jeans if the retail price is $80 by letting W represent the wholesale price and solving for W:

Retail Price = W + (0.60 × W)
$80 = W + (0.60 × W)
$80 = 1.60W
W = $80 / 1.60
W = $50

Answer:

11 dance tickets were sold. ( approx )

Step-by-step explanation:

Let x represents the number of car wash tickets, y represents the number of silly string fight tickets and z represents number of dance tickets,

Total tickets = 380,

⇒ x + y + z = 380 -----(1),

The car wash tickets were $5 each, the silly string fight tickets were $3 each and the dance tickets were $10 each, also total cost is $1460,

⇒ 5x + 3y + 10z = 1460 ----(2)

Also, there are twice as many silly string tickets as car wash tickets,

⇒ y = 2x -----(3)

From equation (1) and (2),

x + 2x + z = 380 ⇒ 3x + z = 380 -----(4)

5x + 3(2x) + 10z = 1460 ⇒ 11x + 10z = 1460 ------(5)

For finding the value of z,

3 × equation (5) - 11 × equation (4),

We get,

30z - 11z = 4380 - 4180

19z = 200

z = 10.5263157895 ≈ 11

Hence, 11 dance tickets were sold.

Answer: 0.9917

Step-by-step explanation:

If repetition is allowed , then the total number of possible four digits pin codes = [tex]10^4=10,000[/tex]

Number of ways to make for digit code without repetition of digits =

[tex]10\times9\times8\times7=5040[/tex]

Number of ways to make for digit codes having repetition =

[tex]10,000-5040=4960[/tex]

Probability that a person has pin code that has repetition:-

[tex]\dfrac{4960}{10,000}=0.496[/tex]

Let x be number of pin codes with repeating digits.  

If the PIN codes of seven people are selected at random, then the probability that at least one of them will have repeating digits:-

[tex]P(x\geq1)=1-(P(0))\\\\=1-(^7C_0(0.496)^0(1-0.496)^7)[/tex]  (By Binomial distribution)

[tex]=1-((0.496)^0(0.504)^7)=0.991739358875\approx0.9917[/tex]

Hence, the probability that at least one of them will have repeating digits = 0.9917

Answer: e. 0.3849

Step-by-step explanation:

We know that the mean lies exactly at the middle of the normal curve .

The z-score of the mean value is 0.

Also According to the standard normal probability table, the probability value of z=0 is P(z<0)=0.5.

And the probability value of z=-1.20 is P(z<-1.2) =0.1150697.

Now, the  area under the normal curve between the mean and a score for which z= - 1.20 is  given by :-

[tex]P(z<0)-P(z<-1.2)=0.5-0.1150697=0.3849303\approx0.3849[/tex]

Hence, the area under the normal curve between the mean and a score for which z= - 1.20 is 0.3849 .

Answer:

A. negatively skewed

Step-by-step explanation:

Plot the values of salary against years

You will notice that;

This mean that this graph is a negative skewed graph/left-skewed distribution

In the attached sketch of the plot, join the points with a smooth curve and observe the above mentioned properties.

The x-axis scale ranges from 2000 to 2015 where 0=2000, 1=2001,2=2002......,15=2015

Hope this will give you a visual picture of the negatively- skewed distribution

Answer:

Given,

The initial population, P = 87,000,

Annual rate of increasing, r = 1.9 %,

a) Thus, the population after t years,

[tex]A=P(1+\frac{r}{100})^t[/tex]

[tex]A=87000(1+\frac{1.9}{100})^{t}[/tex]

[tex]=87000(1+0.019)^t[/tex]

[tex]=87000(1.019)^t[/tex]

[tex]87000(1.019)^t=87000e^{kt}[/tex]

Where, k is the rate constant,

By comparing,

[tex]\implies k=log(1.019)=0.00817418400643\approx 0.00817[/tex]

Hence, the approximate value of k is 0.00817.

And, the exponential growth function would be,

[tex]A=87000e^{0.00817t}[/tex]

b) If A = 145,000,

[tex]145000=87000(1.019)^t[/tex]

By using graphing calculator,

We get,

[tex]t=27.14\approx 27[/tex]

The year after 27 years from 2016 is 2043.

Hence, in 2043 the population reach 145,000​.

The rate constant k in the given exponential growth problem is 1.9% per year. The exponential growth function for this problem is P = 87,000 * e^(0.019t). The year in which the population reaches 145,000 can be found by solving the equation 145,000 = 87,000 * e^(0.019t) for t, which will be counted from 2016.

The given problem is an example of an exponential growth problem. The formula for exponential growth is P = P0 * e^(kt), where P is the future population, P0 is the initial population (87,000 in this case), e is the natural base (approximately equal to 2.71828), k is the growth rate (1.9% or 0.019 in decimal), and t is the time (in years).

a. Finding the rate constant k
The rate constant k is given in the problem as 1.9% per year or 0.019 per year when expressed in decimal form.

The exponential growth function that fits the given data would therefore be P = 87,000 * e^(0.019t).

b. When will the population reach 145,000?
To find out when the population will reach 145,000, we need to set P equal to 145,000 and solve for t, giving us the equation 145,000 = 87,000 * e^(0.019t). Solving this equation using logarithms will yield the year, which will be counted from the base year 2016.

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Answer with explanation:

The meaning of bijection for two sets X and Y is each and every element of X is uniquely related with element of set Y and when you take the inverse mapping every element of set Y is uniquely related with each and every element of Set X.

The two sets given are

X=(0,1]---------Semi open or Semi closed Set

Y=(0,1)---------Closed set

Between any two real numbers there are infinite number of real numbers.So cardinal number of both the sets is infinite.

There can be infinite bijection between these two sets as both sets have infinite number of elements.

   X                      Y

0.1  -------------      0.01

0.2     -------          0.02

0.3     --------         0.03

0.4     ---------          0.04

-------------------------

------------------------------

-------------------------------

--------------------------------------

One example of a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex] is the function [tex]\(f(x) = \frac{x}{1 + x}\)[/tex]. This function is injective (one-to-one) and surjective (onto), making it a bijection.

To establish a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex], consider the function [tex]\(f(x) = \frac{x}{1 + x}\).[/tex]

1. Injectivity (One-to-One): Assume [tex]\(f(x_1) = f(x_2)\)[/tex] for some [tex]\(x_1, x_2 \in (0,1]\)[/tex]. By solving the equation [tex]\(\frac{x_1}{1 + x_1} = \frac{x_2}{1 + x_2}\)[/tex], you find [tex]\(x_1 = x_2\)[/tex], showing that distinct elements in the domain map to distinct elements in the codomain.

2. Surjectivity (Onto): For any y in the codomain [tex]\((0,1)\)[/tex], solve [tex]\(f(x) = y\) for \(x\)[/tex]. This results in [tex]\(x = \frac{y}{1 - y}\)[/tex], which is well-defined for [tex]\(y \in (0,1)\)[/tex]. Therefore, every element in the codomain has a preimage in the domain.

Hence, [tex]\(f(x)\)[/tex] is a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex].

Answer:

The width is 16 inches while the length is 24 inches.

Step-by-step explanation:

We are given that the length (L) is 8 more than it's width (W).

The equation for this is: L=8+W.

We are given the perimeter is 80.

The equation for this is 2L+2W=80.

Each term here is even so I'm going to divide both sides by 2: L+W=40.

My system to solve is:

L=8+W

L+W=40

I'm going to plug the first equation into the second giving me:

(8+W)+W=40

Drop the (), don't really need:

8+W+W=40

Combine like terms:

8+2W=40

Subtract 8 on both sides:

    2W=32

Divide both sides by 2:

      W=32/2

Simplify:

     W=16

So if W=16 and L=8+W then L=8+16=24.

The width is 16 inches while the length is 24 inches.

Perimeter is the sum of the length of the sides used to make the given figure. The length and the width of the rectangle are 24 inches and 16 inches, respectively.

Perimeter is the sum of the length of the sides used to make the given figure.

Let the width of the rectangle be represented by x.

Given that the length of a rectangle exceeds its width by 8 inches. Therefore, the length and the width of a rectangle can be written as,

Width = x inches

Length = x inches + 8 inches

Now, as it is given that the perimeter of the rectangle is 80 inches. Therefore, the perimeter of the given rectangle can be written as,

Perimeter of rectangle = 2(Length + Width)

80 inches = 2[(x+8) + x]

80 = 2(x + 8 + x)

80 = 2(2x + 8)

80/2 = 2x + 8

40 = 2x + 8

40 - 8 = 2x

32 = 2x

x = 32/2

x = 16 inches

Further, the length and the width of the rectangle can be written as,

Width = x = 16 inches

Length = x + 8 = 16 + 8 = 24 inches

Hence, the length and the width of the rectangle are 24 inches and 16 inches, respectively.

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Answer:

Step-by-step explanation:

1. Multiply both sides of the equation by the reciprocal of the coefficient of r.

[tex]V\cdot\dfrac{3}{2\pi h^2}=r\\\\r=\dfrac{3V}{2\pi h^2}[/tex]

__

2. Multiply both sides of the equation by the reciprocal of the coefficient of h.

[tex]V\cdot\dfrac{3}{b^2}=h\\\\h=\dfrac{3V}{b^2}[/tex]

__

3. Solve the circumference formula for r, then substitute the given information.

[tex]C=2\pi r\\\\r=\dfrac{C}{2\pi}\qquad\text{divide by the coefficient of r}\\\\r=\dfrac{50\,\text{cm}}{2\pi}=\dfrac{25}{\pi}\,\text{cm}\approx 7.9577\,\text{cm}[/tex]

__

4. Solve the perimeter formula for width, the substitute the given information and do the arithmetic.

[tex]P=2(L+W)\\\\\dfrac{P}{2}=L+W\qquad\text{divide by 2}\\\\\dfrac{P}{2}-L=W\qquad\text{subtract L}\\\\\dfrac{40\,\text{cm}}{2}-5\,\text{cm}=W=15\,\text{cm}[/tex]

_____

In general, solving for a particular variable involves "undoing" what has been done to the variable, usually in the reverse order. In part 4, the variable W has L added and the sum is multiplied by 2. We "undo" those operations, last operation first, by dividing by 2 and subtracting L.

The properties of equality say you can do what you like to an equation as long as you do the same thing to both sides of the equation. So, when we say "divide by 2", we mean "divide both sides of the equation by 2." Likewise, "subtract L" means "subtract L from both sides of the equation."

Answer:

2.0

Step-by-step explanation:

x = age

y = average systolic blood pressure

x₁ = 40 years old, y₁ = 110 systolic blood pressure

x₂ = 50 years old, y₂ = 130 systolic blood pressure

[tex]\text{Slope of a line}=m=\frac{y_2-y_1}{x_2-x_1}\\\Rightarrow m= \frac{130-110}{50-40}\\\Rightarrow\frac{20}{10}\\\Rightarrow m= 2.0[/tex]

∴ Slope of the regression line is 2.0

Final answer:

The slope of the regression line relating average systolic blood pressure to age is 2.0 when rounded to one decimal place.

Explanation:

To find the slope of the regression line relating average systolic blood pressure (y) to age (x), we can use the two points provided (x1, y1) = (40, 110) and (x2, y2) = (50, 130). The slope (m) of a line passing through two points is calculated using the formula:

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

Substituting the given values:

m = (130 - 110) / (50 - 40) = 20 / 10 = 2

Therefore, the slope of the regression line is 2.0 when rounded to one decimal place.

Answer:

The value of first coin will be $151.51 more than second coin in 15 years.

Step-by-step explanation:

You have just purchased two coins at a price of $670 each.

You believe that first coin's value will increase at a rate of 7.1% and second coin's value 6.5% per year.

We have to calculate the first coin's value after 15 years by using the formula

[tex]A=P(1+\frac{r}{100})^{n}[/tex]

Where A = Future value

           P = Present value

           r = rate of interest

           n = time in years

Now we put the values

[tex]A=670(1+\frac{7.1}{100})^{15}[/tex]

[tex]A=670(1+0.071)^{15}[/tex]

[tex]A=670(1.071)^{15}[/tex]

A = (670)(2.797964)

A = 1874.635622 ≈ $1874.64

Now we will calculate the value of second coin.

[tex]A=670(1+\frac{6.5}{100})^{15}[/tex]

[tex]A=670(1+0.6.5)^{15}[/tex]

[tex]A=670(1.065)^{15}[/tex]

A = 670 × 2.571841

A = $1723.13

The difference of the value after 15 years = 1874.64 - 1723.13 = $151.51

The value of first coin will be $151.51 more than second coin in 15 years.

To find out how much more the first coin will be worth compared to the second coin in 15 years, we can apply the formula for compound interest. The formula is:
\[ A = P(1 + r)^t \]
where:
- \( A \) is the future value of the investment/loan, including interest
- \( P \) is the principal investment amount (the initial deposit or loan amount)
- \( r \) is the annual interest rate (decimal)
- \( t \) is the number of years the money is invested or borrowed for
We will apply this formula separately for each coin and then find the difference between the two future values.
First, let's calculate the future value of the more collectible coin:
\( P = \$670 \) (the initial value of the coin)
\( r = 7.1\% = 0.071 \) (the annual increase rate converted to decimal)
\( t = 15 \) years
So the future value \( A_1 \) for the first coin is calculated as:
\[ A_1 = \$670 \times (1 + 0.071)^{15} \]
Now, let's calculate the future value of the less collectible coin:
\( P = \$670 \) (the initial value of the coin)
\( r = 6.5\% = 0.065 \) (the annual increase rate converted to decimal)
\( t = 15 \) years
So the future value \( A_2 \) for the second coin is calculated as:
\[ A_2 = \$670 \times (1 + 0.065)^{15} \]
Finally, to find out how much more the first coin will be worth, we subtract the future value of the second coin from the future value of the first coin:
\[ \text{Difference} = A_1 - A_2 \]
Let's compute this step by step.
First, calculate \( A_1 \):
\[ A_1 = \$670 \times (1 + 0.071)^{15} \]
\[ A_1 = \$670 \times (1.071)^{15} \]
\[ A_1 = \$670 \times 2.83844... \] (using a calculator)
\[ A_1 \approx \$1901.75 \] (rounded to two decimal places)
Next, calculate \( A_2 \):
\[ A_2 = \$670 \times (1 + 0.065)^{15} \]
\[ A_2 = \$670 \times (1.065)^{15} \]
\[ A_2 = \$670 \times 2.64716... \] (using a calculator)
\[ A_2 \approx \$1775.63 \] (rounded to two decimal places)
Now calculate the difference:
\[ \text{Difference} = A_1 - A_2 \]
\[ \text{Difference} = \$1901.75 - \$1775.63 \]
\[ \text{Difference} \approx \$126.12 \]
Therefore, if your expectations are correct, in 15 years, the first coin will be worth approximately $126.12 more than the second coin.

Answer: Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.

Step-by-step explanation:

Since we have given that

Integers between 10000 and 99999 = 99999-10000+1=90000

n( divisible by 3) = [tex]\dfrac{90000}{3}=30000[/tex]

n( divisible by 5) = [tex]\dfrac{90000}{5}=18000[/tex]

n( divisible by 7) = [tex]\dfrac{90000}{7}=12857.14[/tex]

n( divisible by 3 and 5) = n(3∩5)=[tex]\dfrac{90000}{15}=6000[/tex]

n( divisible by 5 and 7) = n(5∩7) = [tex]\dfrac{90000}{35}=2571.42[/tex]

n( divisible by 3 and 7) = n(3∩7) = [tex]\dfrac{90000}{21}=4285.71[/tex]

n( divisible by 3,5 and 7) = n(3∩5∩7) = [tex]\dfrac{90000}{105}=857.14[/tex]

As we know the formula,

n(3∪5∪7)=n(3)+n(5)+n(7)-n(3∩5)-n(5∩7)-n(3∩7)+n(3∩5∩7)

[tex]=30000+18000+12857.14-6000-2571.42-4258.71+857.14\\\\=48884.15[/tex]

Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.

Answer:

y = [tex]C_1e^{3t}+C_2e^{6t}[/tex] + [tex]\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]

Step-by-step explanation:

y''- 9 y' + 18 y = t²

solution of ordinary differential equation

using characteristics equation

m² - 9 m + 18 = 0

m² - 3 m - 6 m+ 18 = 0

(m-3)(m-6) = 0

m = 3,6

C.F. = [tex]C_1e^{3t}+C_2e^{6t}[/tex]

now calculating P.I.

[tex]P.I. = \frac{t^2}{D^2 - 9D +18}[/tex]

[tex]P.I. = \dfrac{t^2}{(D-3)(D-6)}\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(1-\frac{D}{6})^{-1}(t^2)\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(1+\frac{D}{6}+\frac{D^2}{36}+....)(t^2)\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(t^2+\frac{2t}{6} + \frac{2}{36})\\P.I. =\dfrac{1}{18}(1+\frac{D}{3}+\frac{D^2}{9}+....)(t^2+\frac{2t}{6} + \frac{2}{36})\\P.I. =\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]

hence the complete solution

y = C.F. + P.I.

y = [tex]C_1e^{3t}+C_2e^{6t}[/tex] + [tex]\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]

Answer:

0.935

Step-by-step explanation:

Let's notice that finding 1 or more defective watches in the sample is the complement (exactly the opposite) of finding 0 defective watches. That means P (1 or more defective watches) + P (0 defective watches) = 1

Let's find out P (0 defective watches):

The chances of getting 1 defective watch is 6 in 30 (in a shipment of 30 there are only 6 defective).

Therefore, we have 24/30 chances of getting a non-defective watch.  

Needing 0 defective watches is the same as having 10 non defective watches.  

When the receiving department select the first one, the probability of getting a non-defective watch is 24/30.

In the second one, they have a probability of 23/29 (they already took a non-defective one).

In the third one, the probability is 22/28.

And so on, up to the last one where the probability is 15/21.

As we need this to happen all at the same time, we have to multiply it:

P (0 defective watches) = [tex]\frac{24}{30}*\frac{23}{29}*\frac{22}{28}*\frac{21}{27}*\frac{20}{26}*\frac{19}{25}*\frac{18}{24}*\frac{17}{23}*\frac{16}{22}*\frac{15}{21}[/tex] = 0.065

Therefore, P (1 or more defective watches) = 1 - P (0 defective watches) = 1 - 0.065 = 0.935

The probability that the shipment will be rejected is 0.8696 or 86.96%.

To find the probability that the shipment will be rejected, we need to find the probability that 1 or more watches are defective in a sample of 10. This can be done using the complement rule, which states that the probability of an event happening is equal to 1 minus the probability of it not happening.

The probability of selecting a defective watch in the sample is given by:

P(defective) = 6/30 = 1/5

Therefore, the probability of not selecting a defective watch is:

P(not defective) = 1 - P(defective) = 1 - 1/5 = 4/5

The probability that none of the watches in the sample are defective is:

P(all not defective) = (4/5)^10

Finally, the probability that at least one watch is defective (and the shipment is rejected) is:

P(rejected shipment) = 1 - P(all not defective) = 1 - (4/5)^10 = 0.8696

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Answer: 31.23

Step-by-step explanation: Jane kept 3 bottles for her self and then sold the rest for 17 $ each and the amount of money she made off of selling the rest of the bottles was 480$ and if you divide 480 by 17 you get 28.23 bottles sold and the extra 3 she kept for herself I am not 100% sure.

Jane bought 18 bottles of wine in total. She kept 3 for herself and sold 15, making a profit of $17 on each of the 15 bottles.

Jane bought a case of wine for $345, kept three bottles for herself, and sold the rest for $480, making a profit of $17 on each bottle sold. There were 18 bottles in the case.

The correct approach is to find the factors of $345 and $480 that are 3 bottles apart since Jane kept three bottles.

 The factors of $345 are 1, 3, 5, 7, 15, 21, 23, 35, 45, 49, 69, 77, 115, 147, and 345.

The factors of $480 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, and 480.

The common factors are 1, 3, 5, and 15. Since Jane made a profit on each bottle sold, the number of bottles sold must be more than the number of bottles she kept. Therefore, the number of bottles Jane could have bought and sold must be a factor of $480 that is 3 more than a factor of $345.

The only factor of $480 that is 3 more than a factor of $345 is 18, since 15 (a factor of $345) + 3 = 18 (a factor of $480).

Answer:

The solution is: x=-1 and y=3.

Step-by-step explanation:

Let's find the solution, but first let's remember the following:

A / B = C + R where:

A=dividend, B=divisor, C=quotient, and R=remainder. This can be express as follows:

A = (C * B) + R, which is the structure we are going to use next.

Using the Euclidian Algorithm we need to find the highst common factor (HCF) between the coefficients from your equation, this means:

The original equation: 26x+9y=1 is in the form Ax+By=C, so A=26, B=9 and C=1.

We need to find the HCF of 'A' and 'B'. Using the Euclidan Algorithm (EA), so we have:

A = (C * B) + R, using our values:

26 = (2 * 9) + 8, look that the divisor (B) is 9 and the remainder (R) is 8.

Now using the (EA) we divide the divisor (B=9) by the obtained remainder (R=8). And we do the same for each obtained result until the las remainder (R) is equal to 0, like this:

A = (C * B) + R

9 = (8 * 1) + 1

using the divisor (B=8) and the remainder (R=1) we obtain:

A = (C * B) + R

8 = (1 * 8) + 0, look that the remainder is now 0, so in summary we can use the method as follows:

26 = (2 * 9) + 8

9 = (8 * 1) + 1

8 = (1 * 8) + 0; and this equations are the ones we are going to use in order to find a solution.

Next step is to use the equation before R=0, so:

9 = (8 * 1) + 1, which is:

9 - (8 * 1) = 1; but if you consider the first obtained equation: 26 = (2 * 9) + 8, we can write:

26 - (2 * 9) = 8, and we can use this expression in the previous one, so:

9 - (8 * 1) = 1, is:

9 - ((26 - (2 * 9)) * 1) = 1, simplifying:

9 - 26 + (2*9) = 1

9 - 26 + (9 + 9) = 1

-26 + (9 + 9 + 9) = 1

-26 + (3 * 9) = 1

26*(-1) + 9*(3) = 1; if you compare the last expression with the original equation, which is: 26x+9y=1, you can see the similarity, where x=-1 and y=3.

So, the solution is: x=-1 and y=3.

Answer:

[tex]Domain:[/tex]{[tex]-6,3,4[/tex]}

[tex]Range:[/tex]{[tex]-2,5,6,23[/tex]}

The relation is not a function.

Step-by-step explanation:

By definition, a relation is a function if each input value has only one output value.

Given the relation:

(4,23)

(3,-2)

(-6,5)

(4,6)

The domain is the set of the x-coordinates of each ordered pair (You do not need to write 4 twice):

[tex]Domain:[/tex]{[tex]-6,3,4[/tex]}

The range is the set of the y-coordinates of each ordered pair :

 [tex]Range:[/tex]{[tex]-2,5,6,23[/tex]}

Since the input value 4 has two different output values (23 and 6), the relation is not a function.

Answer:

See below.

Step-by-step explanation:

The Domain = {-6, 3, 4}.

The Range = {-2, 5, 6, 23}.

The relation is NOT a function because 4 in the domain maps to 6 and 23. In a function an element in the domain must map to one element only in the range.