A particular dam contains approximately 1,100,000,000,000 cubic feet of water. For a​ week-long spike​ flood, water was released at a rate of 25,100 cubic feet per second.


The amount of water released during the​ week-long flood was?

Answer:

a) 0.8508

b) 0.1389

c) 0.0099

Step-by-step explanation:

Total number of calculators bought = 100

Number of calculators with defect = 2

Probability of selecting a calculator with defect = p = 2 out of 100 = [tex]\frac{2}{100}=0.02[/tex]

Probability of selecting a calculator without defect = q = 1 - p = 1 - 0.02 = 0.98

Part a)

The bookstore buys 8 calculators. This means the number of trials is fixed ( n =8). The calculator is either with defect or without defect. This means there are 2 outcomes only. The outcome of one selection is independent of other selections. This means all the events(selections) are independent of each other.

Hence, all the conditions of a Binomial Experiment are being satisfied. So we will use Binomial Probability to solve this problem.

We need to find the probability that all calculators are free of defects. i.e. number of success is 0 i.e. P( x = 0 )

The formula of Binomial Probability is:

[tex]P(x) =^{n}C_{x} (p)^{x} q^{n-x}[/tex]

Using the values, we get:

[tex]P(0)=^{8}C_{0}(0.02)^{0}(0.98)^{8-0}\\\\ P(0)=0.8508[/tex]

Thus, the probability that all calculators are free of defects is 0.8508

Part b) Exactly 1 calculator will have defect

This means the number of success is 1. So, we need to calculate ( x = 1)

Using the formula of binomial probability again and using x = 1, we get:

[tex]P(1)=^{8}C_{1}(0.02)^{1}(0.98)^{8-1}\\\\ P(1)=0.1389[/tex]

Thus, the probability that exactly two calculators will be having a defect is 0.1389

Part c) Exactly 2 calculator will have defect

This means the number of success is 2. So, we need to calculate ( x = 2 )

Using the formula of binomial probability again and using x = 2, we get:

[tex]P(2)=^{8}C_{2}(0.02)^{2}(0.98)^{8-2}\\\\ P(2)=0.0099[/tex]

Thus, the probability that exactly two calculators will be having a defect is 0.0099

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.

The width of the rectangular field is 139 feet and its length is 320 feet.

To solve this problem, we can use the formula for the perimeter of a rectangle, which is 2*(length + width). We know that the perimeter is 918 feet, and it's given that the length is 181 feet more than the width.

Let's denote the width as w, therefore the length would be w + 181.

By substituting these expressions into the formula for the perimeter, we get:
2*(w + w + 181)=918,
which simplifies to 2*(2w + 181)=918,
further simplifies to 4w + 362 = 918.

To isolate 4w, subtract 362 from both sides:
4w = 918 - 362 = 556.
Finally, divide both sides by 4 to solve for w:
w = 556/4 = 139 feet.

Substitute w = 139 into the length equation to get the length:
length = w + 181 = 139 + 181 = 320 feet.

So the width is 139 feet, and the length is 320 feet.

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Length is 320 feet.

Let's define the width of the rectangle as w feet and the length as l feet.

According to the problem,

the perimeter of the rectangle is 918 feet and the length is 181 feet more than the width.

We can express these conditions using the following equations:

Perimeter equation: 2l + 2w = 918

Length equation: l = w + 181

First, we can solve the perimeter equation for l+w:

2l + 2w = 918
Divide both sides by 2:

l + w = 459

Next, we substitute the length equation l = w + 181 into the perimeter equation:

w + 181 + w = 459 Simplify:

2w + 181 = 459
Subtract 181 from both sides:

2w = 278
Divide by 2:

w = 139 feet

Now substitute the width back into the length equation:

l = w + 181
l = 139 + 181

l = 320 feet

So, the width is 139 feet, and the length is 320 feet.

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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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:   The required solution of the given IVP is

[tex]x(t)=\dfrac{1}{9}(1-\cos3t).[/tex]

Step-by-step explanation:  We are given to use Laplace transforms to solve the following initial value problem :

[tex]x^{\prime\prime}+9x=1,~~~x(0)=0=x^\prime(0).[/tex]

We will be using the following formula for the Laplace transform :

[tex](i)~L\{t^n\}=\dfrac{n!}{s^{n+1}},\\\\\\(ii)~L\{\coskt\}=\dfrac{s}{s^2+k^2}.[/tex]

Applying Laplace transform on both sides of the above equation, we have

[tex]L\{x^{\prime\prime}+9x\}=L\{1\}\\\\\\\Rightarrow s^2X(s)-sx(0)-x^\prime(0)+9X(s)=\dfrac{1}{s}\\\\\\\Rightarrow s^2X(s)-s\times0-0=\dfrac{1}{s}\\\\\\\Rightarrow (s^2+9)X(s)=\dfrac{1}{s}\\\\\\\Rightarrow X(s)=\dfrac{1}{s(s^2+9)}\\\\\\\Rightarrow X(s)=\dfrac{1}{9s}-\dfrac{s}{9(s^2+9)}.[/tex]

Taking inverse Laplace transform on both sides of the above equation, we get

[tex]L^{-1}\{X(s)\}=L^{-1}\left(\dfrac{1}{9s}\right)-L^{-1}\left(\dfrac{s}{9(s^2+9)}\right)\\\\\\\Rightarrow x(t)=\dfrac{1}{9}\times1-\dfrac{1}{9}\times \cos3t\\\\\\\Rightarrow x(t)=\dfrac{1}{9}(1-\cos3t).[/tex]

Thus, the required solution of the given IVP is

[tex]x(t)=\dfrac{1}{9}(1-\cos3t).[/tex]

To solve the given initial value problem using Laplace transforms, apply the transforms to the equation, simplify to get the equation in the s-domain, and then take the inverse Laplace transform to get the solution in the time domain.

To solve the initial value problem x" + 9x = 1 where x(0) = 0 = x'(0) using Laplace transforms, we firstly apply it to the whole equation. The Laplace transform of a second derivative x''(t) is s^2 X(s) - s x(0) - x`(0) and that of x(t) is X(s), where X(s) is the Laplace transform Of x(t). Since the Laplace transform of a constant is the constant itself divided by s, the Laplace transform of the equation becomes s^2 X(s) - 0 - 0 + 9X(s) = 1/s. Simplifying the above gives X(s) = 1/(s^3 + 9s), as a solution in the s-domain. Apply a combination of basic formulas and decomposition techniques to convert this equation into a zero-state solution. After this, take the inverse Laplace transform to get the solution in the time domain. It should be simple given that the initial values are zero.

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Answer:

[tex]f (x) = (x-4)(x + 5)^2(x-9)(x+7)[/tex]

Step-by-step explanation:

The zeros of the polynomial are all the values of x for which the function [tex]f (x) = 0[/tex]

In this case we know that the zeros are:

[tex]x = 4,\ x-4 =0[/tex]

[tex]x = 9,\ x-9=0[/tex]

[tex]x = -5[/tex], [tex]x + 5 = 0[/tex] (multiplicity 2)

[tex]x = -7,\ x+7=0[/tex]

Now we can write the polynomial as a product of its factors

[tex]f (x) = (x-4)(x + 5)^2(x-9)(x+7)[/tex]

Note that the polynomial is of degree 5 because the greatest exponent of the variable x that results from multiplying the factors of f (x) is 5

The polynomial of degree 5 with the zeros 4, 9, -5 (multiplicity 2), and -7 is given by f(x) = (x - 4)(x - 9)(x + 5)² (x + 7).

To find a polynomial f(x) of degree 5 that has the zeros 4, 9, -5 (multiplicity 2), -7, you would begin by setting up a polynomial that has these zeros. From the given roots, we can form the polynomial in factored form as follows:

f(x) = (x - 4)(x - 9)(x + 5)²(x + 7).

Multiplicity refers to how many times a root is repeated. In this case, -5 is repeated twice, hence the exponent of 2.

So, the polynomial in factored form with the zeros provided is:

f(x) = (x - 4)(x - 9)(x + 5)² (x + 7).

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Answer:

c > 9

Step-by-step explanation:

Isolate the variable, c. Treat the greater than sign as an equal, what you do to one side, you do to the other. Add 7 to both sides:

c - 7 > 2

c - 7 (+7) > 2 (+7)

c > 2 + 7

c > 9

c > 9 is your answer.

~

Answer:

[tex]\Huge \boxed{C>9}\checkmark[/tex]

Step-by-step explanation:

Add by 7 from both sides.

[tex]\displaystyle c-7+7>2+7[/tex]

Simplify, to find the answer.

[tex]\displaystyle 2+7=9[/tex]

[tex]\huge \boxed{c>9}[/tex], which is our answer.

Answer:

  about 12.75%

Step-by-step explanation:

Let r represent the annual rate of return. Compounded annually for 18 years, the account multiplier is (1+r)^18. Then we have ...

  26,000 = 3,000(1+r)^18

  (26,000/3,000)^(1/18) = 1+r . . . . . . divide by 3000, take the 18th root

  (26/3)^(1/18) -1 = r ≈ 12.7465%

Ezio's account earned about 12.75% annually.

Ezio's $3,000 investment grew to $26,000 over 18 years at this rate.

To calculate the annual rate of return Ezio earned on his investment for his son, Flavia's, college tuition, we'll need to use the compound interest formula:

A = P(1 + r)^n

Where:

A is the amount of money accumulated after n years, including interest.

P is the principal amount (the initial amount of money).

r is the annual interest rate (decimal).

n is the number of years the money is invested.

We know that Ezio deposited $3,000 (P = 3000), the amount in the account after 18 years is $26,000 (A = 26000), and that the time period n is 18 years. We need to find the annual interest rate r.

Let's rearrange the formula to solve for r:

(1 + r)^n = A / P

(1 + r)^18 = 26000 / 3000

(1 + r)^18 = 8.6667

Now we need to take the 18th root of 8.6667 to find (1 + r):

1 + r = (8.6667)^(1/18)

The 18th root of 8.6667 is approximately 1.1225. This means:

1 + r = 1.1225

Subtract 1 from both sides to find r:

r = 1.1225 - 1

r = 0.1225 or 12.25%

The annual rate of return Ezio earned on his investment is approximately 12.25%.

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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Step-by-step answer:

Given:

class time uniformly distributed between 48 and 53 minutes (range 5 minutes)

Find probability that the class period runs between 51.5 and 51.75 minutes.

Solution:

Uniformly distributed means that the probability that the class time is the same for any minute (out of 5), i.e. 20%, or probability density is 0.2 / minutes, no matter which minute.

The interval between 51.5 and 51.75 is 0.25 minutes, so the probability of class period within that range is 0.25 minutes * 0.2 / minute = 0.05.

Final answer:

The probability that a class period runs between 51.5 and 51.75 minutes, when class lengths are uniformly distributed between 48.0 and 53.0 minutes, is 0.05 or 5%.

Explanation:

To find the probability that a given class period runs between 51.5 and 51.75 minutes when class lengths are uniformly distributed between 48.0 and 53.0 minutes, we utilize the properties of the uniform distribution. The probability is the area under the uniform probability density function (PDF) between the two specified lengths. For a uniform distribution, this area corresponds to the length of the interval divided by the length of the entire interval in which the variable is uniformly distributed.

Here, we're looking for the probability between 51.5 minutes and 51.75 minutes. The total length of the available interval is 53.0 - 48.0 = 5 minutes. The length of the desired interval is 51.75 - 51.5 = 0.25 minutes. So, we divide the length of the desired interval by the total interval to get the probability.

Probability = (51.75 - 51.5) / (53.0 - 48.0) = 0.25 / 5 = 0.05

Therefore, the probability that a class period runs between 51.5 and 51.75 minutes is 0.05 or 5%.

Circumference: 18.84 m

Area: 28.26[tex]m^{2}[/tex]

Explanation

For circumference, use the formula [tex]\pi[/tex]*d, where d=the diameter(the radius times 2).

For area, use the formula [tex]\pi[/tex][tex]r^{2}[/tex], where r=the radius. Make sure to square the radius first before multiplying by pi!

A good way to remember these formulas is Cherry(circumference) Pi([tex]\pi[/tex]) is Delicious(diameter) and Apple(area) [tex]\pi[/tex] [tex]r^{2}[/tex]("are too")

The circumference of the circle with radius 3 meters is 18.84 meters and the area is 28.26 square meters.

Given that the radius of the circle is 3 meters, we can find the circumference and the area using the respective formulae.

Circumference of a circle with radius r (C) is given by the formula C = 2πr. Substituting π = 3.14 and r = 3 meters, the circumference C = 2*3.14*3 = 18.84 meters.

The Area of a circle with radius r (A) is given by the formula A = πr². Substituting π = 3.14 and r = 3 meters, the area A = 3.14*3² = 28.26 square meters.

So, the circumference of the circle with radius 3 meters would be 18.84 meters and the area would be 28.26 square meters.

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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%.

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:

  2280 kcal

Step-by-step explanation:

The given function R(t) is periodic with a period of 24 hours, so the integral is the product of the average value (95 kcal/h) and the 24-hour interval:

  (95 kcal/h)(24 h) = 2280 kcal

To calculate the total basal metabolism over 24 hours for a young man modeled by R(t) = 95 − 0.18 cos(πt/12), you need to apply integral calculus. First, integrate the function R(t) and then apply a definite integral over the period from 0 to 24 hours. The result will be the total basal metabolism over this time period.

To solve this problem, you need to integrate the function R(t) = 95 − 0.18 cos(πt/12) over the time interval of 0 to 24 hours. In calculus, such an operation is represented as ∫R(t) dt from 0 to 24. This operation will give you the total basal metabolism during these 24 hours.

The first thing you need to do in this case is to integrate the function. As this function is a composition of functions, you would need to use the basics of integral calculus, specifically the trigonometric substitution method or table of integrals for cosine function.

Afterward, apply a definite integral over the period from 0 to 24 hours. This will give you the total basal metabolism for 24 hours, expressed in kcal.

The steps presented above represent a simplification of the process. The actual calculations might be quite complex, so please don't worry if they seem overwhelming - it's a normal part of learning integral calculus.

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Answer:

0.0488%

Step-by-step explanation:

Here we have the probability of different independent events, which means that none of the previous ones affect the next. For this kind of events, the probability 'P' that a series of events occur is the multiplication of the probability of each singular event.

p1: Probability that an H be obtained: 50% (is always 50% because it is independent of the previous results)

P: The probability that H be obtained all of the 10 times. This is the complementary probability to E:(a T comes up at least once).

By the first definition given

[tex]P=0.5^{11}=0.00048828=0.0488%[/tex]

The complementary probability 'P' is the probability that 'E' does not happen, so the probability that E happen is: [tex]P_{E} =1-P[/tex]

The last makes sense if we think about the fact that for the experiment there are just two possibilities, 'E' happen, or 'E' does not happen. Then,

[tex]P_{E} =1-0.000488=99.95[/tex]%

Answer with explanation:

It is given that , a and b are positive integers.

gcd(a,b)=1

We have to prove for any positive integer x and y ,

a x + by =c, for any integer greater than ab-a-b.

Proof:

GCD of two numbers is 1, when two numbers are coprime.

Consider two numbers , 9 and 7

GCD (9,7)=1

So, we have to calculate positive integers x and y such that

⇒ 9 x +7 y > 9×7-9-7

⇒9x +7 y> 47

To prove this we will draw the graph of Inequality.

So the ordered pair of Integers are

x>5 and y>6.

So, for any integers , a and b ,

→ax+ by > a b -a -b, if

[tex]\Rightarrow \frac{x}{\frac{ab-a-b}{a}}+ \frac{y}{\frac{ab-a-b}{b}}>1,\frac{ab-a-b}{a},\text{and},\frac{ab-a-b}{b}[/tex]

⇒Range of x for which this inequality hold

      [tex]=[\frac{ab-a-b}{a},\infty)[/tex]

if,

[tex]\frac{ab-a-b}{a}[/tex]

is an Integer ,otherwise range of x

         [tex]=(\frac{ab-a-b}{a},\infty)[/tex]

⇒Range of y for which this inequality hold

      [tex]=[\frac{ab-a-b}{b},\infty)[/tex]

if,

[tex]\frac{ab-a-b}{b}[/tex]

is an Integer ,otherwise range of y

         [tex]=(\frac{ab-a-b}{b},\infty)[/tex]

Answer:

it would be A 1251

Step-by-step explanation:

The question involves hypothesis testing of proportions. Based on a sample where 80% favour Candidate A, we may infer the true population proportion is significantly greater than 75%, however, actual mathematical calculations are needed for confirmation.

The question examines whether the proportion of the population favouring Candidate A is significantly more than 75% based on a sample of 100 people where 80 favoured Candidate A. This is a problem of hypothesis testing for proportions. The null hypothesis (H₀) is that the true population proportion is 75% (p = 0.75), versus the alternative hypothesis (H₁) stating the true population proportion is more than 75%. Using a significance level of 0.05, we examine the data.

With a sample proportion of 80 out of 100 (p' = 0.80) and given the large sample size, we apply the Normal approximation to the Binomial distribution, followed by a one-sample z-test. If the resulting p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the true population proportion of individuals favouring Candidate A is significantly greater than 75%.

However, without performing the actual calculations, we cannot definitively determine the conclusion. From a practical perspective, an 80% sample proportion showing favour in a sample as large as 100 might indicate a significantly higher proportion than 75%, but an exact mathematical test should be done to confirm this.

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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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Let the smaller odd integer be: a

then the larger odd integer which is consecutive to a will be: a+2

It is given that:

Seven times the first (smaller) of two consecutive odd integers is equal to five times the second (larger) integer.

This means that:

7 times of a is equal to 5 times of (a+2)

i.e.

[tex]7a=5(a+2)\\\\i.e.\\\\7a=5\times a+5\times 2[/tex]

( Since, by using the distributive property of multiplication)

i.e.

[tex]7a=5a+10\\\\i.e.\\\\7a-5a=10\\\\i.e.\\\\2a=10\\\\i.e.\\\\a=5[/tex]

Hence, the smaller number is:  5

and the larger number is: 7

( Since a+2=5+2=7 )

[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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1. (1010 0110)₂ = (166)₁₀

2. (145)₁₀ = (1001 0001)₂

3. (101 000 111 100)₂ = (5074)₈

4. (1111 1100 0011 0110)₂ = (FC36)₁₆

1. Convert the 8-binary binary expansion (1010 0110)₂ to a decimal expansion.

In order to solve this problem we have to use the expansion:

n = aₓbˣ + aₓ₋₁bˣ⁻¹ + ... + a₁b¹ + a₀

where b = 2, x = 8 - 1 = 7 due is a 8-binary

(1010 0110)₂ = 1 x 2⁷ + 0 x 2⁶ + 1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2⁰

(1010 0110)₂ = 128 + 0 + 32 + 0 + 0 + 4 + 2 + 0

(1010 0110)₂ = (166)₁₀

2. Convert the following decimal expansion (145)₁₀ to an 8-bit binary expansion.

To solve this problem we have to use the divide by 2 process.

Since we are dividing by 2, when the dividend is an even number, the remainder will be 0, and when the dividend is an odd number the binary residual will be 1.

145 ---------------> 1  Less significant bit

145/2 = 72 -----> 0

72/ 2 = 36 -----> 0

36/2 = 18 ------> 0

18/2 = 9 -------> 1

9/2 = 4 --------> 0

4/2 = 2 ---------> 0

2/2 = 1 ----------> 1  Most significant bit

Then we order from the most significant bit to the less significant bit (from the bottom to the top) to obtain the 8-binary number:

(145)₁₀ = (1001 0001)₂

3. Convert the following hexadecimal expansion (A3C)₁₆ to an octal expansion.

To convert a hexadecimal expansion to an octal expansion we have to convert  from hexadecimal to binary and then to octal using the table hexadecimal to binary and binary to octal.

Converting from hexadecimal to binary:

(A3C)₁₆

A = 1010, 3 = 0011 and C = 1100

(A3C)₁₆ = (1010 0011 1100)₂

Converting from binary to octal:

To convert binary to octal we have to order the binary expansion into group of 3-bits and use the table to convert binary to octal.

(1010 0011 1100)₂ = (101 000 111 100)₂

101 = 5, 000 = 0, 111 = 7 and 100 = 4

(101 000 111 100)₂ = (5074)₈

4. Convert the following binary expansion (1111 1100 0011 0110)₂ to a hexadecimal expansion.

To solve this exercise we have to use the binary to hexadecimal table.

(1111 1100 0011 0110)₂

1111 = F, 1100 = C, 0011 = 3 and 0110 = 6

(1111 1100 0011 0110)₂ = (FC36)₁₆

Parameterize the first line segment [tex]C_1[/tex] by

[tex]\vec r(t)=(1,0,1)(1-t)+(2,4,1)t=(1+t,4t,1)[/tex]

and the second line segment [tex]C_2[/tex] by

[tex]\vec s(t)=(2,4,1)(1-t)+(2,6,4)t=(2,4+2t,1+3t)[/tex]

both with [tex]0\le t\le1[/tex]. Then

[tex]\displaystyle\int_{C_1}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz[/tex]

[tex]\displaystyle=\int_0^1\bigg((1+5t)\cdot1+2(1+t)\cdot4+(1-t)4t\cdot0\bigg)\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^1(9+13t)\,\mathrm dt=\frac{31}2[/tex]

and

[tex]\displaystyle\int_{C_2}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz[/tex]

[tex]\displaystyle=\int_0^1\bigg((2+(4+2t)(1+3t))\cdot0+2(2)\cdot2+2(4+2t)(1+3t)\cdot3\bigg)\,\mathrm dt[/tex]

[tex]\displaystyle=\int_0^1(32+84t+36t^2)\,\mathrm dt=86[/tex]

Then

[tex]\displaystyle\int_C(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz=\frac{31}2+86=\boxed{\frac{203}2}[/tex]

To evaluate the line integral, parameterize each segment of the curve separately, substitute the parameterization into the line integral expression, evaluate the integral for each segment, and sum up the results.

To evaluate the line integral, we need to first parameterize the curve C. The given curve consists of two line segments, so we will parameterize each segment separately.

For the first segment from (1, 0, 1) to (2, 4, 1), we can parameterize it as r(t) = (1 + t, 4t, 1), where t ranges from 0 to 1.  

For the second segment from (2, 4, 1) to (2, 6, 4), we can parameterize it as r(t) = (2, 4 + 2t, 1 + 3t), where t ranges from 0 to 1.

Next, we substitute the parameterization into the line integral expression and evaluate the integral for each segment separately. Finally, we sum up the individual results to get the value of the line integral.

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Answer:

Secretariat's average speed in the Kentucky Derby of 1973 was of 47.05 miles per hour.

Step-by-step explanation:

Secretariat set the current record of 1 minute 59.40 seconds for the Kentucky Derby, running 10 furlongs, which are 1.25 miles.

To determine the speed at which the horse ran, we must first isolate the time in which he ran, to determine the fraction of time in which he traveled the 1.25 mile.

An hour has 60 minutes. To determine the speed, whe have to divide 60 by the minutes the horse ran, and then multiply it by the number of miles he ran: 60 / 1.5940 = 37,64 x 1.25 = 47,05.

The horse ran at 47.05 miles per hour.

Secretariat's record average speed during the Kentucky Derby was an impressive 37.68 mi/hr.

The question asks to calculate Secretariat's average speed during the Kentucky Derby in miles per hour (mi/hr). First, we need to express the distance of the Kentucky Derby in miles. The Derby is 10 furlongs long, and since 1 furlong is 1/8 mile, this translates to 1.25 miles (10 furlongs × 1/8 mile/furlong). Understanding this conversion is crucial because speed is typically measured in miles per hour in the United States.

Next, Secretariat's record-setting time for completing this distance is 1 minute and 59.40 seconds, which is equivalent to 119.40 seconds. To find the average speed, we use the formula:

Speed = Distance / Time

Hence, Secretariat's average speed is:

1.25 miles / (119.40 seconds × (1 hour / 3600 seconds)) = 1.25 miles / 0.03317 hours = 37.68 mi/hr, when the time is converted from seconds to hours.

Therefore, Secretariat's record average speed during the Kentucky Derby was an impressive 37.68 mi/hr.

Answer:

Viatical settlement

Step-by-step explanation:

Here Barney sold his life insurance to 3rd party for less net death benefit value.

He entered into the arrangement called - viatical settlement

A viatical settlement is the sale of a life insurance policy to a third party. The owner of the life insurance policy sells the policy for cash benefit, thus making the buyer the new owner of the policy who will get death benefits.

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:

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:

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: 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:

(x+8)^2 + (y-4)^2 = 3^2

or

(x+8)^2 + (y-4)^2 = 9

Step-by-step explanation:

The diameter is 6, so the radius would be d/2 =3

The equation of a circle is

(x-h)^2 + (y-k)^2 = r^2

where (h,k) is the center and r is the radius

(x- -8)^2 + (y-4)^2 = 3^2

(x+8)^2 + (y-4)^2 = 3^2

or

(x+8)^2 + (y-4)^2 = 9