A shipment of 30 inexpensive digital​ watches, including 6 that are​ defective, is sent to a department store. The receiving department selects 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found defective. What is the probability that the shipment will be​ rejected?

Answer:

See below.

Step-by-step explanation:

1.   Suppose that the sum is rational  then we can write:

a/b + i = c/d     where i is irrational and by definition a/b and c/d are rational.

Rearranging:

i = c/d - a/b

Now the sum on the right is rational  so 'irrational' = 'rational' which is a contradiction.

So  the original supposition is false and the sum must be irrational.

2. Proof of For all integers m if m is even then 3m + 7 is odd:

If m is even then 3m is even.

Suppose 3m + 7 is even, then:

3m + 7 = 2p  where p is an integer.

3m - 2p  = -7

But 3m and 2p are both even so their result is even  and -7 is odd.

Therefore the original supposition is false because it leads to a contradiction,  so 3m + 7 is odd.

Answer:

13/4

Step-by-step explanation:

The slope of the line between 2 points is found by

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

   = (6--7)/(-3--7)

   = (6+7)/(-3+7)

   = 13/4

Answer:

The slope is 13/4.

Step-by-step explanation:

Slope formula:

[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\displaystyle \frac{6-(-7)}{(-3)-(-7)}=\frac{13}{4}[/tex]

[tex]\huge\boxed{\frac{13}{4}}[/tex], which is our answer.

Answer:

$333.17

Step-by-step explanation:

Use the compounding formula

[tex]A(t)=P(1+r)^t[/tex]

where A(t) is the amount at the end of the compounding,

P is the initial deposit,

r is the interest rate in decimal form, and

t is the time in years.

Filling in our info:

[tex]A(t)=180(1+.08)^8[/tex]

Simplify a bit to

[tex]A(t)=180(1.08)^8[/tex]

Raise 1.08 to the 8th power and get

A(t) = 180(1.85093021) and then multiply to get

A(t) = $333.17

Answer:

C

Step-by-step explanation:

The melons are 3 for 8 dollars, this means each melon is worth 8/3 dollars, this is the first proportion in the expression.  In this proportion the numerator is the value of money so 25 will be the numerator in the other proportion.

8/3=25/x, C

Answer:

monthly payment is $377

C is the correct option.

Step-by-step explanation:

The formula for the monthly payment is given by

[tex]C=\frac{Prt(1+r)^n}{(1+r)^n-1}[/tex]

Given that,

P =  $20,000

n = 5 years = 60 months

r = 0.05

Substituting these values in the formula

[tex]C=\frac{20000\cdot \frac{0.05}{12}(1+\frac{0.05}{12})^{60}}{(1+\frac{0.05}{12})^{60}-1}[/tex]

On simplifying, we get

[tex]C=\$377.425\\\\C\approx \$377[/tex]

Therefore, the monthly payment is $377

C is the correct option.

Answer:

[tex]g(4) = 8 \pi h\\\\g(\frac{3}{2}) = 3 \pi h\\\\ g(2c) = 4 \pi ch\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

Step-by-step explanation:

You need to substitute [tex]r=4[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(4) = 2 \pi(4)h\\\\g(4) = 8 \pi h[/tex]

Substitute [tex]r=\frac{3}{2}[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(\frac{3}{2}) = 2 \pi(\frac{3}{2})h\\\\g(\frac{3}{2}) = 3 \pi h[/tex]

Substitute [tex]r=2c[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(2c) = 2 \pi(2c))h\\\\g(2c) = 4 \pi ch[/tex]

Substitute [tex]r=c+3[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(c+3) = 2 \pi (c+3)h\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

For this case we have the following function:

[tex]g (r) = 2 \pi * r * h[/tex]

We must evaluate the function for different values:

[tex]g (4) = 2 \pi * (4) * h = 8 \pi*h\\g (\frac {3} {2}) = 2 \pi * (\frac {3} {2}) * h = 3 \pi*h\\g (2c) = 2 \pi * (2c) * h = 4 \pi * c * h\\g (c + 3) = 2 \pi * (c + 3) * h = 2 \pi * c * h + 6 \pi * h[/tex]

Answer:

[tex]g (4) = 8 \pi*h\\g (\frac {3} {2}) =3 \pi*h\\g (2c) = 4 \pi * c * h\\g (c + 3) = 2 \pi * c * h + 6 \pi * h[/tex]

First compute the probability that a bill would exceed $115. Let [tex]X[/tex] be the random variable representing the value of a monthly utility bill. Then transforming to the standard normal distribution we have

[tex]Z=\dfrac{X-100}{10}[/tex]

[tex]P(X>115)=P\left(Z>\dfrac{115-100}{10}\right)=P(Z>1.5)\approx0.0668[/tex]

Then out of 300 randomly selected bills, one can expect about 6.68% of them to cost more than $115, or about 20.

Answer:

75.

Step-by-step explanation:

Let x be the original price. 20% = 0.2. The price with the discount is 60, that is

x-x*0.2 = 60

x(1-0.2) = 60 using common factor,

x (0.8) = 60

x = 60/0.8

x = 75.

So, the original price is $75.

The original price of an item discounted 20% with a sale price after the discount of $60 is $75.

A discounted price is the marked-down price of an item.

The discounted price represents the selling price after reducing it with the discount.

Discount rate = 20%

Discounted price = $60

Original price = $75 ($60/1-20%)

Thus, the original price of an item discounted 20% with a sale price after the discount of $60 is $75.

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Let [tex]y_2(t)=tv(t)[/tex]. Then

[tex]{y_2}'=tv'+v[/tex]

[tex]{y_2}''=tv''+2v'[/tex]

and substituting these into the ODE gives

[tex]t^2(tv''+2v')+3t(tv'+v)-3tv=0[/tex]

[tex]t^3v''+5t^2v'=0[/tex]

[tex]tv''+5v'=0[/tex]

Let [tex]u(t)=v'(t)[/tex], so that [tex]u'(t)=v''(t)[/tex]. Then the ODE is linear in [tex]u[/tex], with

[tex]tu'+5u=0[/tex]

Multiply both sides by [tex]t^4[/tex], so that the left side can be condensed as the derivative of a product:

[tex]t^5u'+5t^4u=(t^5u)'=0[/tex]

Integrating both sides and solving for [tex]u(t)[/tex] gives

[tex]t^5u=C\implies u=Ct^{-5}[/tex]

Integrate again to solve for [tex]v(t)[/tex]:

[tex]v=C_1t^{-6}+C_2[/tex]

and finally, solve for [tex]y_2(t)[/tex] by multiplying both sides by [tex]t[/tex]:

[tex]tv=y_2=C_1t^{-5}+C_2t[/tex]

[tex]y_1(t)=t[/tex] already accounts for the [tex]t[/tex] term in this solution, so the other independent solution is [tex]y_2(t)=t^{-5}[/tex].

Answer:

The amount he should borrow is 19.25 hundreds dollars.

Step-by-step explanation:

Given : The cost (in hundreds of dollars) of tuition at the community college is given by [tex]T = 1.25c + 3[/tex], where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits.

To find : Use the model to determine how much money he should borrow?

Solution :

The model is given by [tex]T = 1.25c + 3[/tex]

Where,

c is the number of credits the student has registered.

T is the cost of tuition at the community college.

If a student is planning to take out a loan to cover the cost of 13 credits.

The amount he should borrow will get by putting the value of c in the model,

[tex]T = 1.25c + 3[/tex]

[tex]T = 1.25(13) + 3[/tex]

[tex]T = 16.25 + 3[/tex]

[tex]T =19.25[/tex]

Therefore, The amount he should borrow is 19.25 hundreds dollars.

Answer:

Equation is 4x + 6 = 598, where x represents smaller number.

Numbers are 148, 152, 156 and 160

Step-by-step explanation:

Let x be the smaller natural number,

So, the other consecutive natural numbers are x+1, x+2, x+3,

According to the question,

Sum of x, x+1, x+2 and x+3 is 598,

⇒ x + x + 1 + x + 2 + x + 3 = 598

⇒ 4x + 6 = 598

Which are the required equation,

Subtract 6 on both sides,

4x = 592

Divide both sides by 4,

x = 148

Hence, the numbers are 148, 152, 156 and 160

Final answer:

The equation to find four consecutive natural numbers with a sum of 598 is 4x + 6 = 598. Solving for x gives the first number as 148, which leads to the sequence: 148, 149, 150, and 151.

Explanation:

The student is tasked with finding four consecutive natural numbers whose sum is 598.

To solve this problem, we introduce a variable to represent the first number in the sequence, and then express the following three numbers in terms of this variable.

Let's denote the first number as x. Then the next three numbers will be x+1, x+2, and x+3, respectively. Our equation to find the numbers is:

x + (x+1) + (x+2) + (x+3) = 598

Combining like terms, we get:

4x + 6 = 598

We then solve for x:

4x = 598 - 6

4x = 592

x = 592 / 4

x = 148

So the four consecutive numbers are 148, 149, 150, and 151.

a. Recall that

[tex]\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C[/tex]

For [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

By integrating both sides, we get

[tex]\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

If [tex]x=0[/tex], then

[tex]\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0[/tex]

so that

[tex]\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

We can shift the index to simplify the sum slightly.

[tex]\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n[/tex]

b. The power series for [tex]x\ln(1-x)[/tex] can be obtained simply by multiplying both sides of the series above by [tex]x[/tex].

[tex]\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n[/tex]

c. We have

[tex]\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)[/tex]

[tex]\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}[/tex]

The power series of f(x) = ln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex], the power series of xln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex] and the value of ln(2) is [tex]\rm \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex].

Given :

f(x) = ln (1−x)

a) The integration of 1/(1 - x) is given by:

[tex]\rm \int \dfrac{1}{1-x}dx=-ln|1-x| + C[/tex]

When |x| >1 :

[tex]\dfrac{1}{1-x} = \sum^{\infty}_{n=0} x^n[/tex]

Now, integrate on both sides in the above equation.

[tex]\rm -ln(1-x) = C+\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]   --- (1)

Now, at (x = 0) the above expression becomes:

[tex]\rm -ln(1-0) = C+\sum^{\infty}_{n=0}\dfrac{0^{n+1} }{n+1}[/tex]

By simplifying the above expression in order to get the value of C.

C = 0

Now, substitute the value of C in expression (1).

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]

Now, by shifting the index the above expression becomes:

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex]

b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).

[tex]\rm xln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex]

c) Now, substitute the value x = 1/2 in the expression (1).

[tex]\rm ln2 = \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex]

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19 and 42 are coprime, so we can use the CRT right away. Start with

[tex]x=19+42[/tex]

Taken mod 42, we're left with a remainder of 19. Notice that

[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]

so we need to multiply the first term by 3 to get the remainder we want.

[tex]x=19\cdot3+42[/tex]

Next, taken mod 19, we're left with a remainder of 4. Notice that

[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]

so we need to multiply the second term by 6.

Then by the CRT, we have

[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]

so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.

Answer: a) The probability is approximately = 0.5793

b) The probability is approximately=0.8810

Step-by-step explanation:

Given : Mean : [tex]\mu= 62.5\text{ in}[/tex]

Standard deviation : [tex]\sigma = \text{2.5 in}[/tex]

a) The formula for z -score :

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

Sample size = 1

For x= 63 in. ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{1}}}=0.2[/tex]

The p-value = [tex]P(z<0.2)=[/tex]

[tex]0.5792597\approx0.5793[/tex]

Thus, the probability is approximately = 0.5793

b)  Sample size = 35

For x= 63 ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{35}}}\approx1.18[/tex]

The p-value = [tex]P(z<1.18)[/tex]

[tex]= 0.8809999\approx0.8810[/tex]

Thus , the probability is approximately=0.8810.

Answer:

[tex]P(t)=80(10^{0.1t})[/tex]

Step-by-step explanation:

The 'y' axis represent log(P), so it may be modeled as a line (or linear function), where its slope is 0.1:

[tex]log(P)=0.1t+C[/tex]

Pow each part of the equation by 10:

[tex]10^{log(P)}=10^{0.1t+C}\\ P=10^{0.1t+C}[/tex]

Evaluate at t=0, where the population is known.

[tex]P(0)=10^{C}=80[/tex]

Applying logarithmic properties:

[tex]P=10^{0.1t+C}=10^{0.1t}*10^{C}[/tex]

So, the final function is:

[tex]P(t)=80(10^{0.1t})[/tex]

The population size over time follows an exponential growth function described by P(t) = 80 * [tex]e^(0.23026t)[/tex], where the slope on a semilog plot of 0.1 is converted to the natural log base to find the growth rate constant.

When dealing with population growth, plotting the data on a semilog plot where the logarithm of the population size is plotted against time can simplify the analysis. If the resulting plot is a straight line with a slope of 0.1, this indicates exponential growth, because plotting an exponential function on a semilog plot yields a straight line.

The general form of an exponential growth function is:

Where:

Given that the slope of the line on the semi log plot is 0.1, this slope represents the constant k in the context of logs with base 10.

To convert this to a natural logarithm base (e), use the conversion factor ln(10):

k = 0.1 * ln(10)

Since ln(10) ≈ 2.3026, we have:

k = 0.1 * 2.3026

≈ 0.23026

Given the initial population size (P0) at time t = 0 is 80, the function describing the population size over time t is:

P(t) = 80 * [tex]e^(0.23026t)[/tex],

Answer: 0.1310

Step-by-step explanation:

Given : Mean : [tex]\mu = \text{131 millimeters}[/tex]

Standard deviation :  [tex]\sigma = \text{7 millimeters}[/tex]

Sample size : [tex]n=31[/tex]

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters i.e. less than 129.1 milliliters and less than 132.9 milliliters.

The formula for z-score :-

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

For x = 129.1 milliliters

[tex]z=\dfrac{129.1-131}{\dfrac{7}{\sqrt{31}}}\approx-1.51[/tex]

For x = 132.9 milliliters

[tex]z=\dfrac{132.9-131}{\dfrac{7}{\sqrt{31}}}\approx1.51[/tex]

The P-value= [tex]P(x<-1.51)+P(x>1.51)[/tex]

[tex]=2P(z>1.51)=2(1-P(z<1.15))\\=2(1-0.9344783)\\=0.1310434\approx0.1310[/tex]

Hence, the required probability = 0.1310

The probability that the sample mean would differ from the population mean by more than 1.9 millimeters is approximately 0.1744.

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters, we can use the Z-score formula. The Z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. Once we have the Z-score, we can use a Z-table or calculator to find the probability.

In this case, the Z-score is (1.9 - 0) / (7 / sqrt(31)) = 0.935. To find the probability of a Z-score greater than 0.935, we can look up the corresponding area under the normal distribution curve in a Z-table or use a calculator. The probability is approximately 0.1744.

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The nominal value - without discounting the inflation rate - of income was $ 3000.

If the interest rate was 6%, a rule of three is enough to find the value of the original investment.

3000 - 6%

x - 100%

x = 50,000

The value of the investment was $ 50,000

In this case, the inflation rate also requires a simple calculation.

Inflation corroded $ 1000 dollars of income of $ 3000

Therefore the inflation rate will be 1000/3000 = 33.3%

Answer:

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Step-by-step explanation:

We are given the following expression and we are to simplify the given function:

[tex] f ( x ) = x ^ 2 - 8 [/tex]

Applying the function [tex]f(2x)[/tex] on [tex] f ( x ) = x ^ 2 - 8 [/tex] to get:

[tex] f ( 2 x ) = ( 2 x ) ^ 2 - 8 [/tex]

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Answer:

433 minutes

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

The nurse is infusing 1.5 ml of solution every x minutes, Since there are a total of 650 ml we can divide the amount the nurse is infusing every x minutes by the total in order to calculate the amount of time it will take the nurse to finish.

[tex]\frac{650}{1.5} = 433.33min[/tex]

So we can see that it will take the nurse 433 min to infuse all of the solution.

Answer:

In 433.33 minutes will it take for the medicine to be given.

Step-by-step explanation:

Given : A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution.

To find : How many minutes will it take for the medicine to be given ?

Solution :

As 1.5 ml of solution infuse in x minutes

i.e. the amount of solution she has = [tex]1.5x[/tex]

Now, she has 650 ml of solution.

which means  [tex]1.5x=650[/tex]

Solving the equation,

[tex]x=\frac{650}{1.5}[/tex]

[tex]x=433.33[/tex]

Therefore, in 433.33 minutes will it take for the medicine to be given.

To find the cost of resurfacing the path, we first calculate the area of the path which is 300 square feet. We then multiply this by the unit cost of resurfacing which comes out to be $600.

This is a problem in area calculation and application of unit cost. Firstly, we need to calculate the area for the path surrounding the pool. The total area of the pool and the path is (14+2*3) feet by (30+2*3) feet = 20 feet by 36 feet, which equals 720 square feet. The area of the pool itself is 14 feet by 30 feet = 420 square feet. So, the area of just the path is 720-420 = 300 square feet. With a cost of $2 per square foot to resurface the path, the total cost would be 300*$2 = $600.

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The fourth point that will complete the quadrilateral is a rhombus is -3.

In a rhombus, two parallel sides must be equal.

            slope(m)=tan∅=y axis/x axis.

Calculation:-

1st coordinate- side=[tex]\sqrt{10}[/tex]

2nd coordinate- side= [tex]\sqrt{18}[/tex]

3rd coordinate  must be [tex]\sqrt{10}[/tex]

so,y=-3

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Answer:

Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?

(–1, -5 )

Step-by-step explanation:

Answer:

64

Step-by-step explanation:

[tex]mean=\frac{sum\ of\ total\ number\ of\ score}{total\ number\ of\ students}[/tex]

we have given that mean of 9 students is 84

so total score of 9 students = mean×9

                                              =84×9=756

and we have given mean score of exam is 82 and there is total 10 students so the total score of 10 students =10×82

                                                      =820

so the unreadable score = score of 10 students -score of 9 students =820-756=64

The solution of the given differential equation involves rearranging it into the standard form of a first-order linear differential equation, determining the integrating factor, and subsequently solving for the dependent variable y(x) via integration.

To solve the given differential equation, we can rewrite it in the form of dy/dx = f(x, y). That gives us (2(y-4x^2))/x = dy/dx. The resulting equation is a first-order linear differential equation, which can be solved using an integrating factor.

Here, the standard form of the differential equation is dy/dx + P(x)y = Q(x). Comparing this with our equation, we find P(x) = -2/x and Q(x) = -8x. We know that μ(x) = exp(∫P(x) dx) is the integrating factor. On solving we get μ(x) = 1/x2. We then multiply through our differential equation by μ(x) and integrate both sides to solve for y(x).

These steps on how to solve the differential equation involve certain knowledge in differential equation theory, namely about first-order linear differential equations, integrating factors, and the process of integration.

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Answer:

[tex]4\sqrt{17}[/tex]

Step-by-step explanation:

Let's find the answer by using the arc length formula which is:

[tex]\int\limits^a_b {\sqrt{1+(\frac{dy}{dx})^{2} } } \, dx[/tex]

First, let's find dy/dx which is:

y=4x-5

y'=4*(1)-0

y'=4, now let's use the formula:

[tex]\int\limits^3_{-1} {\sqrt{1+4^{2}} } \, dx=\sqrt{17} *(3-(-1))=4\sqrt{17}[/tex]

Now, using the distance formula we have:

[tex]d=\sqrt{(x2-x1)^{2} +(y2-y1)^{2} }[/tex]

[tex]y(-1)=4*(-1)-5=-9 \\y(3)=4*(3)-5=7[/tex]

So we have two points (-1, -9) and (3, 7) so:

[tex]d=\sqrt{(3-(-1))^{2} +(7-(-9))^{2} }=4\sqrt{17}[/tex]

Notice both equations gave the same length [tex]4\sqrt{17}[/tex].

Final answer:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, integrate sqrt(1 + (dy/dx)^2) from x = -1 to x = 3. The length of the curve is 4√17. The result can be confirmed by calculating the length using the distance formula.

Explanation:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, we need to integrate the square root of 1 + (dy/dx)^2 from x = -1 to x = 3. The derivative of y = 4x - 5 is dy/dx = 4. Substituting this into the arc length formula, we have:

L = ∫sqrt(1 + (dy/dx)^2) dx = ∫sqrt(1 + 4^2) dx = ∫sqrt(17) dx = x√17 + C

Now, plugging in the limits of integration, we have:

L = [(3√17) + C] - [(-1√17) + C] = (3√17) - (-1√17) = 4√17

To check our answer, we can use the distance formula. The endpoints of the line segment are (-1, -9) and (3, 7). Using the distance formula:

D = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - (-1))^2 + (7 - (-9))^2) = sqrt(4^2 + 16^2) = sqrt(272) = 16√17

As we can see, the length of the curve obtained using the arc length formula (4√17) matches the length calculated using the distance formula (16√17), confirming our answer.

We are to find the surface integral of [tex]\mathbf{\int \int _S \ F.ds}[/tex]

where;

∴

The plane x + y + z = 1

z = 1 - x - y

[tex]\dfrac{\partial z}{\partial x} = -1[/tex]

[tex]\dfrac{\partial z}{\partial y} = -1[/tex]

Since the surface is oriented downward

∴

[tex]dS = \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy[/tex]

[tex]dS = (-i-j-k) dxdy[/tex]

However,

Flux = [tex]\mathbf{\int \int _S \ F.ds}[/tex]

[tex]= \int \int_R F. \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy \\ \\ \\ = \int^1_{ x=0} \int ^{1-x}_{y=0} ( y^3i + 8j -xk)*(-i-j-k) dx dy \\ \\ \\ =\int^1_{ x=0} \int ^{1-x}_{y=0} (-y^3 -8+x) dxdy \\ \\ \\ = \int^1_{ x=0} \Big( \int ^{1-x}_{y=0} \Big(x-y^3 -8\Big) dy \Big) dx \\ \\ \\ = \int^1_{ x=0} \Bigg [xy - \dfrac{y^4}{4}-8y \Bigg]^{1-x}_{y=0} \ dx \\ \\ \\[/tex]

[tex]= \int^1_{ x=0} \Bigg [x(1-x) - \dfrac{1}{4}(1-x)^{4} - 8(1-x) \Bigg ] dx \\ \\ \\ = \int^1_{ x=0} \Bigg [(x-x^2) - \dfrac{1}{4}(x-1)^4+8(x-1)\Bigg] dx[/tex]

[tex]= \Bigg [ \dfrac{x^2}{2}-\dfrac{x^3}{3} - \dfrac{1}{4} \dfrac{(x-1)^5}{5}+(x-1)^2\Bigg] ^1_0[/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} -0+0-0+0+ \dfrac{1}{4}\times \dfrac{(-1)^5}{5}-(0-1)^2\Bigg][/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} - \dfrac{1}{20}-1\Bigg][/tex]

[tex]= \Bigg [ \dfrac{30-20-3-60}{60}\Bigg][/tex]

[tex]= \Bigg [ \dfrac{-53}{60}\Bigg][/tex]

Therefore, we can conclude that the surface integral is [tex]\mathbf{-\dfrac{53}{60}}[/tex]

Learn more about surface integral here:

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To compute a surface integral, take the antiderivatives of both dimensions defining the area with the surface edges as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

A surface integral over the given oriented surface can be computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

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1) x-intercept:

x-intercept is the point where the graph of the equation crosses the x-axis. From the given figure, we can see that the line is crossing the x-axis at -10. Thus the x-intercept is -9

2) y-intercept:

y-intercept is the point where the graph of the equation crosses the y-axis. From the given figure, we can see that the line is crossing the y-axis at -10. Thus the y-intercept is -9

3) Slope:

Slope of a line is calculated as:

[tex]slope=m=\frac{\text{Difference in y coordinates}}{\text{Difference in x coordinates}}[/tex]

For calculating the slope we can use both intercepts. x-intercept is ordered pair will be (-9, 0) and y-intercept will be (0, -9). So the slope of the line will be:

[tex]m=\frac{-9-0}{0-(-9)}=-1[/tex]

Therefore, the slope of the line is -1.

4) Slope intercept form of the line:

The slope intercept form of the line is represented as:

[tex]y=mx+c[/tex]

where,

m = slope of line = -1

c = y-intercept = -9

Using these values, the equation becomes:

[tex]y=-x- 9[/tex]

Answer:

x-intercept: [tex]-9[/tex].

y-intercept: [tex]-9[/tex].

Slope: [tex]-1[/tex]

Equation: [tex]y=-x-9[/tex]

Step-by-step explanation:

We have been given a graph of a line on coordinate plane. We are asked to find the x-intercept, y-intercept, and slope.          

We know that x-intercept of a function is a point, where graph crosses x-axis.

Upon looking at our given graph, we can see that graph crosses x-axis at point [tex](-9,0)[/tex], therefore, x-intercept is [tex]-9[/tex].

We know that y-intercept of a function is a point, where graph crosses y-axis.

Upon looking at our given graph, we can see that graph crosses y-axis at point [tex](0,-9)[/tex], therefore, y-intercept is [tex]-9[/tex].

We have two points on the line. Let us find slope of line using points  [tex](-9,0)[/tex] and  [tex](0,-9)[/tex].

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{-9-0}{0-(-9)}=\frac{-9}{0+9}=\frac{-9}{9}=-1[/tex]

Therefore, the slope of the line is [tex]-1[/tex].

Now, we will substitute [tex]m=-1[/tex] and y-intercept [tex]-9[/tex] in slope form intercept of equation as:

[tex]y=mx+b[/tex], where,

m = Slope,

b = The y-intercept.

[tex]y=-1(x-(-9))[/tex]

[tex]y=-1(x+9)[/tex]

[tex]y=-x-9[/tex]

Therefore, the equation of the line would be [tex]y=-x-9[/tex].

Answer:

The given expression can be written as -1=19(-1)+18.

Step-by-step explanation:

According to quotient remainder theorem:

For any integer a and a positive integer d, there exist unique integers q and r such that

[tex]a=d\times q+r[/tex]

It can also written as

[tex]a\text{mod }d=r[/tex]

The given expression is

[tex]-1\text{mod }19[/tex]

Here a=-1 and d=19.

[tex]\frac{-1}{19}=19(-1)+18[/tex]

[tex]-1\text{mod }19=18[/tex]

If -1 is divides by 19, then the remainder is 18. The value of r is 18.

Therefore the given expression can be written as -1=19(-1)+18.

-1 mod 19 is equal to 18.

Finding -1 mod 19

To determine -1 mod 19, we use the formula a = dq + r, where a = -1, d = 19, and q and r are the quotient and remainder, respectively.

First, note that any number mod 19 will yield a remainder between 0 and 18.

Since -1 is a negative number, we rewrite it in terms of 19: -1 = -1 + 19k for some integer k.

To find a positive equivalent, we choose k such that -1 + 19k is positive and falls within the range of 0 to 18. When k = 1, we get -1 + 19(1) = 18.

Therefore, -1 mod 19 is equal to 18.

This means that if you divide -1 by 19, the remainder that falls within the range of 0 to 18 is 18.

No idea what the cited section's method is, but this ODE is linear:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]

[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]

Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:

[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]

[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]

Integrating both sides gives

[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]

[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]

Answer:

(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

(B) f(x + h) - f(x) = 8xh + 4h² - 6h

(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

Step-by-step explanation:

* Lets explain how to solve the problem

- The function f(x) = 4x² - 6x + 6

- To find f(x + h) substitute x in the function by (x + h)

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) = 4(x + h)² - 6(x + h) + 6

- Lets simplify 4(x + h)²

∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²

∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²

- Lets simplify 6(x + h)

∵ 6(x + h) = 6(x) + 6(h)

∴ 6(x + h) = 6x + 6h

∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6

- Remember (-)(+) = (-)

∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

- Lets find f(x + h) - f(x)

∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)

- Remember (-)(-) = (+)

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6

- Simplify by adding the like terms

∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)

∴ f(x + h) - f(x) = 8xh + 4h² - 6h

* (B) f(x + h) - f(x) = 8xh + 4h² - 6h

- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]

∵ f(x + h) - f(x) = 8xh + 4h² - 6h

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]

- Simplify by separate the three terms

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]

∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

Answer:  The required number of different possible committees is 81172.

Step-by-step explanation:    Given that a department contains 9 men and 15 women.

We are to find the number of different committees of 6 members that are possible if the committee must have strictly more women than men.

Since we need committees of 6 members, so the possible combinations are

(4 women, 2 men), (5 women, 1 men) and (6 women).

Therefore, the number of different committees of 6 members is given by

[tex]n\\\\\\=^{15}C_4\times ^9C_2+^{15}C_5\times ^9C_1+^{15}C_6\\\\\\=\dfrac{15!}{4!(15-4)!}\times \dfrac{9!}{2!(9-2)!}+\dfrac{15!}{5!(15-5)!}\times \dfrac{9!}{1!(9-1)!}+\dfrac{15!}{6!(15-6)!}\\\\\\\\=\dfrac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!}\times\dfrac{9\times8\times7!}{2\times1\times7!}+\dfrac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!}\times\dfrac{9\times8!}{1\times8!}+\dfrac{15\times14\times13\times12\times11\times10\times9!}{6\times5\times4\times3\times2\times1\times9!}\\\\\\=1365\times36+3003\times9+5005\\\\=49140+27027+5005\\\\=81172.[/tex]

Thus, the required number of different possible committees of 6 members is 81172.