What is the average rate of change of the function on the interval from x = 0 to x = 5 f(x)=
1/2(3)x

Answer:

76302 grams

Step-by-step explanation:

Diameter of the balloon = 9 m

Density of the gas (weight occupied per m³) = 200 g per m³

Volume of the spherical balloon = [tex]\frac{4}{3}\pi r^{3}[/tex]

Here r = 4.5 m

Then V= [tex]\frac{4}{3}\times\ 3.14 \times (4.5)^{3}[/tex]

V = 381.51 m³

Total weight = Total volume occupied * Density

                     = 381.51*200 = 76302 grams.

Final answer:

The total weight of the gas in a filled balloon with a 9-meter diameter is 76,340.25 grams, calculated by first finding the volume of the balloon and then multiplying it by the weight of the gas per cubic meter.

Explanation:

To find the total weight of the gas in a filled spherical balloon, first, we need to find the volume of the balloon using the formula for the volume of a sphere, which is V = (4/3)πr³. The diameter of the balloon is given as 9 meters, so the radius r is 9/2 meters or 4.5 meters.

Calculating the volume of the balloon:

Now, we'll multiply the volume of the balloon by the weight per cubic meter of the gas to get the total weight of the gas:

Answer:

216 sq. units

Step-by-step explanation:

From the figure, we can see that only one pair of opposite angles are equal. So, the quadrilateral is a kite.

Formula to find the area of a kite:

Area, A = [tex]$ \frac{1}{2} \times d_1 \times d_2 $[/tex]

where, [tex]$ d_1 $[/tex] and [tex]$ d_2 $[/tex] are the lengths of the diagonals.

Here, [tex]$d_1 = 18 $[/tex] units.

And, [tex]$ d_2 = 24 $[/tex] units.

Therefore, the area A = [tex]$ \frac{1}{2} \times 18 \times 24 $[/tex]

= [tex]$ \frac{432}{2} $[/tex]

= 216 sq. units which is the required answer.

Explanation:

To achive that you have to piant a ring in red, which will have its big diameter equal to the sphere diameter and its small diameter equal to the cube diagonal.

The diagonal of the cube can be calculated using Pithagoras:

[tex]D^2=L^2+L^2[/tex]

Where D is the diagonal and L is the side of the cube

Answer:

The least possible value is -2, obtained for a = -1 and b = 1.

Step-by-step explanation:

We want the minimum of [tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) [/tex] , where

First, lets simplify the expression given, we use common denominator on the second part, using ab as the common denominator. We obtain

[tex] \frac{1}{b} - \frac{1}{a} = \frac{a-b}{ab} [/tex]

As a result

[tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) = \frac{a-b}{(ab)^2} [/tex]

we need the minimum of the function

[tex] f(a,b) = \frac{a-b}{(ab)^2} [/tex]

with the restrictions [tex] -5 \leq a \ -1, 1 \leq b \leq 3 [/tex]

First, we calculate the gradient of f and find where it takes the zero value.

[tex]\nabla{f} = (f_a,f_b) [/tex]

with

[tex]f_a = \frac{(ab)^2 - (a-b) 2ab^2}{(ab)^2} = -1 + 2 \frac{b}{a}[/tex]

Since it has the reversed sign, we get

[tex]f_b = - (-1 + 2 \frac{a}{b}) =1 - 2 \frac{a}{b}[/tex]

In order for [tex] \nabla{f} [/tex] to be zero, we need both [tex] f_a [/tex] and [tex] f_b [/tex] to be zero, observe that

[tex]f_b = 0 \, \rightarrow 1 -2\frac{a}{b} = 0 \, \rightarrow 1 = 2 \frac{a}{b} \, \rightarrow b = 2a [/tex]

Which is impossible with the given restrictions. Hence, the minimum is realized in the border.

If we fix a value a₀ for a, with a₀ between -5 and -1 the function g(b) = f(a₀,b) wont have a minimum for b in [1,3] because the partial derivate of f over b didnt reach the value 0 in the restrictions given. On the other hand. by making a similar computation that before, we can obtain that the partial derivate of f over the variable a doesnt reach the value 0 either. This means that f doesnt reach the minimum on the sides. As a consecuence, it reach a minimum on the corners.

The 4 possible corner values are (-5,1), (-5,3), (-1,1) and (-1,3)

[tex] f(-5,1) = \frac{-6}{25} =  -0.24 [/tex]

[tex] f(-5,3) = \frac{-8}{225} = -0.0355 [/tex]

[tex] f(-1,1) = \frac{-2}{1} = -2 [/tex]

[tex] f(-1,3) = \frac{-4}{9} = -0.444 [/tex]

Clearly the least possible value between the four corners is -2.

Answer:the length of the rectangle is 100m

the width of the rectangle is 80m

Step-by-step explanation:

Let L represent the length of the rectangle.

Let W represent the width of the rectangle.

The formula for determining the perimeter of a rectangle is expressed as

Perimeter = 2(L + B)

The perimeter of a rectangle is 360m. This means that

2(L + B) = 360

L + B = 180 - - - - - - - - - -1

If it's length is decreased by 20%, its new length would be

L - 0.2L = 0.8L

it's breadth is increased by 25%, it means that the new breadth

B + 0.25B = 1.25B

Since the perimeter remains the same,

2(0.8L + 1.25B) = 360

0.8L + 1.25B = 180 - - - - - - - - -2

Substituting L = 180 - B into equation 2, it becomes

0.8(180 - B) + 1.25B = 180

144 - 0.8B + 1.25B = 180

- 0.8B + 1.25B = 180 - 144

0.45B = 36

B = 36/0.45 = 80

L = 180 - B = 180 - 80

L = 100

Answer:

Step-by-step explanation:

Total number of students = 68

Let history = H

Maths = M

English = E

n(H) = 25

n(M) = 25

n(E) = 34

n(HnMnE) = 3

Total = n(H) + n(M) + n(E) - people in exactly two groups + 2(people in exactly 3 groups) + people in none of the groups

68 = 25 + 25 + 34 - people in exactly two groups - 6 +0

68 = 84 -6 - people in exactly two groups

68 = 78 - people in exactly two groups

People in exactly two groups = 78 - 68

= 10

OR

From the venn diagram, people in exactly two groups are represented by x, y and z

Total = 25 - x - y - 3 + 25 - x - z - 3 + 34 - y - z - 3 + x + y + z + 3

68 = 50 - x - 3 + 34 - y - z - 3

68 = 84 - 6 - x - y - z

68 = 78 - x - y - z

68 - 78 = - x - y - z

-10 = -(x + y + z)

x+y+z = -10/-1

x+y+z = 10

The number of students that registered for exactly two courses = 10

Answer:

[tex[ two groups=-68+25+25+34 -6=10[/tex]

Step-by-step explanation:

For this case we have a diagram of the situation on the figure attached.

And for this case we can us the following rule from probability:

[tex]P(AUBUC) = P(A)+P(B) +P(C) -P(A and B)- P(A andC)-P(B and C) +P(A and B and C)[/tex]

Or equivalently:

[tex] total= A+ B +C - two groups -2*[people in3 groups]+ nonegroups[/tex]

We know the total on this case total=68, the people for history is A=25, for math B=25 and for english C=34, and the people in all the groups 3, so we can replace:

[tex]68=25+25+34- two groups -2*3 +0[/tex]

And if we solve for the poeple in two groups we got:

[tex[ two groups=-68+25+25+34 -6=10[/tex]

To calculate the rate at which the distance between the bicycle and the balloon is changing, take into account the speeds of the balloon and bicycle and use Pythagoras's theorem and the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2). This calculates out to approximately 17.03 ft/sec.

This problem can be solved using the concepts of relative velocity and Pythagoras' theorem. It's a classic problem of a balloon moving vertically upwards at a speed of 1 ft/sec and a bicycle moving horizontally at a speed of 17 ft/sec.

After 3 seconds, the balloon would have ascended an additional 3 feet (because it's rising at 1 foot per second), so it will be at a height of 68 feet. Meanwhile, in those same 3 seconds, the bicycle will have travelled 51 feet (because it's going at 17 feet per second).

At this point, you can use Pythagoras's theorem to find the distance (d) between the bicycle and the balloon. The distance d is the hypotenuse of a right triangle where one side (the vertical side) is 68 feet and the other side (the horizontal side) is 51 feet. Using Pythagoras' theorem, d² = 68² + 51². So, d = sqrt(68^2+51^2) = 85 feet.

Then, find the change in d, known as dd/dt, which tells us how fast the distance between the balloon and the bicycle is changing. Given the constant velocities of the balloon and the bicycle, dd/dt is also a constant. You can use the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2), where dx/dt = 17 ft/sec (the speed of the bicycle), and dy/dt = 1 ft/sec (the speed of the balloon). So, dd/dt = sqrt(1^2 + 17^2) = sqrt(290) = approximately 17.03 ft/sec. So, the distance between the balloon and the bicycle is increasing at a rate of 17.03 ft/sec after 3 seconds.

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The distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec, 3 seconds after the bicycle has passed under the balloon.

We need to determine how fast the distance between the bicycle and the balloon is changing 3 seconds after the bicycle passes under the balloon.

Let's denote:

We are given:

After 3 seconds:

Apply the Pythagorean theorem to find z:

z = √(x² + y²)
z = √(51² + 68²)
z = √(2601 + 4624)
z = √7225
z = 85 ft

To find the rate of change of the distance z, we differentiate with respect to time (t):

2z (dz/dt) = 2x (dx/dt) + 2y (dy/dt)

Continue with substituting the known values:

85 (dz/dt) = 51 (17) + 68 (1)
85 (dz/dt) = 867 + 68
dz/dt = 935 / 85
dz/dt ≈ 11 ft/sec

Thus, the distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec after 3 seconds.

Answer: check explanation

Step-by-step explanation:

The question is not complete but nonetheless, gross profit is easy to calculate. Let us start by the definition of gross profit, to how to calculate gross profit and, calculation of gross profit margin.

GROSS PROFIT: gross profit can be defined as the profit gained after subtracting the costs of making and selling the products. Gross profit is also known as gross margin.

HOW TO CALCULATE GROSS PROFIT: Gross profit can be calculated by subtracting the total revenue from the cost of goods sold. That is, gross profit= total revenue - cost of goods sold.

So, for example; Company A makes women handbags. Assuming the company made $10million in total revenue for the year and cost of goods sold is $5 million. We can use the formula above to find the company's gross profit margin.

Hence, company A's gross profit= total revenue($10,000,000) - cost of goods sold ($5,000,000).

= $5,000,000.

That is, the gross profit for company A= $5,000,000.

GROSS PROFIT MARGIN: This can be calculated using the formula below;

Gross profit margin= (total revenue - cost of goods)/ revenue.

Hence, from the example above;

Gross profit margin= $10,000,000 - $5,000,000) / $10,000,000.

= 50%

Answer:

The common difference of given AP is Option 2) -10.

Step-by-step explanation:

We are given the following information in the question:

First term of AP, a  = 100

The sum of its first 6 terms = 5(the sum of its next six terms)

We have to find the common difference of AP.

The sum of n terms of AP is given by:

[tex]S_n = \dfrac{n}{2}\big(2a + (n-1)d\big)[/tex]

where a is the first term and d is the common difference.

Thus, we can write:

[tex]S_6 = 5 (S_{12}-S_6)\\\dfrac{6}{2}\big(200 + (6-1)d\big) = 5\bigg(\dfrac{12}{2}\big(200 + (12-1)d\big)-\dfrac{6}{2}\big(200 + (6-1)d\big)\bigg)\\\\600 + 15d =5(1200+66d-600-15d)\\600+15d=3000+255d\\2400 = -240d\\d = -10[/tex]

Thus, the common difference of given AP is -10.

Answer:

B) Mean

Step-by-step explanation:

Standard Deviation is the measure of the amount of variation or dispersion of a set of values. Standard Deviation is represented by lower case Greek alphabet sigma σ. Standard Deviation is the square root of variance. Variance is the average of the squared differences or variation or dispersion from the Mean. Therefore, to compute standard deviation, the mean of the given data must be known.

    Standard Deviation, σ =  [tex]\sqrt{Variance}[/tex]

                     σ = [tex]\sqrt({\frac{1}{N-1}}[/tex] ∑[tex]_{i = 1}^{N}(x_{i} - X)^{2})[/tex]

where

[tex]x_{1},x_{2},x_{3}, ...x_{N},[/tex] are the values of the sample observed,

X is the mean value of these observations, and

N is the number of observations in the sample.

Why is this the answer?

The x value x = 4 shows up in the point (4, 3), which is in the given function set. Adding (4, -3) to this set will have x = 4 show up twice. We cannot have one x value pair up with more than one y value. In other words, any input cannot map to more than one output. Visually, the two points (4,3) and (4,-3) will fail the vertical line test, which means we wouldnt have a function.

Answer:

It's D; degree- 5; leading coefficient-10 :)

Step-by-step explanation:

I just took the test and got it right

The given polynomial function f(x) = 10x⁵ + 7x⁴ + 5 has Degree: 5; leading coefficient: 10 which is the correct answer would be option (D).

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.

The degree of a polynomial is the highest exponent of the variables in the polynomial. In this case, the highest exponent is 5, so the degree of the polynomial is 5.

The leading coefficient is the coefficient of the term with the highest degree.

In this case, the coefficient of the term with the highest degree (x⁵) is 10, so the leading coefficient is 10. Therefore, the correct answer is D).

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Answer:

x = 3, x = -3

Step-by-step explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non zero for these values then they are vertical asymptotes.

Solve : x² - 9 = 0 ⇒ x² = 9 ⇒ x = ± 3

The vertical asymptotes are x = - 3 and x = 3

The vertical asymptotes of the rational function F(x) = (x - 6)(x + 6)/(x^2 - 9) are x = -3 and x = 3.

The rational function F(x) = (x - 6)(x + 6)/(x^2 - 9) has vertical asymptotes at x = -3 and x = 3.

To find the vertical asymptotes, we need to determine the values of x that make the denominator equal to zero. In this case, the denominator is x^2 - 9, which can be factored as (x - 3)(x + 3).

Therefore, the vertical asymptotes occur at x = -3 and x = 3, since these are the values that make the denominator zero.

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Answer:

a.) y = 3/(2ˣ)

b.) y = 1/(2ˣ)

c.) y = (π^π)ˣ

d.) y = (1/27)(1/√(3))ˣ

e.) y = .002908/(.119025ˣ)

f.) y = .00000004808/(.0413ˣ)

Step-by-step explanation:

Concept need to know is:

Answer:

The data file is included in attached excel file.

Step-by-step explanation:

There are seven variables in the data file for four persons interviewed. Person type, marital status, ethnic group and gender are categorical variables while age, number of pets and GPA are quantitative variables. Person type is classified as A,B,C and D. Marital status consists of category married and single. Age of persons lies between 20-50. Ethic group has three categories that are white, Asian and Hispanic. Gender of persons has two categories male and female. Number of pets for the persons interviewed lies in the range of 2 to 5. Last GPA variable ranges from 2.2 to 3.8.

Answer:

  b.  is $34 per ounce

Step-by-step explanation:

If the production cost were less, a competitor would drive the price down. If the production cost were more, the supplier would go out of business.

Since we're at equilibrium, the production cost must be equal to $34 per ounce.

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

[tex]Time=\frac{Distance}{Speed}[/tex]

Time took by James for climbing = [tex]\frac{2}{3}\ hours[/tex]

Time took for running back = [tex]\frac{2}{4}\ hours[/tex]

Total time = [tex]\frac{2}{3}+\frac{2}{4}=\frac{8+6}{12}=\frac{14}{12}[/tex]

Total time taken by James = [tex]\frac{7}{6}\ hours[/tex]

1 hour = 60

Total time taken by James = [tex]\frac{7}{6}*60=70\ minutes[/tex]

Time took by Lucas for climbing = [tex]\frac{2}{2}=\ 1\ hour[/tex]

Time took by Lucas for climbing = 60 minutes

Time took on return = [tex]\frac{2}{5} of\ an\ hour=\frac{2}{5}*60=24\ minutes[/tex]

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Keywords: distance, speed

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Answer:

Step-by-step explanation:

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

Time took by James for climbing =

Time took for running back =

Total time =

Total time taken by James =

1 hour = 60

Total time taken by James =

Time took by Lucas for climbing =

Time took by Lucas for climbing = 60 minutes

Time took on return =

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.®

Answer:

Price of  hamburgers = $2

Price of white-dogs = $3

Step-by-step explanation:

Let

The number price of hamburgers be x

The number price of white-dogs  be  y

Then the first family buys 3 hamburgers and 2 white -dogs for $13

3x + 2y = 13-----------------------------(1)

Another first family buys 2 hamburgers and 5 white -dogs for $16

2x + 5y = 16----------------------------(2)

To solve let us multiply eq(1) by 2 and eq(2) by 3, we get

6x + 4y = 26-----------------------------(3)

6x + 15y = 48----------------------------(4)

subracting (3) from (4)

6x + 15y = 48

6x + 4y = 26

(-)

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

        11y  = 22

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

[tex]y =\frac{22}{11}[/tex]

y =2

Now substituting the value of y in eq(1),

3x + 2(2) = 13

3x + 4 = 13

3x = 13-4

3x = 9

x =[tex]\frac{9}{3}[/tex]

x =3

Answer:

a) [tex]5xz + xy + 4yz[/tex]

b) 10

Step-by-step explanation:

a) Here [tex]F(x,y,z)=(5z+y)i+(4z+x)j+(4y+5x)k[/tex]

Since the case [tex]F[/tex]  =  ∇[tex]f[/tex] holds, then

∇[tex]f = f_xi+f_yj+f_zk[/tex] = [tex](5z+y)i+(4z+x)j+(4y+5x)k[/tex]

So, [tex]f_x = 5z + y[/tex]

If we integrate [tex]f_x[/tex] with respect to x, we will get an integration constant C which is also a function that depends to y and z.

Hence,

[tex]f = \int f_xdx = 5xz + xy + g(y,z)[/tex]

Now we need to find g(y,z).

So first let's take the derivative of g(y,z) with respect to y.

[tex]f_y = x + g_y(y,z) = 4z + x[/tex]

Hence, [tex]g_y(y,z) = 4z[/tex]

So now, if we integrate [tex]g_y[/tex] with respect to y to find g(y,z)

[tex]g = \int g_ydy = 4yz + C[/tex]

Thus,

[tex]f = 5xz + xy + g(y,z) = 5xz + xy + 4yz + C[/tex]

And since [tex]f(0,0,0)=0[/tex], then [tex]C=0[/tex]

Thus,

[tex]f = f(x,y,z) = 5xz + xy + 4yz[/tex]

b) By the Fundamental Theorem of Line Integrals, we know that

[tex]\int\limits^a_b F. dr = F[r(b)]-F[r(a)][/tex]

Hence,

[tex]\int\limits^a_b F. dr = F(1,1,1)-F(0,0,0) =[(5+1+4)-(0+0+0)]=10[/tex]

To find ????, solve the system of partial differential equations. Use the function ???? from part a to compute the line integral.

To find a function ???? such that ????=∇???? and ????(0,0,0)=0, we can solve the system of partial differential equations. Let ????=????????+????????+????????, then compute the partial derivatives of ???? with respect to each variable. Equating these partial derivatives to the given vector field components, we can solve for the unknown function ???? and find its value at the point (0,0,0).

To compute the line integral ∫????????⋅????????, we can use the fundamental theorem of calculus for line integrals. Since ????(x,y,z) is the gradient of ????, the line integral is equal to the change in ???? along the curve C from (0,0,0) to (1,1,1). We can use the function ???? found in part a to evaluate this change in ????.

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Answer:

Step-by-step explanation:

Initial amount that was deposited into the savings account is $600 This means that the principal,

P = 600

The account earns 2.1 % interest compounded annually.. This means that it was compounded once in a year. So

n = 1

The rate at which the principal was compounded is 2.1%. So

r = 2.1/100 = 0.021

It was compounded for t years. So

t = t

a) The formula for compound interest is

A = P(1+r/n)^nt

A = total amount in the account at the end of t years. Therefore

A = 600 (1+0.021/1)^1×t

A = 600(1.021)^t

b)when A =$800, it becomes

800 = 600(1.021)^t

Dividing both sides by 600, it becomes

1.33 = (1.021)^t

Taking the tth root of both sides

t = 14 years

It will take 14 years

Complete answer:

Fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of the 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.

(a) How many choices are there for which 7 courses to take?

(b) Explain intuitively why the answer to (a) is not [tex] \binom{5}{1}\binom{19}{6} [/tex]

Answer:

a) 71085 choices

b) See below

Step-by-step explanation:

a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's: [tex]\binom{20}{7} [/tex], but now we should subtract from that all the possibilities when none of the courses chose are a statistic course, that's is [tex]\binom{15}{7} [/tex] because 15 courses are not statistics and 7 are the ways to arrange them. So finally, the choices for which 7 courses to take with the constraint that at least 1 of the 7 courses must be a statistics course are:

[tex]\binom{20}{7}-\binom{15}{7}=71085 [/tex]

b) It's important to note that the constraint at least 1 of the 7 courses must be a statistics course make the possible events dependent, we can not only fix an statistic course and choose the others willingly ( that is what [tex] \binom{5}{1}\binom{19}{6} [/tex] means) because the selection of one course affect the other choices.

In this exercise, we have to use our knowledge of statistics to calculate how many options can be chosen for a course, so we find that:

a) 169 choices

b) 1 of the 7 courses must be a statistics course, because the selection of one course affect the other choices.

So from the data reported in the exercise we can say that:

a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's.  So we have that;

[tex]C=\frac{m!}{p!(m-p)!}\\C=\frac{20!}{7!(20-7)!}\\C=169[/tex]

b) It's influential to note that the restraint not completely 1 of the 7 courses must be a enumeration course form the attainable occurrence determined by, we can not only fix a statistic course and select the possible choice gladly because the preference from among choices of individual course influence the added selection.

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Answer:

8:30 AM

Step-by-step explanation:

Given:

Bus 143 stops every 30 minutes on its express route to downtown

and  Bus 62 stops there every 8 minutes on its west side route

both meet at the bus stop at 6.30 AM

to find when they would meet the second time we need to find LCM( 30,8)

= 120

Hence the two buses would meet after 120 minutes that is 2 hours.

Hence they would be meet at 6.30+2= 8.30 AM

The probability of getting two consecutive ones in a string of 1s and 0s of length n can be calculated as (n-1) / (2n).

The probability of getting two consecutive ones in a string of 1s and 0s of length n can be calculated by considering the number of possible outcomes and the number of favorable outcomes.

Let's assume that the probability of getting a one is p and the probability of getting a zero is q. The favorable outcomes are when two consecutive ones occur, which can be represented as (1,1).

To calculate the probability, we need to determine the number of ways we can arrange the numbers in the string. Since we are only interested in the position of the ones, we can ignore the zeros.

Therefore, the number of favorable outcomes is n-1, because we have n-1 places where two consecutive ones can occur in a string of length n.

The number of possible outcomes is 2n, because each digit in the string can be either a one or a zero. Therefore, the probability of getting two consecutive ones is:

p = (n-1) / (2n)

Answer:

6

Step-by-step explanation:

Answer:

Step-by-step explanation:

The initial length of Josh's rope was 43 feet. He cuts off a piece that is 10 feet long and gives it to his brother. This means that the length of the remaining rope would be

43 - 10 = 33 feet.

He wants to cut the rest of the rope into pieces measuring exactly 5 feet each. To determine the number of pieces that he can cut, it becomes

33/5 = 6.6

Therefore, only 6 pieces measuring exactly 5 feet each can be cut.

Average speed of slower car = 60 mph

Average speed of faster car = 70 mph

Solution:

Given that Two cars enter the Delaware turnpike at shoreline drive at 8 am, each heading for ocean city

Given that One cars average speed is 10 mph faster than the other

Let "x" be the average speed of slower car

Then x + 10 is the average speed of faster car

The faster car arrives at ocean city at 11 am, a half hour before the slower car

Time taken by faster car:

The faster car arrives at ocean city at 11 am but given that they start at 8 a.m

So time taken by faster car = 11 am - 8am = 3 hours

Time taken by slower car:

The faster car arrives at ocean city at 11 am, a half hour before the slower car

So slower car takes half an hour more than faster car

Time taken by slower car = 3 hour + half an hour = [tex]3\frac{1}{2} \text{ hour}[/tex]

Now the distance between Delaware turnpike at shoreline drive and ocean city will be same for both cars

Let us equate the distance and find value of "x"

The distance is given by formula:

[tex]distance = speed \times time[/tex]

Distance covered by faster car:

[tex]distance = (x + 10) \times 3 = 3x + 30[/tex]

Distance covered by slower car:

[tex]distance = x \times 3\frac{1}{2} = x \times \frac{7}{2} = 3.5x[/tex]

Equating both the distance,

3x + 30 = 3.5x

3x - 3.5x = -30

-0.5x = -30

x = 60

Thus average speed of slower car = 60 mph

Average speed of faster car = x + 10 = 60 + 10 = 70 mph

Answer:

2 hours

Step-by-step explanation:

45x + 40x = 170

85x = 170

x = 2

Answer:

40 mph: 4 hours 15 minutes

45 mph: 3 hours 46 minutes 40 seconds

Step-by-step explanation:

The rate of separation = 45 + 40 = 85 mph

speed = distance / time

85 = 170 / t

t= 170/85 hour

Answer:

Choice B

Step-by-step explanation:

Looks like interest isn't being used, so that makes it simple.

Every week the same amount is added, that means  we can use a linear equation, which is what it is asking us to pick between so it all works out, awesome.

Now, we need two things to make an equation.  the slope and two points.  Definitely have points so we need the slope.  

The slope is found by taking any two points and finding the difference of their y values and dividing that by the distance of their x values.  so find two points (x1, y1) and (x2, y2) and then use the formula(y2-y1)/(x2-x1)  Also, we will say number of weeks is the x and the amount of money is y

No matter which point you use you will get the slope is 55

From there we find the function with the formula y - y1 = m(x - x1)

We know m and I reccomend using (0, 100) as our x1 y1 because 0 will usually make things easier.

y - y1 = m(x - x1)

y - 100 = 55(x-0)

y = 55x + 100

you could have used any point again, 0 just means it goes away if you add 0 or subtract 0.  Of course your problem uses n so just replace x with n.  and y is the value we want to end at, which is 825

825 = 55n + 100, or arranging it so it's the same as the option, 55*n + 100 = 825, so that's choice B

The equation that calculates the weeks it would take to reach the target is : 55n = 825.

Given is the table shows the amount of money in your savings account over a period of 6 weeks and you plan to keep saving at the same rate until you have $825 in the account.

Assume that it would take [n] weeks to reach the target. The unit rate from the given table can be calculated as -

m = (155 - 100)/(1 - 0)

m = 55

So, we can write the equation as -

55n = 825

Therefore, the equation that calculates the weeks it would take to reach the target is : 55n = 825.

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Answer:

The function

{\ displaystyle f (z) = {\ frac {z} {1- | z | ^ {2}}}} {\ displaystyle f (z) = {\ frac {z} {1- | z | 2}

It is an example of real and bijective analytical function from the open drive disk to the Euclidean plane, its inverse is also an analytical function. Considered as a real two-dimensional analytical variety, the open drive disk is therefore isomorphic to the complete plane. In particular, the open drive disk is homeomorphic to the complete plan.

However, there is no bijective compliant application between the drive disk and the plane. Considered as the Riemann surface, the drive disk is therefore different from the complex plane.

There are bijective conforming applications between the open disk drive and the upper semiplane and therefore determined as Riemann surfaces, are isomorphic (in fact "biholomorphic" or "conformingly equivalent"). Much more in general, Riemann's theorem on applications states that the entire open set and simply connection of the complex plane that is different from the whole complex plane admits a bijective compliant application with the open drive disk. A bijective compliant application between the drive disk and the upper half plane is the Möbius transformation:

{\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}} {\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}}

which is the inverse of the transformation of Cayley.

if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

To understand why a power series diverges on the unit circle when the analytic function it represents has a pole on the unit circle, we can use the concept of analytic continuation and the properties of poles and singularities.

Here's a step-by-step explanation:

1.Analytic function on the unit disk: Let's consider an analytic function defined on the open unit disk, denoted by[tex]\(D = \{z \in \mathbb{C} : |z| < 1\}\)[/tex]. This means the function is holomorphic (complex differentiable) at every point within this disk.

2.Power series representation: Since the function is analytic on[tex]\(D\)[/tex], it can be represented by a power series expansion around any point [tex]\(z_0\) in \(D\)[/tex]. Let's denote this function by[tex]\(f(z)\)[/tex], and its power series representation centered at[tex]\(z_0\) by \(\sum_{n=0}^{\infty} a_n (z - z_0)^n\)[/tex].

3.Pole on the unit circle: Suppose[tex]\(f(z)\)[/tex] has a pole (a point where the function becomes unbounded) on the unit circle[tex]\(|z| = 1\)[/tex], i.e., there exists a point [tex]\(z_1\)[/tex] on the unit circle such that [tex]\(f(z_1)\)[/tex] is infinite. Without loss of generality, let's assume [tex]\(z_1 = 1\)[/tex] (since the unit circle is symmetric about the origin).

4.Behavior near the pole: Near the pole at [tex]\(z = 1\)[/tex], the function[tex]\(f(z)\)[/tex]can be expanded in a Laurent series, which includes negative powers of [tex]\((z - 1)\)[/tex]. This expansion will have infinitely many terms with negative powers, indicating the singularity at [tex]\(z = 1\)[/tex].

5.Radius of convergence: The radius of convergence of the power series representation of [tex]\(f(z)\)[/tex]is at least the distance from the center of convergence to the nearest singularity. In this case, since the singularity (pole) is on the unit circle, the radius of convergence of the power series cannot exceed 1.

6.Divergence on the unit circle: Since the radius of convergence of the power series representation of [tex]\(f(z)\)[/tex] is at most 1, the power series diverges at every point on the unit circle (except possibly at the point of singularity itself, where it may converge by definition). This divergence occurs because the function has a singularity (pole) on the unit circle.

Therefore, if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

The correct equilibrium at one atmosphere pressure associated with its Kelvin temperature is the steam-water equilibrium at 373 K.

The equilibrium at one atmosphere pressure that is correctly associated with the Kelvin temperature at which it occurs is option d. steam-water equilibrium at 373 K. To explain, in the Kelvin temperature scale, the freezing point of water is 273.15 K and the boiling point is 373.15 K, both under standard atmospheric conditions (1 atmosphere pressure). So, at 373 K, the situation would be a steam-water equilibrium, not an ice-water equilibrium as in options a and b. The Kelvin temperature for ice-water equilibrium is 273.15 K and not 0 K and 32 K as stated in options a or b. Similarly, steam-water equilibrium does not occur at 212 K as suggested in option c.

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Option d. steam-water equilibrium at 373 K. The steam-water equilibrium at one atmosphere pressure occurs at 373 K.

The correct answer is:

Let's break down the reasons:

Thus, the correct association is the steam-water equilibrium occurring at 373 K at 1 atmosphere pressure.

Answer:

256

Step-by-step explanation:

a₁ = 2

aₙ = (aₙ₋₁)²

a₂ = (a₂₋₁)²

a₂ = (a₁)²

a₂ = (2)²

a₂ = 4

a₃ = (a₃₋₁)²

a₃ = (a₂)²

a₃ = (4)²

a₃ = 16

a₄ = (a₄₋₁)²

a₄ = (a₃)²

a₄ = (16)²

a₄ = 256