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Using the distance formula, d = √(x2 - x1)2 + (y2 - y1)2, what is the distance between point (-5, -2) and point (8, -3) rounded to the nearest tenth?

10.3 units

12.6 units

1 unit

13 units

The number of pennies and nickels that has a worth of $0.45 is 15 and 6 respectively

Given:

total worth = $0.45

Total coins = 21

let

number of pennies = x

number of nickels = y

x + y = 21 (1)

0.01x + 0.05y = 0.45 (2)

multiply (1) by 0.01

0.01x + 0.01y = 0.21 (3)

0.01x + 0.05y = 0.45 (2)

subtract (2) from (1)

0.05y - 0.01y = 0.45 - 0.21

0.04y = 0.24

y = 0.24 / 0.04

y = 6

substitute y = 6 into (1)

x + y = 21 (1)

x + 6 = 21

x = 21 - 6

x = 15

Therefore, the number of pennies and nickels that has a worth of $0.45 is 15 and 6 respectively.

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Answer:

(a) V(0) = 50 gal, V(20) = 0 gal

(b)At t= 0 the tank is full.

At t=0 the tank is empty

(c)

Time       volume

 0               50 gal

 5                37.5 gal

 10              25 gal

 15             12.5 gal

20                0 gal

(d)

Net change of volume = 50 gal

Step-by-step explanation:

Given that the capacity of the tank is 50 gal.

Torricelli's Law gives the volume of water remaining in the tank after t minutes as

[tex]V(t)=50(1-\frac{t}{20})^2[/tex]

(a)

To find V(0), we put t = 0 in the above equation

[tex]V(0)=50(1-\frac{0}{20})^2[/tex]

        [tex]=50(1-0)^2[/tex]

        = 50 gal

To find V(20), we put t =2 0 in the above equation

[tex]V(20)=50(1-\frac{20}{20})^2[/tex]

        [tex]=50(1-1)^2[/tex]

        = 0 gal

(b)

At t= 0 the tank is full.

At t=0 the tank is empty.

(c)

Time                                          V(t)

  0                                  [tex]50(1-\frac{0}{20})^2=50 \ gal[/tex]

  5                                  [tex]50(1-\frac{5}{20})^2=37.5 \ gal[/tex]

 10                                  [tex]50(1-\frac{10}{20})^2=25 \ gal[/tex]

 15                                 [tex]50(1-\frac{15}{20})^2=12.5 \ gal[/tex]

 20                                [tex]50(1-\frac{20}{20})^2=0[/tex]

(d)

Net change of volume = V(0) -V(20)

                                     =(50-0) gal

                                    = 50 gal

The volume V(t) of water remaining in the tank after t minutes is given by V(t) = 50(1−t/20). V(0) represents the initial volume of water in the tank, which is 50 gallons. V(20) represents the volume of water remaining in the tank after 20 minutes, which is 0 gallons.

(a) To find V(0), substitute t = 0 into the equation V(t) = 50(1−t/20).

V(0) = 50(1−0/20) = 50(1−0) = 50(1) = 50

Similarly, to find V(20), substitute t = 20 into the equation V(t) = 50(1−t/20).

V(20) = 50(1−20/20) = 50(1−1) = 50(0) = 0

(b) V(0) represents the initial volume of water in the tank, which is 50 gallons. V(20) represents the volume of water remaining in the tank after 20 minutes, which is 0 gallons.

(c) Creating a table of values of V(t) for t = 0, 5, 10, 15, 20:

t | V(t)

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

0 | 50

5 | 37.5

10 | 25

15 | 12.5

20 | 0

(d) The net change in volume V as t changes from 0 min to 20 min is V(20) - V(0).

V(20) - V(0) = 0 - 50 = -50 gallons

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Answer: U' = (0, 1)

Step-by-step explanation:

U = (3, -5)

rotate 90° counterclockwise means (x, y) = (-y, x)

new U = (5, 3)

down 2 units means subtract 2 from the y-coordinate

newer U = (5, 1)

left 5 units means subtract 5 from the x-coordinate

U' = (0, 1)

Answer: The volume is 143

Step-by-step explanation:

Answer:

D)191 feet

Step-by-step explanation:

Let the height of the light house be |AB| and the Boat be at point C as shown in the diagram.

The angle of depression of the boat from the top A of the lighthouse is given as 25 degrees

Angle BCA = 25 degrees (Alternate Angles are Equal)

We want to determine the distance of the boat C from the base of the lighthouse B i.e. |BC|

[tex]Tan\alpha =\frac{opposite}{adjacent}[/tex]

Tan 25=[tex]\frac{89}{|BC|}[/tex]

Cross multiply

|BC| X tan 25 =89

|BC| = [tex]\frac{89}{tan 25}[/tex]=190.86 feet

The distance of the boat C from the base of the lighthouse B is 191 feet (to the nearest feet).

From the given info (and the linked question) we find

[tex]\cos\alpha=-\dfrac5{13}[/tex]

[tex]\sin\alpha=\dfrac{12}{13}[/tex]

[tex]\sin\beta=-\dfrac45[/tex]

Then using the angle-sum identity for cosine, we have

[tex]\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta[/tex]

[tex]\cos(\alpha-\beta)=\left(-\dfrac5{13}\right)\dfrac35+\dfrac{12}{13}\left(-\dfrac45\right)=-\dfrac{63}{65}[/tex]

Final answer:

To find the exact value of cos(α-β), we use the cosine sum and difference identity and the respective sine and cosine values calculated from the given tangent and cosine values for angles α and β. Using this approach, we find that cos(α-β) equals 63/65.

Explanation:

The exact value of cos(α-β) can be found by using the sum and difference identities for cosine. Since tanα = -12/5, in quadrant II, we can find the corresponding sine and cosine values for α using the Pythagorean identity sin²α + cos²α = 1. For cosβ = 3/5, in quadrant IV, we do a similar procedure to find the sine of β. With both sine and cosine for α and β, we use the identity cos(A-B) = cosA cosB + sinA sinB to find cos(α-β).

To find the sine and cosine for α, given that tanα = -12/5, we know that the opposite side is -12, and the adjacent side is 5, so the hypotenuse using the Pythagorean theorem is √(12² + 5²) = √(144+25) = √169 = 13. Thus sinα = -12/13 (negative because α lies in the second quadrant where sine is negative) and cosα = 5/13 (positive because cosine in the second quadrant is positive).

For β, we already have cosβ = 3/5. The sine can be found using the Pythagorean identity 1 - cos²β = sin²β, which gives sinβ = -√(1 - (3/5)�) = -√(1 - 9/25) = -√(16/25) = -4/5 (negative because β is in the fourth quadrant where sine is negative).

Now, we can find the exact value of cos(α-β) by plugging in the values:

cos(α-β) = cosα cosβ + sinα sinβ = (5/13)*(3/5) + (-12/13)*(-4/5)
= 15/65 + 48/65 = 63/65.

Therefore, the exact value of cos(α-β) is 63/65.

Step-by-step explanation:

Below is an attachment containing the solution.

Paulina and Aliya's equations, 4x+2y=6 and 12x+6y=18, are multiples of each other. This means they represent the same line and indicate an infinite number of solutions rather than a single intersection point typically sought after in a system of distinct linear equations.

Paulina and Aliya have created linear equations to solve a mathematical problem. Paulina has the equation 4x+2y=6 and Aliya has 12x+6y=18. At a glance, these equations may look different, but upon closer inspection, Aliya's equation is actually just Paulina's equation multiplied by 3. This realization is pertinent because it suggests both equations represent the same line. Therefore, these two equations should have the same solution set.

To analytically solve simultaneous equations, one could use methods such as substitution, elimination, or matrix and determinant-based approaches. Using elimination or substitution, we aim to isolate one variable and solve for it. For example, because Aliya's equation is a multiple of Paulina's, if they were meant to be a system of separate lines, one way to solve them would be to simplify Aliya's equation by dividing by 3, revealing it to be identical to Paulina's, which indicates that this system has an infinite number of solutions (all points on the line represented by the equation).

If a system has two distinct equations, elimination involves adding or subtracting equations from one another to eliminate one of the variables, and substitution involves solving for one variable in terms of the other and then substituting this expression into the other equation. When equations are actually multiples of each other, this indicates either an infinite number of solutions or no solutions dependant if the equations are consistent or inconsistent respectively.

Answer:

0.900

Step-by-step explanation:

The easiest way to solve this problem is by putting it into a calculator.

When put into a calculator, the answer comes out to be 0.9004304526, which can be rounded to 0.900, or just 0.9.

Good morning ☕️

Answer:

0.900

Step-by-step explanation:

using a calculator you’ll find:

sin⁻¹(-0.435) = -25.785293878311

now

cos(sin⁻¹(-0.435)) = cos(-25.785293878311)

                           = 0.900430452617

If we round 0.900430452617 to nearest thousandth we get: 0.900

:)

Answer: There are 15 nickles and 20 dimes.

Step-by-step explanation:

Since we have given that

Let the number of nickles be 'x'

Let the number of dimes be '2x-10'.

Total amount = $2.75

According to question, we get that

[tex]0.1(2x-10)+0.05x=2.75\\\\0.2x-1+0.05x=2.75\\\\0.25x=2.75+1\\\\0.25x=3.75\\\\x=\dfrac{3.75}{0.25}\\\\x=15[/tex]

Hence, there are 15 nickles and [tex]2x-10=2(15)-10=30-10=20[/tex] dimes.

Therefore, there are 15 nickles and 20 dimes.

Final answer:

Mary has 15 nickels and 20 dimes.

Explanation:

Let's solve this problem step-by-step:

Answer:

Explanation:

The "Shortest Route Algorigtm" aims to determine the most efficient or short route, when a several alternative pahtways can connect or be used to implement a solution.

A graph is drawn with the different nodes and paths that connect them. The distance between every pair of consecutive nodes is written.

The picture shows that for the step #1, there are, in principle, three routes: AB, AC, and AD.

AB must be discarded because it is not viable (a negative distance is not possible).

AC is more efficient than AD because the distance of AC is 3 and the distance of AD is 8. Thus AC is selected and circled.

To continue from AC, the possible routes are shown in step #2. They are ACB; 3 and ACE; 6.

ACB i s shorter, thus ACB is circled.

In step #3, the possible routes are ACBE; 8 and ACBD; 7. Thus, route ACBD is shorter, and it shall be circled.

The conclusion of the algorithm is that the route ACBD is the shoretes (most efficient).

The route to circle next is route ACBD; 7

From the question, we understand that she wants to determine the shortest route.

This means that, she has to circle the node with the smallest value in each step.

From the diagram, the smallest node in step 3 is ACBD; 7

Hence, the route to circle next is route ACBD; 7

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Answer:

She makes 7.8% portion of sales collectively.

Step-by-step explanation:

Let us assume she makes total of 100 calls.

The percentage of calls which are busy or unanswered = 22 %

So, now calculating 22% of 100 , we get:

[tex]\frac{22}{100} \times 100 = 22[/tex]

So, 22 of her calls are UNANSWERED.

Now, let us find out the number of calls successfully made by telemarketer.

Total successful calls = Total Calls made - Number of unanswered calls

                                    = 100 - 22  = 78

So, she makes a total of 78 calls successfully.

Now, he sells subscriptions to 10% of the 78 calls made successfully.

So, now calculating 10% of 78 , we get:

[tex]\frac{10}{100} \times 78 = 7.8[/tex]

So, she sold 7.8 subscriptions in total out of 100 attempts.

Also, as we know 7.8 out of 100  = 7.8% of 100.

Hence, she makes 7.8% portion of sales collectively.

Final answer:

The antique purchased by Juan increases in value by 10% each year. The value of the antique after 2 years can be found by calculating the expression $200a, where a is a constant. The value of [tex]\( a \) is \( 1.21 \).[/tex]

Explanation:

To find the value of a, we need to represent the annual increase of 10% in terms of multiplication.

After the first year, the value of the antique increases by [tex]\( 10\% \) of its previous value, which is \( 0.10 \times 200 \) dollars.[/tex]

After the second year, the value of the antique increases by [tex]\( 10\% \) of its value at the end of the first year, which is \( 0.10 \times (200 + 0.10 \times 200) \) dollars.[/tex]

Generally, after \( n \) years, the value of the antique will be [tex]\( 200 \times (1 + 0.10)^n \) dollars.[/tex]

The expression given for the value of the antique 2  years after purchase is 200a , where a is a constant. This represents the value of the antique after 2  years.

Equating the expression to the value of the antique after 2 years, we have:

[tex]\[ 200a = 200 \times (1 + 0.10)^2 \][/tex]

Now, let's solve for \( a \):

[tex]\[ 200a = 200 \times (1.10)^2 \][/tex]

200a = 200 \times 1.21

200a = 242

Dividing both sides by 200:

[tex]\[ a = \frac{242}{200} \][/tex]

[tex]\[ a = 1.21 \][/tex]

Therefore, the value of [tex]\( a \) is \( 1.21 \).[/tex]

Answer:

Step-by-step explanation:

The relationship is negative, negative correlation

Answer:   [tex]\bold{(a)\quad \dfrac{32}{3}\qquad (b)\quad \dfrac{32}{3}}[/tex]

Step-by-step explanation:

(a) First, find the x-coordinates where the two equations cross

                            y = -1    and   y = 3 - x²

  -1 = 3 - x²

 -4 =     -x²

  4 =       x²

± 2 =       x       → These are the upper and lower limits of your integral

Then subtract the two equations and integrate with upper bound of x = 2 and lower bound of x = -2

[tex]\int_{-2}^{+2}[(3-x^2)-(-1)]dx\\\\\\=\int_{-2}^2(4-x^2)dx\\\\\\=4x-\dfrac{x^3}{3}\bigg|_{-2}^{+2}\\\\\\=\bigg(8-\dfrac{8}{3}\bigg)-\bigg(-8+\dfrac{8}{3}\bigg)\\\\\\=\large\boxed{\dfrac{32}{3}}[/tex]

(b) We know the upper and lower bounds of the y-axis as y = 3 and y = -1

Next, find the equation that we need to integrate by solving for x.

      y = 3 - x²

x² + y = 3

x²       = 3 - y

x         [tex]=\pm\sqrt{3-y}\\[/tex]

[tex]\rightarrow \qquad x=\sqrt{3-y}\quad and \quad x=-\sqrt{3-y}[/tex]

Now, subtract the two equations and integrate with upper bound of y = 3 and lower bound of y = -1

[tex]\int_{-1}^{+3}[(\sqrt{3-y})-(-\sqrt{3-y})]dy\\\\\\=\int_{-1}^{+3}(2\sqrt{3-y})dy\\\\\\=\dfrac{-4\sqrt{(3-y)^3}}{3}\bigg|_{-1}^{+3}\\\\\\=\bigg(0\bigg)-\bigg(-\dfrac{32}{3}\bigg)\\\\\\=\large\boxed{\dfrac{32}{3}}[/tex]

Answer:

More. The product of 25 and 3/5 is 15 which is greater than 14.

Step-by-step explanation:

Answer:

  see below

Step-by-step explanation:

As always, the coordinate transformations are ...

  a. 180° — (x, y) ⇒ (-x, -y)

  b. 270° CCW or 90° CW — (x, y) ⇒ (y, -x)

  c. 90° CCW or 270° CW — (x, y) ⇒ (-y, x)

___

It can be quick and easy to use a transparency or tracing paper to locate the rotated position of the image.

Answer: [tex]\frac{2}{4}[/tex]  or [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

You need to analize the information given in the exercise.

Let be "x" represents the whole distance part that Jolene have driven to the state park.

According the the explained in the problem, in the first day Jolene drove [tex]\frac{1}{4}[/tex] of the distance.

 Knowing that, you can identify that the whole distance (or the value of "x"), is the following:

[tex]x=\frac{4}{4}[/tex]    

(If you simplify it, you get: [tex]x=1[/tex])

You also know that the second day Jolene drove farther than the first day; therefore, there are two possible cases for the part of the distance she might have driven the second day.  These cases are:

Case 1: [tex]\frac{2}{4}[/tex] of the distance the second day.

Case 2: [tex]\frac{3}{4}[/tex] of the distance the second day.

Answer:

the answer is 7

Step-by-step explanation:

angle 30°

x=14/2

x=7

The value of x is 7 cm.

If you have the hypotenuse, multiply it by sin(θ) to get the length of the side opposite to the angle.

y = 14* sin 30°

y = 14* 0.5

y = 7

y = 7cm

The bottom line of a triangle is the base of the triangle, and it can be one of the three sides of the triangle. In a triangle, one side is the base side and the remaining two sides can be the height or the hypotenuse side.

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Answer:

The given data is not normal.

Step-by-step explanation:

We are given the following data:

30, 59, 69, 50, 58, 71, 55, 43, 3,  66, 52, 56, 62, 36, 13, 29, 17, 31

Condition for normality:

Mean = Mode = Median

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{800}{18} = 44.44[/tex]

Mode is the most frequent observation of the data.

Since all the value appeared once, there is no mode.

[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]

Sorted data: 3, 13, 17, 29, 30, 31, 36, 43, 50, 52, 55, 56, 58, 59, 62, 66, 69, 71

Median =

[tex]=\dfrac{9^{th}+10^{th}}{2} = \dfrac{50+52}{2}=51[/tex]

Since the mean, mode and median of data are not equal, the data is not normal.

Answer:

e) Trend Analysis

The term is 'trend analysis'. This refers to a statistical method used to evaluate and predict future trends based on historical data. This technique is specifically referenced in prediction of labor demand using prior year’s data.

The concept referred to in the question is e. Trend analysis. Trend analysis is a statistical method used to evaluate and predict future trends based on historical data. In the context of labor demand, a trend analysis would involve examining labor demand data from the previous year, identifying patterns and trends within that data, and using statistical models to make predictions about labor demand for the upcoming year.

For example, suppose a company has seen a steady increase in labor demand over the past five years. Based on this trend, they can build a statistical model that predicts an increase in labor demand for the next year as well. This strategy helps companies plan for their future staffing needs, ensuring they have the necessary resources to meet their objectives.

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Answer: [tex]m\angle A=116\°[/tex]

Step-by-step explanation:

The missing figure is attached.

For this exercise it is important to remember that, by definition, the opposite interior angles of an inscribed quadrilateral are supplementary, which means that their sum is 180 degrees.

Based on this, you can identify that the angle D and the angle B are opposite and, therefore, supplementary.

Knowing that, you can write the following equation:

[tex]x+28\°=180\°[/tex]

Now you must solve for "x" in order to find its value. This is:

[tex]x=180\°-28\°\\\\x=152\°[/tex]

Then:

[tex]m\angle D=152\°[/tex]

You know that:

[tex]m\angle A=(x-36)\°[/tex]

Therefore, since you know the value of "x", you can substitute it into   [tex]m\angle A=(x-36)\°[/tex] and then you must evaluate, in order to find the measure of the angle A. This is:

 [tex]m\angle A=152\°-36\°\\\\m\angle A=116\°[/tex]

Step-by-step explanation:

Here, given:

Let us assume the number on Tens place  = M

The number in Units place  = N

So, the actual value of the number   = 10 M + N

The number can be written as MN.

Now, The value of M ( Tens digit) is Less than 5.

So, M  = 1, 2 , 3 or 4  ( 0 not included)

Also, The value of N(Unit digit)  is greater than 7.

So, N  = 8, 9 or 0  ( 0  included)

So, by combining all possibilities for M and N , the possible number chosen by Andy can be expressed as MN :

Number = {18,19,10,28,29,20,38,39,30,48,49 or 40}

Any number can be chosen from the above set as all the numbers match the given restrictions.

Answer:

Step-by-step explanation:

Look at the info given as (x, y) coordinates with the number of hours as x and the number of miles as y.  The first coordinate then is (2, 22) and the second is (5, 55).  The rate at which the bee flies is the same as the slope of the coordinates.

[tex]\frac{55-22}{5-2}=11[/tex]

This means that the bee flies 11 miles per hour.  Use either one of the coordinates now to find the equation for the line.  I pick (2, 22) and point-slope form:

y - 22 = 11(x - 2) and

y - 22 = 11x - 22 so

y = 11x

That's the equation.  If we want to use it as a model, we can find how many miles it will fly in a given time, or how long it will take to fly a given number of miles.  We are asked to find how far it can fly in 4 hours.  So we will use our equation and replace x with 4:

y = 11(4) so

y = 44 miles

Answer:

general admission tickets  1170

courtside tickets  985

tickets bench seats.  130

Step-by-step explanation:

To solve the problem, it is necessary to generate a system of equations with the information provided by the statement.

First be

x = # general admission tickets

y = # courtside tickets

z = # tickets bench seats.

The first equation would be that they sold nine times more general admission tickets than bench seats tickets.

That is: x = 9 * z (1)

And the second equation is that the number of general admission tickets sold was 55 more than the sum of the number of courtside tickets and bench seats tickets.

That is: x = 55 + y + z (2)

Now the third equation would be the money raised.

28 * x + 43 * y + 173 * z = 97605 (3)

Now if I replace I rearrange (1):

z = x / 9 (4) and replacement in 2, I have:

x = 55 + y + x / 9, rearranging:

y = (8/9) * x - 55 (5)

Now replacing (4) and (5) in 3, we have:

28 * x + 43 * ((8/9) * x - 55) + 173 * (x / 9) = 97605

Solving the above:

x = 1170, therefore

y = (8/9) * 1170 55 = 985

z = 1170/9 = 130

We can confirm this with equation (3)

28 * x + 43 * y + 173 * z = 97605

28 * 1170 + 43 * 985 + 173 * 130 = 97605

450 general admission tickets, 171 courtside tickets, and 50 bench tickets were sold for the Harlem Globetrotter show.

Let's assign variables to represent the number of tickets sold for each category:

Lets  x  = number of general admission tickets sold

Lets  y   = number of courtside tickets sold

Lets  z  = number of bench tickets sold

According to the given information:

The cost of the general admission ticket = $28

The cost of the courtside ticket = $43

The cost of the bench ticket = $173

There were 9 times as many general admission tickets sold as bench tickets: x = 9z

The number of general admission tickets sold was 55 more than the sum of the courtside and bench tickets: x = y + z + 55

The total sales for all three types of tickets was $97,605: 28x + 43y + 173z = 97605

Now, we have a system of three equations with three variables:

x = 9z

x = y + z + 55

28x + 43y + 173z = 97605

We can use substitution or elimination method to solve this system of equations. Solving this system, we find that x = 450, y = 171, and z = 50. Therefore, 450 general admission tickets, 171 courtside tickets, and 50 bench tickets were sold for the Harlem Globetrotter show.

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Answer:

Answer:

a) k = 0.1875 m

b) r2 = 0.46875 m

c) q = -1.125*10^-8 C

Step-by-step explanation:

Given:

- The total Length of rod L = 1.5 m

- The total charge of the rod Q = -9 * 10^8 C

- Total section of a rod n = 8

Find:

1. What is the length of one of these pieces?

2. What is the location of the center of piece number 2?

3. How much charge is on piece number 2?

Solution:

- The entire rod is divided into 8 pieces, so the length of each piece would be k:

                                     k = L / n

                                     k = 1.5 / 8

                                     k = 0.1875 m

- The distance from center of entire rod and center of section 2 is 2.5 times the section length

                                     r2 = 2.5*k

                                     r2 = 2.5*(0.1875)

                                     r2 = 0.46875 m

- Assuming the charge on the rod is uniformly distributed. The the charge for each section of rod is given by q:

                                     q = Q / n

                                     q = -9 * 10^8 / 8

                                     q = -1.125*10^-8 C

The length of one of the eight pieces of the rod is 0.1875 m. The center of piece number 2 is at <0.28125, 0, 0> m. The charge on piece number 2 is -1.125e-08 Coulombs.

The problem involves the concepts of electric field, charge distribution, and coordinate system in Physics. Let's answer the question part by part:

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Answer:

Therefore the wire should be anchored at 10 ft away from pole which is 10 ft long.

Step-by-step explanation:

Given that , The distance between two poles is 30 ft.

The length of 1st pole is = 20 ft

The length of second pole is = 10 ft.

Let the wire anchored to the ground at a distance x ft from the second pole.

Then, the distance of anchored from the first pole is = (30-x)

The total length of the wire is L = m+n

We know the pythagorean theorem,

Height²+base² = hypotenuse²

To find the value of m and n we use  pythagorean theorem

From the left side triangle in the picture we get,

10²+x²= m²

⇒m²=100+x²

[tex]\Rightarrow m= \sqrt {100+x^2[/tex]

and right side  triangle in the picture we get,

20²+(30-x)² = n²

⇒n²= x²-60x+1300

[tex]\Rightarrow n= \sqrt {x^2 -60x+1300}[/tex]

Then ,

[tex]L= \sqrt{(100+x^2)}+\sqrt{(x^2-60x+1300) }[/tex]

Differentiating with respect to x

[tex]L'= \frac {2x}{2\sqrt{100+x^2}}+ \frac{2x-60}{2\sqrt {x^2-60x+1300}}[/tex]

For minimize, L' =0

[tex]\frac {2x}{2\sqrt{100+x^2}}+ \frac{2x-60}{2\sqrt {x^2-60x+1300}}=0[/tex]

[tex]\Rightarrow \frac {x}{\sqrt{100+x^2}}=- \frac{x-30}{\sqrt {x^2-60x+1300}}[/tex]

Squaring both sides

[tex]\Rightarrow( \frac {x}{\sqrt{100+x^2}})^2=(- \frac{x-30}{\sqrt {x^2-60x+1300}})^2[/tex]

[tex]\Rightarrow x^2(x^2-60x+1300)= (x^2-60x+900)(100+x^2)[/tex]

[tex]\Rightarrow x^4 -60x^3+1300x^2= 100x^2-6000x+90000+x^4-60x^3+900x^2[/tex]

[tex]\Rightarrow 300x^2+6000x-90000=0[/tex]

[tex]\Rightarrow x^2+20x-300=0[/tex]

[tex]\Rightarrow x=10,-30[/tex]

Therefore x = 10. [x=-30 negligible, since distance can not negative]

Therefore the wire should be anchored at 10 ft away from pole which is 10 ft long.

The problem can be solved geometrically through the principles of trigonometry. By setting up two right triangles formed by the telephone poles and the anchoring point, we can create two equations by Pythagorean Theorem. By taking the derivative of the total wire length and setting it to zero, we can find the optimal value for 'x' (location of the anchoring point) which results in the minimal amount of wire used.

To solve for the minimal amount of wire needed, we can use the principles of mathematics. More specifically, we will use the concept of trigonometry and geometry to create two right triangles. The taller pole (20ft), the shorter pole (10ft) and the point on the ground where the wire is anchored form the two right triangles, one with 20ft height and another with 10ft height.

Let's denote the length of wire between the taller pole and ground as 'a', between the shorter pole and the ground as 'b', and the distance between the point on the ground where the wire is anchored and the base of the first pole as 'x'. We have:

The total length of wire used (which we want to minimize) is a + b.

To find the minimal length, we can take the derivative of 'a+b' with respect to 'x' and set the derivative equation to 0 then solve for 'x'. This will give you where to place the anchor on the ground (minimal amount of wire used) between the two poles. You may find out an optimal 'x' value that is less than 30ft, ensuring that the anchoring point is between the two poles.

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Answer:

The distance is 6.3 units

Answer: P = Po ( 1 + 0.075)

Step-by-step explanation: let Po = initial population

P = final population.

The increase in population is by 7.5%, which implies that if the initial population Increases by 7.5%, we would have a new (current) population.

Final population = initial population + increament of initial population.

Where increment of initial population = 7.5% of Po = 0.075 Po

P = Po + 0.075Po

P = Po ( 1 + 0.075)

Answer:

  -8

Step-by-step explanation:

Let n represent the number. Then we have ...

  n -3 . . . . the difference of a number and 3

  = . . . . . .  equals

  5 +2n . . . 5 added to twice the number

Adding -n-5 to both sides of the equation gives ...

  n -3 -n -5 = 5 +2n -n -5

  -8 = n . . . . . simplify

The number is -8.

Modular division, also known as modular arithmetic, has various applications in cryptography, computer science, and number theory. It is used in encryption algorithms like RSA and in solving congruence equations.

Modular division, also known as modular arithmetic, is a mathematical operation that involves finding the remainder when one number is divided by another. Modular division is commonly used in various mathematical applications such as cryptography, computer science, and number theory.

One application of modular division is in the encryption and decryption of data using modular arithmetic. For example, the RSA encryption algorithm relies on modular division to encode and decode messages.

Another application of modular division is in solving congruence equations. Congruence equations represent the idea of equivalence of numbers modulo a given number. By using modular division, we can determine the solutions to these equations.

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Modular division is used to find remainders in mathematics, but it is not used for dividing polynomials or dividing by zero. It's also not useful for dividing vectors in physics.

Modular division is a mathematical method used to find the remainder of a division problem, but it is not necessarily applicable in all contexts. For example, dividing a polynomial by a polynomial generally does not result in a polynomial, indicating that other methods must be used in this situation. Furthermore, it's important to note that modular division cannot be used to divide by zero, since this is undefined in mathematics. Additionally, within the discipline of physics, modular division is not applicable for operations such as dividing a vector by another vector component by component, as there are no useful physics applications for this type of operation.