Every week, cross country team members run more than 15 miles. Write an inequality that represents this situation. Let m represent the number of miles ran each week by cross country team members.

Answer:

The demand value of time lead is

a) 40

Step-by-step explanation:

X  Y Z company Uses "Continuous Review System for an item

Currently demand = 100 units

standard deviation =  20 units

lead time increase = 4 weeks

Apply Z statistic we get the value of standard deviation.

Answer:

The answer to your question is 74 u²

Step-by-step explanation:

Process

1.- Find the 4 vertices

  A (-2, 8)

  B (0, -4)

  C (4, 9)

  D (6, -3)

2.- Find the length of the base and the height

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

Distance AB =  \sqrt{(0 + 2)^{2} + (-4 - 8)^{2} }[/tex]

              dAB = [tex]\sqrt{4 + 144}[/tex]

              dAB = [tex]\sqrt{148}[/tex]

Distance BD = \sqrt{(6 - 0)^{2} + (-3 + 4)^{2} }[/tex]

             dBD = [tex]\sqrt{36 + 1}[/tex]

             dBD = [tex]\sqrt{37}[/tex]

3.- Find the area

Area = base x height

Area = [tex]\sqrt{148} x \sqrt{37}[/tex]

Area = [tex]\sqrt{5476}[/tex]

Area = 74 u²

Answer:

a) [tex]f(x)=\frac{2}{3}e^{-\frac{2}{3}x}[/tex] when [tex]x\geq 0[/tex]

[tex]f(x)=0[/tex] otherwise

b) [tex]P(X<2)=0.2636[/tex]

Step-by-step explanation:

First of all we have a Poisson process with a mean equal to :

μ = λ = [tex]\frac{2}{3}[/tex] (Two phone calls every 3 minutes)

Let's define the random variable X.

X : ''The waiting time until the first call that arrives after 10 a.m.''

a) The waiting time between successes of a Poisson process is modeled with a exponential distribution :

X ~ ε (λ)    Where λ is the mean of the Poisson process

The exponential distribution follows the next probability density function :

I replace λ = a for the equation.

[tex]f(x)=a(e)^{-ax}[/tex]

With

[tex]x\geq 0[/tex]

and

[tex]a>0[/tex]

[tex]f(x)=0[/tex] Otherwise

In this exercise λ= a = [tex]\frac{2}{3}[/tex] ⇒

[tex]f(x)=(\frac{2}{3})(e)^{-\frac{2}{3}x}[/tex]

[tex]x\geq 0[/tex]

[tex]f(x)=0[/tex] Otherwise

That's incise a)

For b) [tex]P(X>2)[/tex] We must integrate between 2 and ∞ to obtain the probability or either use the cumulative probability function of the exponential

[tex]P(X\leq x)=0[/tex]

when [tex]x<0[/tex]

and

[tex]P(X\leq x)=1-e^{-ax}[/tex] when [tex]x\geq 0[/tex]

For this exercise

[tex]P(X\leq x)=1-e^{-\frac{2}{3}x}[/tex]

Therefore

[tex]P(X>2)=1-P(X\leq 2)[/tex]

[tex]P(X>2)=1-(1-e^{-\frac{2}{3}.2})=e^{-\frac{4}{3}}=0.2636[/tex]

(A) The pdf of X, the waiting time until the first call after 10 a.m., is f(x; 2/3) = (2/3) * e^(-(2/3) * x), and, (B) the probability that X > 2 (the first call arrives more than 2 minutes after 10 a.m.) is approximately 0.264.

(a) To find the probability density function (pdf) of X, we first need to understand the arrival rate of the calls, which follows a Poisson process. In our case, the arrival rate (λ) is two calls every 3 minutes, which could also be expressed as 2/3 of a call per minute.

For a Poisson process, the waiting times between arrivals are exponentially distributed. Therefore, the pdf for X, the waiting time until the first call, is given by the exponential distribution function.

The exponential distribution has the following pdf:

f(x; λ) = λ * e^(-λ * x)

In our case, substituting λ = 2/3 (the arrival rate per minute), the pdf of waiting time X becomes:

f(x; 2/3) = (2/3) * e^(-(2/3) * x)

(b) The second part of the question asks for the probability that the waiting time until the first call, X, is greater than 2 minutes.

For an exponential distribution, the cumulative distribution function (CDF), which gives the probability that a random variable is less than or equal to a certain value, is as follows:

F(x; λ) = 1 - e^(-λ * x)

We need P(X > 2), but it's easier to compute P(X <= 2), and then subtract that from 1.

So, we first find the cumulative probability that the waiting time is 2 minutes or less, using our given λ and x = 2:

P(X <= 2) = F(2; 2/3) = 1 - e^(-(2/3) * 2)

After calculating, this probability is approximately 0.736.

Therefore, the probability that waiting time X is greater than 2 minutes, P(X > 2), is simply 1 minus this result, which approximately equals to 0.264.

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Answer:

B. Customer Dimension

Step-by-step explanation:

Custom dimensions is used to collect and analyze data that Analytics doesn't capture. You can send value to custom dimensions with a variable that pulls data from web page or use layer to pass specific values.

If you want to view data by different user such as Bronze , Gold , Platinum level Google Analytics feature set up the Custom Dimensions to collect the data.

The G. Analytics feature I would  set up to collect this data is  B. Customer Dimension

Custom dimensions are used to gather and examine information that Analytics is unable to. A variable that retrieves information from a web page can be used to deliver value to custom dimensions, or a layer can be used to provide certain values. Sales data is broken down into individual customers via the customer hierarchy in the customer dimension.

To comply with reporting standards, the hierarchy between the root element All Customers and the individual customer might be arranged arbitrarily.

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For this case we must find the solution set of the given inequalities:

Inequality 1:

[tex]3 (2x + 1)> 21[/tex]

Applying distributive property on the left side of inequality:

[tex]6x + 3> 21[/tex]

Subtracting 3 from both sides of the inequality:

[tex]6x> 21-3\\6x> 18[/tex]

Dividing by 6 on both sides of the inequality:

[tex]x> \frac {18} {6}\\x> 3[/tex]

Thus, the solution is given by all the values of "x" greater than 3.

Inequality 2:

[tex]4x + 3 <3x + 7[/tex]

Subtracting 3x from both sides of the inequality:

[tex]4x-3x + 3 <7\\x + 3 <7[/tex]

Subtracting 3 from both sides of the inequality:

[tex]x <7-3\\x <4[/tex]

Thus, the solution is given by all values of x less than 4.

The solution set is given by the union of the two solutions, that is, all real numbers.

Answer:

All real numbers

Answer:

  x = 34

Step-by-step explanation:

The figure will be a rhombus if the diagonals cross at right angles. That is ...

  (3x -12)° = 90°

  3x = 102

  x = 34

The figure is a rhombus when x=34.

The value of x will make parallelogram ABCD a rhombus when all sides are of equal length, which corresponds to the situation where the diagonals have slopes of +1 and -1 and bisect each other at right angles.

To determine the value of x that will make parallelogram ABCD a rhombus, we can consider the geometric properties that define a rhombus. A rhombus is a type of parallelogram with all sides of equal length, which also means its diagonals bisect each other at right angles. Given that the diagonals of the parallelogram must have slopes of +1 and -1 to maintain the properties of bisection, x would be the length making the sides of the parallelogram equilateral.

In the scenario where the original shape is a unit square, changes in frame of reference should preserve the affine property of bisection. Hence, parallelogram ABCD will become a rhombus when all sides are of equal length, which can be determined through equilateral properties of the parallelogram when the diagonals bisect each other at right angles and have slopes of +1 and -1.

Answer:

The area of the sumo wrestling ring is [tex]5.29 \pi[/tex]

Step-by-step explanation:

The circumference of the circular sumo wrestling ring is [tex]4.6\pi[/tex], that means its radius [tex]r[/tex] is:

[tex]2\pi r=4.6\pi[/tex]

[tex]r=\frac{4.6}{2} =\boxed{2.3\:meters.}[/tex]

Now once we have the radius [tex]r[/tex] of the sumo wrestling ring we can find its area [tex]A[/tex] by the following formula:

[tex]A=\pi r^2[/tex]

Putting in the value of [tex]r=2.3\:meters[/tex] we get:

[tex]A=\pi (2.3m)^2=\boxed{5.29\pi\:\:m^2}[/tex]

Therefore the area of the sumo wrestling ring is [tex]{5.29\pi\:\:m^2[/tex]

Answer:

5.29pi

Step-by-step explanation:

The equation of the linear relationship given the x and y coordinates is calculated in slope-intercept form by finding the slope and y-intercept. In this case, the equation of the line is y = -0.5x + 95.

In mathematics, the equation of a linear relationship can be represented in the slope-intercept form, which is y = mx + c.

Where, 'm' is the slope of the line and 'c' is the y-intercept.

Given the x and y coordinates, we can calculate the slope 'm' using the formula, m = (y2 - y1) / (x2 - x1).

For example: m = (87.5-92.5) / (35-25) = -5 / 10 = -0.5. So the slope 'm' is -0.5.

Now we can find the y-intercept 'c' by substituting the known x,y coordinates and the slope into the equation and solving for 'c'. Let's take x = 25 and y = 92.5, substituting these values, we will get c = y - mx =  92.5 - (-0.5 * 25) = 95.

So, the equation of the straight line in slope-intercept form is y = -0.5x + 95.

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Answer:

1296 windows

Step-by-step explanation:

HALF of the floors, means

HALF of 48, that is:

48 * 0.5 = 24

Thus, we can say:

24 floors each have 18 windows, and

24 floors each have 36 windows

Total Number of Windows:

24 * 18 = 432 windows

24 * 36 = 864 windows

Total = 432 + 864 = 1296 windows

Answer:

1296 windows are present in the building

Explanation:

Given the office building is 48 floors high

Half of floors have 18 windows each

Then , half of floors =[tex]\frac{48}{2}[/tex] = 24 floors

Total windows on half of the floors, that is 24 floors

= [tex]18\times 24[/tex]

= 432 windows

Also, half of the floors have 36 windows each

Total windows on rest half floors (24 floors)

=[tex]36 \times 24[/tex]

= 864 windows

Total windows = 432 + 864 = 1296 windows

Therefore, 1296 windows are present in the building

Answer:

Step-by-step explanation:

|t − 56|= 3 states that the temperature, t, can be as low as (56-3)°F, or 53°F, and as high as (56+3)°F, or 59°F.

On a number line, plot a dark dot at both 53°F and 59°F, and then connect these two dots with a solid line.

The maximum and minimum values of temperature are 59°F and  53°F respectively.

A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.

A mathematical phrase in which the sides are not equal is referred to as being unequal. In essence, a comparison of any two values reveals whether one is less than, larger than, or equal to the value on the opposite side of the equation.

Given the inequality

|t − 56|= 3

Now,

Taking positive value ;

t - 56 = 3

t = 59

Now taking negative value

-(t-56) = 3

t = -3 + 56 =  53

Hence "The maximum and minimum values of temperature are 59°F and  53°F respectively".

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Answer:

12

Step-by-step explanation:

x=(-10×1+23×2)÷(2+1)=36/3=12

Final answer:

The coordinate that divides the line segment from -10 at J to 23 at K in the ratio of 2 to 1 is C) 12.

Explanation:

The coordinate that divides the line segment from -10 at J to 23 at K in the ratio of 2 to 1 is 12.

To find this coordinate, we can use the concept of a section formula. Let the ratio be m:n. The coordinate divided is [tex](\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})[/tex]. Substituting the values, we get [tex](\frac{2 ( 23) + 1 ( -10)}{2+1}, \frac{2 (0) + 1 ( 2)}{2+1})[/tex] = (12, 0).

Therefore, the required coordinate that divides the line segment in the ratio of 2 to 1 is C) 12.

Answer:

  y = -3x - 2

Step-by-step explanation:

Parallel lines have the same slope. The only answer choice with the same slope (x-coefficient = -3) as the given line is the one shown above.

Answer: The estimated margin of error = 0.5 centimeter

The sample mean = 4.3 centimeters

Step-by-step explanation:

The confidence interval for population  mean is given by :-

[tex]\overline{x}\pm E[/tex]

or [tex](\overline{x}-E,\ \overline{x}+E)[/tex]

, where [tex]\overline{x}[/tex] = sample mean.

E = Margin of error .

The given confidence interval : (3.8,4.8 )

Lower limit : [tex]\overline{x}-E=3.8[/tex]                (1)

Upper limit =  [tex]\overline{x}+E=4.8[/tex]                (2)

Eliminate equation (1) from (2) , we get

[tex]2E=1.0\\\\\Rightarrow\ E=\dfrac{1}{2}=0.5[/tex]

⇒ The estimated margin of error = 0.5 centimeter

Add (1) and (2) ,we get

[tex]2\overline{x}-E=8.6\\\\\Rightarrow\ \overline{x}=\dfrac{8.6}{2}=4.3[/tex]  

⇒ The sample mean = 4.3 centimeters

Answer:

It is significant because when they migrate from one place to other it becomes a source of genetic diversity between them and other population.

Step-by-step explanation:

Answer:

[tex]25x+10y+18=0[/tex]

Step-by-step explanation:

We are given that a rectangle in which the equation of one side is given by

[tex]2x-5y=9[/tex]

We have to find the equation of another side of the rectangle.

We know that the adjacent sides of rectangle are perpendicular to each other.

Differentiate the given equation w.r.t.x

[tex]2-5\frac{dy}{dx}=0[/tex]   ([tex]\frac{dx^n}{dx}=nx^{n-1}[/tex])

[tex]5\frac{dy}{dx}=2[/tex]

[tex]\frac{dy}{dx}=\frac{2}{5}[/tex]

Slope of the given side=[tex]m_1=\frac{2}{5}[/tex]

When two lines are perpendicular then

Slope of one line=[tex]-\frac{1}{Slope\;of\;another\;line}[/tex]

Slope of another side=[tex]-\frac{5}{2}[/tex]

Substitute x=0 in given equation

[tex]2(0)-5y=9[/tex]

[tex]-5y=9[/tex]

[tex]y=-\frac{9}{5}[/tex]

The equation of given side is passing through the point ([tex]0,-\frac{9}{5})[/tex].

The equation of line passing through the point [tex](x_1,y_1)[/tex] with slope m is given by

[tex]y-y_1=m(x-x_1)[/tex]

Substitute the values then we get

[tex]y+\frac{9}{5}=-\frac{5}{2}(x-0)=-\frac{5}{2}x[/tex]

[tex]y=-\frac{5}{2}x-\frac{9}{5}[/tex]

[tex]y=\frac{-25x-18}{10}[/tex]

[tex]10y=-25x-18[/tex]

[tex]25x+10y+18=0[/tex]

Hence, the equation of another side of rectangle is given by

[tex]25x+10y+18=0[/tex]

Answer:

y=2/5x-9

I just answered this and got it right.

Step-by-step explanation:

The difference between the fastest and slowest time in the 40-yard dash is 0.71 seconds.

The difference between the fastest and slowest time in the 40-yard dash can be found by subtracting the slowest time from the fastest time. In this case, the fastest time was 4.37 seconds and the slowest time was 5.08 seconds. To find the difference, we subtract 5.08 seconds from 4.37 seconds.

The difference between the fastest and slowest time is 0.71 seconds.

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Answer:

x = 8,49 ft

y  = 4,24  ft  

Step-by-step explanation:

Let  x be the longer side of rectangle   and  y  the shorter

Area of rectangle     =    36 ft²     36  =  x* y   ⇒ y =36/x

Perimeter of rectangle:

P  =  2x   +   2y    for convinience we will write it as    P  = ( 2x + y ) + y

C(x,y)   =  1 * ( 2x  +  y  )  +  3* y

The cost equation as function of x is:

C(x)  =  2x  + 36/x   + 108/x

C(x)  =  2x  + 144/x

Taking derivatives on both sides of the equation

C´(x)  = 2  - 144/x²

C´(x)  = 0         2  - 144/x² = 0       ⇒  2x²  -144 = 0    ⇒  x² =  72

x = 8,49 ft       y  = 36/8.49    y  = 4,24  ft    

How can we be sure that value will give us a minimun

We get second derivative

C´(x)  = 2  - 144/x²      ⇒C´´(x)  = 2x (144)/ x⁴

so C´´(x) > 0

condition for a minimum

The number of adult tickets sold for the school play was 225.

This is a problem of simple algebra. Let's denote the number of adult tickets sold as a. It is stated in the problem that 67 more student tickets were sold than adult tickets. Therefore, we can denote the number of student tickets sold as a + 67. The problem also tells us that a total of 517 tickets were sold. Hence, we can form an equation: a + a + 67 = 517. Simplifying this equation gives us 2a + 67 = 517. And solving for a (the number of adult tickets) we subtract 67 from both sides to get 2a = 450, then divide by 2, gives us a = 225. So, 225 adult tickets were sold.

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Amaya will have $18.90 store credit left.

Step-by-step explanation:

Available store credit = $50.86

Cost of video game = $24.97

Cost of golf club accessory = $6.99

Total amount spent = Cost of video game + cost of golf club accessory

[tex]Total\ amount\ spent=24.97+6.99\\Total\ amount\ spent=\$31.96[/tex]

Remaining store credit = Available store credit - total amount spent

[tex]Remaining\ store\ credit=50.86-31.96\\Remaining\ store\ credit=\$18.90[/tex]

Amaya will have $18.90 store credit left.

Keywords: Addition, subtraction

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Answer:

The correct option is B) 720.

Step-by-step explanation:

Consider the provided information.

We have 10 candidates those qualified for 3 of the positions.

There are three nursing positions to be filled at Lilly Hospital. Position 1 is the day nursing supervisor, position 2 is the night nursing supervisor; and position 3 is the nursing coordinator position.

For Position 1  we have 10 choices, if we select 1 out of 10 candidates we are left with 9 candidates.

For position 2 we have 9 candidates, if we select 1 out of 9 candidates we are left with 8 candidates.

For position 3 we have 8 candidates.

Therefore, the number of ways are: [tex]10\times 9\times 8=720[/tex]

The number of different ways that 3 positions can be filled by these applicants is 720.

Hence, the correct option is B) 720.

Correct Option Is (e. 120.) The number of different ways that 3 positions can be filled by the applicants is 120.

To determine the number of different ways that 3 positions can be filled by these applicants, we can use the concept of combinations. Since there are 10 candidates and the order of the positions does not matter, we can use the combination formula. The number of combinations of 10 candidates taken 3 at a time is given by:

C(10, 3) = 10! / (3!(10-3)!)

Simplifying this expression, we get:

C(10, 3) = 10! / (3!7!)

Calculating the factorial values, we have:

C(10, 3) = 10 * 9 * 8 / (3 * 2 * 1) = 120

Therefore, the number of different ways that 3 positions can be filled by these applicants is 120.

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Answer:

  (a)  fave = 8

  (b)  c = 64/9

  (c)   c ≈ 7.111

Step-by-step explanation:

(a) The average value of the function is its integral over the interval, divided by the width of the interval.

  [tex]f_{ave}=\displaystyle\frac{1}{16-0}\int_0^{16} {3x^{\frac{1}{2}}} \, dx=\left.\frac{x^{3/2}}{8}\right|_0^{16}=8[/tex]

__

(b) We want ...

  f(c) = 8

  3√c = 8 . . . . . use f(c)

  √c = 8/3 . . . . . divide by 3

  c = (8/3)² . . . . square

  c = 64/9

__

(c) c ≈ 7.111

To find the average value of a function, evaluate the definite integral over the interval and divide by the length of the interval.

To find the average value of a function on a given interval, we need to evaluate the definite integral of the function over the interval and divide it by the length of the interval.

For the given function f(x) = 3√x on the interval [0, 16], the average value fave is given by:

fave = (1/[16-0]) * ∫(0 to 16) 3√x dx

Simplifying this integral, we get:

fave = 3/16 * (2/3) * (16^(3/2) - 0^(3/2)) = 4(16 - 0) = 64

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Answer:

  0

Step-by-step explanation:

All of the terms have positive signs, so there are no sign changes. Zero sign changes means there are zero positive real roots.

Answer:

The two coaches will charge the same amount of money after working for 11 hours

Step-by-step explanation:

Let us assume for m hours, they both will charge same amount.

For COACH A:

The initial Fee = $ 6,925

The per hour fee  = $450

So, the fees in m hours = m x ( Per hour fees) = m x ($450)  = 450 m

So, the total fees of Coach A in m hours = Initial Fee + fee for m hours

                                                                  = $ 6,925  + 450 m  

⇒ The total fees of Coach A in m hours  = $ 6,925  + 450 m ....  (1)

For COACH B:

The initial Fee = $ 5,000

The per hour fee  = $725

So, the fees in m hours = m x ( Per hour fees) = m x ($725)  = 725  m

So, the total fees of Coach B in m hours = Initial Fee + fee for m hours

                                                                  = $ 5,000  + 725 m  

⇒ The total fees of Coach B in m hours  =$ 5,000  + 725 m ....  (2)

Now, for m hours , they both charge the SAME AMOUNT fees

⇒$ 6,925  + 450 m  = $ 5,000  + 725 m    ( from (1) and (2))

or, 6925 - 5000 = 725 m - 450 m

or, 1925 = 175 m

or,m = 1925 / 175 = 11

or, m = 11

Hence, the two coaches will charge the same amount of money after working for 11 hours.

Answer:

[tex]\displaystyle \iint_S {\text{curl \bold{F}} \cdot} \, dS = \boxed{\bold{0}}[/tex]

General Formulas and Concepts:

Calculus

Integration Rule [Reverse Power Rule]:

[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:

[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:

[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:

[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Methods: U-Substitution + U-Solve

Multivariable Calculus

Partial Derivatives

Triple Integrals

Cylindrical Coordinate Conversions:

Integral Conversion [Cylindrical Coordinates]:

[tex]\displaystyle \iiint_T {f(r, \theta, z)} \, dV = \iiint_T {f(r, \theta, z)r} \, dz \, dr \, d\theta[/tex]

Vector Calculus

Surface Area Differential:

[tex]\displaystyle dS = \textbf{n} \cdot d\sigma[/tex]

Del (Operator):

[tex]\displaystyle \nabla = \hat{\i} \frac{\partial}{\partial x} + \hat{\j} \frac{\partial}{\partial y} + \hat{\text{k}} \frac{\partial}{\partial z}[/tex]

Stokes’ Theorem:

[tex]\displaystyle \oint_C {\textbf{F} \cdot } \, d\textbf{r} = \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma[/tex]

Divergence Theorem:

[tex]\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {\nabla \cdot \textbf{F}} \, dV[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \textbf{F} (x, y, z) = x^2z^2 \hat{\i} + y^2z^2 \hat{\j} + xyz \hat{\text{k}}[/tex]

[tex]\displaystyle \text{Region:} \left \{ {{\text{Paraboloid:} \ z = x^2 + y^2} \atop {\text{Cylinder:} \ x^2 + y^2 = 16}} \right[/tex]

Step 2: Integrate Pt. 1

Step 3: Integrate Pt. 2

Convert region from rectangular coordinates to cylindrical coordinates.

[tex]\displaystyle \text{Region:} \left \{ {{\text{Paraboloid:} \ z = x^2 + y^2} \atop {\text{Cylinder:} \ x^2 + y^2 = 16}} \right \rightarrow \left \{ {{\text{Paraboloid:} \ z = r^2} \atop {\text{Cylinder:} \ r^2 = 16}} \right[/tex]

Identifying limits, we have the bounds:

[tex]\displaystyle \left\{ \begin{array}{ccc} 0 \leq z \leq r^2 \\ 0 \leq r \leq 4 \\ 0 \leq \theta \leq 2 \pi \end{array}[/tex]

Step 4: Integrate Pt. 3

We evaluate the Stokes' Divergence Theorem Integral using basic integration techniques listed under "Calculus".

[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \int\limits^{2 \pi}_0 \int\limits^4_0 \int\limits^{r^2}_0 {r \bigg( 2z^2r \cos \theta + 2z^2r \sin \theta +r^2 \cos \theta \sin \theta \bigg)} \, dz \, dr \, d\theta \\& = \frac{1}{3} \int\limits^{2 \pi}_0 \int\limits^4_0 {zr^2 \bigg[ 2z^2 \big( \cos \theta + \sin \theta \big) + 3r \sin \theta \cos \theta \bigg] \bigg| \limits^{z = r^2}_{z = 0}} \, dr \, d\theta \\\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \frac{1}{3} \int\limits^{2 \pi}_0 \int\limits^4_0 {r^5 \bigg[ 2r^3 \big( \cos \theta + \sin \theta \big) + 3 \sin \theta \cos \theta \bigg]} \, dr \, d\theta \\& = \frac{1}{54} \int\limits^{2 \pi}_0 {r^6 \bigg[ 4r^3 \big( \cos \theta + \sin \theta \big) + 9 \sin \theta \cos \theta \bigg] \bigg| \limits^{r = 4}_{r = 0}} \, d\theta \\\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \frac{2048}{27} \int\limits^{2 \pi}_0 {\cos \theta \Big( 9 \sin \theta + 256 \Big) + 256 \sin \theta} \, d\theta \\& = \frac{-1024}{243} \bigg[ 4608 \cos \theta - \bigg( 9 \sin \theta + 256 \bigg)^2 \bigg] \bigg| \limits^{\theta = 2 \pi}_{\theta = 0} \\& = \boxed{\bold{0}}\end{aligned}[/tex]

∴ we have calculated the Stokes' Theorem integral with the given region and function using the Divergence Theorem.

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Topic: Multivariable Calculus

The Stokes' theorem is applied to convert a surface integral of a curl of a vector into a line integral. This is done by identifying the curl of the given vector field F and setting up the limits of the integral based on given bounds. The integral is then evaluated.

Stokes' theorem is used in vector calculus to simplify certain types of surface integrals. It transforms a surface integral of a curl of a vector field into a line integral. F(x, y, z) = x2z2i + y2z2j + xyzk, here, is the given vector field. The surface S is the part of the paraboloid that lies within the cylinder x² + y² = 16. The theorem is used to evaluate the integral S curl F · dS, by treating the surface integral as a line integral. The line integral can be easier to evaluate. The exact process involves identifying the curl of F, setting up the bounds of the integral based on the restrictions given, and then computing the integral.

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It will take approximately 10.986 years for the employee to earn a combined total of $100,000. Rounding to the nearest year, it will take approximately 11 years for the employee to reach this earnings milestone.

To determine how many years it will take for the employee to earn a combined total of $100,000, we need to set up and solve the following integral:

[tex]\[ \int_{0}^{t} 5000e^{0.1\tau} \, d\tau = 100,000 \][/tex]

Here, [tex]\( t \)[/tex] represents the time in years. The integral represents the accumulated earnings from the start (0 years) to t years based on the continuous stream function[tex]\( f(\tau) = 5000e^{0.1\tau} \).[/tex]

Let's solve this integral:

[tex]\[ \int_{0}^{t} 5000e^{0.1\tau} \, d\tau = \left. \frac{5000}{0.1}e^{0.1\tau} \right|_{0}^{t} \][/tex]

Evaluate this at the upper and lower limits:

[tex]\[ \frac{5000}{0.1}e^{0.1t} - \frac{5000}{0.1}e^{0.1 \times 0} \][/tex]

Simplify:

[tex]\[ 50000(e^{0.1t} - 1) \][/tex]

Now, set this expression equal to the target earnings of $100,000 and solve for  t :

[tex]\[ 50000(e^{0.1t} - 1) = 100,000 \][/tex]

Divide both sides by 50000:

[tex]\[ e^{0.1t} - 1 = 2 \][/tex]

Add 1 to both sides:

[tex]\[ e^{0.1t} = 3 \][/tex]

Now, take the natural logarithm (ln) of both sides:

[tex]\[ 0.1t = \ln(3) \][/tex]

Solve for t:

[tex]\[ t = \frac{\ln(3)}{0.1} \][/tex]

Using a calculator:

[tex]\[ t \approx \frac{1.0986}{0.1} \]\[ t \approx 10.986 \][/tex]

Answer:

Both are moving apart with the rate of 8.99 feet per sec.

Step-by-step explanation:

From the figure attached,

Man is walking north with the speed = 4 ft per second

[tex]\frac{dx}{dt}=4[/tex] feet per sec.

Woman starts walking due south with the speed = 5ft per second

[tex]\frac{dy}{dt}=5[/tex] ft per sec.

We have to find the rate of change in distance z.

From the right angle triangle given in the figure,

[tex]z^{2}=(x+y)^{2}+(500)^{2}[/tex]

We take the derivative of the given equation with respect to t,

[tex]2z.\frac{dz}{dt}=2(x+y)(\frac{dx}{dt}+\frac{dy}{dt})+0[/tex] -----(1)

Since distance = speed × time

Distance covered by woman in 15 minutes or 900 seconds = 5(900) = 450 ft

y = 4500 ft

As the man has taken 5 minutes more, so distance covered by man in 20 minutes or 1200 sec = 4×1200 = 4800 ft

x = 4800 ft

Since, z² = (500)² + (x + y)²

z² = (500)² + (4500 + 4800)²

z² = 250000 + 86490000

z = √86740000

z = 9313.43 ft

Now we plug in the values in the formula (1)

2(9313.43)[tex]\frac{dz}{dt}[/tex] = 2(4800 + 4500)(4 + 5)

18626.86[tex]\frac{dz}{dt}[/tex] = 18(9300)

[tex]\frac{dz}{dt}=\frac{167400}{18626.86}[/tex]

[tex]\frac{dz}{dt}=8.99[/tex] feet per sec.

Therefore, both the persons are moving apart by 8.99 feet per sec.

Final answer:

To find the rate at which the people are moving apart 15 minutes after the woman starts walking, calculate the displacements of both individuals and then find the total displacement between them. Answer comes to be 611.52 feet.

Explanation:

Rate at which people are moving apart:

The question asks at what rate are two people moving apart 15 minutes after one of them starts walking, given that one walks north and the other south from different points. To solve this, one has to understand relative velocity and the concept of adding vectors graphically.

Calculate the man's northward displacement after 15 minutes: 4 ft/s * 5 minutes = 20 ft

Calculate the woman's southward displacement after 15 minutes: 5 ft/s * 15 minutes = 75 ft

Find the total displacement between them: ([tex]\sqrt{(500^2 + 20^2)[/tex]) + [tex]\sqrt{(500^2 + 75^2))[/tex] = 611.52 ft

Answer:

4.95

Step-by-step explanation:

You take the 4.50 and multiply it by 1.10 and it equals 4.95. Also I did it and I got it right.

The total price of the pastry is $4.95.

price of the pastry = $4.50

sales tax = 10%

The sales tax on the pastry is 10% of the price of the pastry.

Tax on pastry = price of the pastry x percentage of sales tax

                       [tex]= \$4.50 \times 10\%\\= 4.5\times \dfrac{10}{100}\\= 4.50 \times 0.1\\= 0.45[/tex]          

therefore, the tax on the pastry will be $0.45

Total price of the pastry =  Price of the pastry + tax on the pastry

                                        =   $4.50 + $0.45

                                        =  $4.95

Hence, the total price of the pastry is $4.95.

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Answer:

8 bathroom jobs and 9 kitchen jobs

Step-by-step explanation:

B=60

K=20

8*60=480

9*20=180

that would give harry 660 dollars in a week. HOWEVER- we have to make sure that its equal to or less than 15 hours of work.

8*1h= 8 hours in bathroom

9*45m=6.75hr in kitchen

8 hours+6.75 hours=14.75hr 14.75 hr<15hr so it works.

Answer:

[tex]\displaystyle \frac{5}{12} = cot∠B \\ 2\frac{2}{5} = cot∠A \\ \\ 2\frac{2}{5} = tan∠B \\ \frac{5}{12} = tan∠A \\ \\ \frac{5}{13} = cos∠B \\ \frac{12}{13} = cos∠A[/tex]

Step-by-step explanation:

[tex]\displaystyle \frac{OPPOSITE}{HYPOTENUSE} = sin\:θ \\ \frac{ADJACENT}{HYPOTENUSE} = cos\:θ \\ \frac{OPPOSITE}{ADJACENT} = tan\:θ \\ \frac{HYPOTENUSE}{ADJACENT} = sec\:θ \\ \frac{HYPOTENUSE}{OPPOSITE} = csc\:θ \\ \frac{ADJACENT}{OPPOSITE} = cot\:θ \\ \\ \frac{10}{24} = cot∠B → \frac{5}{12} = cot∠B \\ \frac{24}{10} = cot∠A → 2\frac{2}{5} = cot∠A \\ \\ \frac{24}{10} = tan∠B → 2\frac{2}{5} = tan∠B \\ \frac{10}{24} = tan∠A → \frac{5}{12} = tan∠A \\ \\ \frac{10}{26} = cos∠B → \frac{5}{13} = cos∠B \\ \frac{24}{26} = cos∠A → \frac{12}{13} = cos∠A[/tex]

I am joyous to assist you anytime.