An electric pole 15 m high is supported by a wire fixing its one end on the ground at some distance from the pole. If the wire joining the top of the pole is inclined to the ground at an angle of 60°, find the length of the wire.​

The solution of Ax = b is unique precisely when Ax = 0 has only the trivial solution, because Ax-b is consistent, its solution set is obtained by translating the solution set of Ax-0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.

The correct answer is an option (b)

Let us assume that the equation Ax = b has a solution.

Now we need to show the solution is unique when Ax = 0 has only the trivial solution.

As we know Ax = 0 represents homogenous equation.

If Ax = b has a solution, it is unique if and only if every column of A is a pivot column. If every column of A is a pivot column, there are no free variables, and therefore the homogeneous equation has only the trivial solution.

Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax= 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.

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The complete question is:

Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Choose the correct answer.

A. Since Ax = b is inconsistent, its solution set is obtained by translating the solution set of Ax-0. For Ax-b to be inconsistent, Ax = 0 has only the trivial solution.

B. Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax-0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.

C. Since Ax = b is inconsistent, then the solution set of Ax = 0 is also inconsistent. The solution set of Ax = 0 is inconsistent if and only if Ax = 0 has only the trivial solution.

D. Since Ax = b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax 0 has only the trivial solution.

The population mean speed for automobiles and trucks on the length of road is equal to or less than 45 mph, which is the null hypothesis (H0) for this claim. This can be expressed mathematically as H0: = 45 mph.

where the population's average speed is represented by.

For this claim, the alternative hypothesis (H1) would be that the average population speed for vehicles and trucks on that section of road is higher than 45 mph. This can be expressed mathematically as H1: > 45 mph.

 A random sample of cars and trucks would be chosen in order to test the assertion, and their speeds would be recorded. The probability of detecting the sample mean speed under the null hypothesis would then be calculated using a statistical test like the t-test.

  Therefore,  the alternative hypothesis is that the population mean speed for vehicles and trucks on the length of road is more than 45 mph, whereas the null hypothesis is that it is equal to or less than 45 mph.

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The null hypothesis (H0) for this statement is that the population mean speed for cars and trucks on the length of the road is equal to or less than 45 mph. H0: = 45 mph is a mathematical expression for this.

where is the average speed of the population.

The alternative hypothesis (H1) for this assertion is that the typical population speed for cars and trucks on that stretch of road is greater than 45 mph. Mathematically, this is represented as H1: > 45 mph.

The population's average speed for cars and trucks travelling the distance of the road is equal to or lower than

To test the claim, a sample of randomly selected cars and trucks would be selected, and their speeds would be recorded. An analytical statistical test like the t-test would then be used to determine the chance of detecting the sample mean speed under the null hypothesis.

The null hypothesis is that the population's mean speed for cars and trucks on the length of the road is equal to or less than 45 mph, whereas the alternative hypothesis is that it is more than 45 mph.

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The account balance after five years will be $10355.

The interest that is paid on the principal and interest at regular intervals is known as compound interest. The difference between compound interest and simple interest starts to show after the first year, though.

The formula to calculate the compound interest is given below;-

[tex]A = P ( 1+\dfrac{R}{n})^{nt}[/tex]

Given that deposit at 7% annual interest compounded yearly to have $10,000 in the account at the end of 5 years.

The account balance after five years will be:-

[tex]A = P ( 1+\dfrac{R}{n})^{nt}[/tex]

[tex]A = 10000 ( 1+\dfrac{0.07}{1})^{5}[/tex]

A = 10000 x ( 1.07)⁵

A = $10355

Hence, the amount will be $10355.

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Answer: The answer is D

Step-by-step explanation: I hope this helps!

There are 384 ways to arrange these given letters as per the given situation.

A permutation is an orderly arrangement of things or numbers. Combinations are a means to choose items or numbers from a collection or set of items without worrying about the items' chronological order.

Given, different ways can the 8 letters a, b, c, d, e, f, g, and h be arranged if the letters a and b must be next to one another, the letters c and d must be next to one another, the letters e and f must be next to one another, and the letters g and h must be next to one another.

In this question, we have 2 ways where A and B (AB, BA) are next to each other and 2 ways where C and D (CD, DC) are next to each other.  2 ways where g and h (gh, hg) are next to each other and 2 ways where e and f (ef, fe) are next to each other.

The total number of arrangements is = 4! * 2 * 2 *2*2

The total number of arrangements is = 24 * 16

The total number of arrangements is = 384

Therefore, the Different ways to arrange these letters are 384.

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Mutually exclusive events are events that have no overlap between them and the sum of the probabilities of all mutually exclusive events is equal to 1.

What is the probability theory?

Probability theory is a branch of mathematics that deals with the study of random phenomena. It provides a mathematical framework for analyzing the likelihood of different outcomes in a given situation, and for making predictions about future events based on past data.

Things that have an equal chance of happening are known as equally likely events or mutually exclusive events. In probability theory, mutually exclusive events are events that cannot happen at the same time; if one event occurs, the other cannot.

In general, mutually exclusive events are events that have no overlap between them and the sum of the probabilities of all mutually exclusive events is equal to 1.

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There is sufficient evidence that the valve performs above the specifications.

Null Hypothesis (H0): The mean pressure of the valve is equal to 5.3 lbs/square inch

Alternative Hypothesis (H1): The mean pressure of the valve is greater than 5.3 lbs/square inch

To test whether the valve performs above the specifications, a hypothesis test can be used. The test statistic used will be a t-test since the sample size is below 30. The critical value at the 0.05 level is 1.645. The formula for the t-test is: t = (X-μ)/(s/√n) , where X is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the known values, the t-test is calculated as: t = (5.5-5.3)/(0.7/√100) = 2.571. Since this value is greater than the critical value at the 0.05 level (1.645), we can reject the null hypothesis and conclude that there is sufficient evidence that the valve performs above the specifications.

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A point-slope form is an equation for a line, using a point and the slope of the line. So we have to find one x-coordinate and y-coordinate and write it in the form of (y-y₁) = m(x-x₁).

Here the total after 10 months is said to be 65 and number of months = 10. So we can calculate the set-up fee.

5 ×10 + x = 65

  x = 65-50 = 15.

The only constant here is the set-up fee, 15.

Lets the number of months be x and total amount be y.

 y- 5x = 15

Lets calculate the y intercept. x-term will be 0.

  y+0 = 15

  y = 15

Lets calculate the x-intercept. y term will be 0

-5x = 15

x = -3

Two points on the line are (-3,0) and (0,15)

slope = (15-0)/(0--3)

         =15/3 = 5

So lets find out one point in the equation

If y=10,

10-5x = 15

  -5x = 15-10

    x = 5/-5 =-1

So a point on the line is (-1,10)

Point slope form,

(y-10) = 5 (x- ⁻1)

(y-10) = 5 (x+1)

So this is the point slope form.

y = 5x+15

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Answer:

All parts are below

Step-by-step explanation:

Seeing the first part asking for f(1). The equation f(n) lies above, so right here is plug and play. Since the n was replaced with a 1, we'll go to the table and find n and find 1.

Since it's the first number, looking directly below it is 7. Meaning f(1) = 7

Seeing common ratio... everytime n goes up 1, the value doubles, making r = 2/1 also known as 2

Now that we know f(1) is 7 and r is 2, we can plug it all in

f(n) = f(1) * (r)^n-1

Plugged in, it's f(n) = 7 * 2^1-1. 1-1 = 0. Anything to the power of 0 equals 1, which means 2^0 is 1

Multiplying 7 and 1 gives you 7, so

f(n) = 7

Geometric sequence is that f(n) = f(1) * (r)^n-1

Also known as f(n) = 7 * 2^n-1

For problem #2, try replicating all this and let me know how it goes

Based on the information provided, I have analyzed the three options for hosting a dance for the students of CISE, and have come to the following recommendations:

Ancient Abyss: This location charges a flat fee of $200 and $4 per person. This option may be suitable for smaller events with less than 50 attendees, as the flat fee is relatively high. However, for larger events with more than 50 attendees, the per person cost will become more significant and may make this option less cost-effective.

Beautiful Babylon: This location charges a flat fee of $99 and $6 per person. This option may be more cost-effective for larger events with more than 50 attendees, as the per person cost is lower than Ancient Abyss. However, for smaller events with less than 50 attendees, the flat fee may be relatively high.

Crazy Cleopatra: This location charges a flat fee of $280 and $3 per person. This option has the highest flat fee but the lowest per person cost. This option may be more cost-effective for larger events with more than 50 attendees, as the per person cost is lower than Beautiful Babylon. However, for smaller events with less than 50 attendees, the flat fee may be relatively high.

Based on the above analysis, I recommend that the student council consider Beautiful Babylon as the most cost-effective option for hosting the dance. However, I also recommend that the council consider the number of attendees expected at the event and adjust their decision accordingly.

In summary, The council should consider the number of attendees expected, Beautiful Babylon is a good option for larger events, for smaller events, Crazy Cleopatra is the better option.

The perimeter of the smaller rectangle would be 27 cm.

Given is that two similar rectangles shown below have a scale factor of

3 : 10.

Since the two rectangles are similar, then the ratio of their corresponding perimeters will be constant.

We can write -

P{1}/P{2} = 3/10

P{1}/90 = 3/10

P{1} = (3 x 90)/10

P{1} = 27

Therefore, the perimeter of the smaller rectangle would be 27 cm.

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Answer:

Amount = 800000×(1+0.09)2 = 800000×1.1881 = 946480

Amount = 800000×(1+0.1)2 = 800000×1.21 = 960000

Answer: (2,7) does not satisfy the system:

x - 2y ≤ -16

6x - y < 5

We can substitute the point (2,7) into the inequalities and see if they are true:

2 - 2*7 ≤ -16

-10 ≤ -16 which is false

6*2 - 7 < 5

5 < 5 which is also false

Since (2,7) does not satisfy both inequalities of the system, it is the correct answer.

Step-by-step explanation:

The fraction of the circle that would represent food is given as 34.09

The data we have here is given as

Activities = 600

Food = 450

Souvenirs = 270

We would have to sum this up

600 + 450 + 270 = 1320

The proportion that represents food is given as 450

= 450 / 1320

= 0.3409

Then we would have to convert this to percentage

0.341 * 100

= 34.09

34.09 x 360 = 123 degrees

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If there is a 1/6 chance that a student will be in Miss Jones's math class, if there are 30 students in her class, then 180 students will be there at all.

Probability is a method to know how likely something is going to happen. Whenever we're uncertain about the outcome of an event, we can talk about the probabilities of certain outcomes. The analysis of events overseen by probability is called statistics.

Total number of students present in the miss jones class = 30

The chance that a student will be in Miss Jones's math class = 1/6

Let x be the students are there at all

According to the question,

1/6 × x = 30

⇒x = 30 × 6

⇒x = 180

Hence, 180 students are there at all.

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Total amount of fluid ounces of water used by Sofia to make proportionate amount of slime is equal to 4.8 fluid ounces.

Let us consider 'x' amount of salt solution.

Amount of glue used by Sofia = 7.5 fluid ounces

Amount of glue = 12.5 times of salt solution

⇒ 7.5 = 12.5x

⇒ x = 7.5 / 12.5

⇒ x = 0.6

Fluid ounces of water used by Sofia to make the required batch of slime is:

Now, Amount of salt solution = Fluid ounces of Water - 4.2

⇒Fluid ounces of Water = x + 4.2

⇒ Fluid ounces of Water = 0.6 + 4.2

⇒ Fluid ounces of Water = 4.8 fluid ounces

Therefore, the fluid ounces of water used by Sofia with salt solution and glue to make a slime is equal to 4.8 fluid ounces.

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The value of the house before the increase is £1560

Now, According to the question:

Percentage basically means a part per hundred. It can be expressed in fraction form as well as decimal form. For example, if we say 25%, it means 25 out of 100. So, 25% is equivalent to the fraction 25/100 or 0.4 in decimal form.

"Information available from the question"

The value of a house increased by 6% the house then had a value of £26000.

Now, The expression will be :

= 6% of £26000.

= 6/100 × £26000.

= £1560

Hence,  the value of the house before the increase is £1560.

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The granite countertop costs $555

From the question, we have the following parameters that can be used in our computation:

Length = 3 meters

Width = 1 meter

Cost per square meter = $185.00

The cost of the granite counter is then calculated as

Cost = Length * width * Cost per square meter

substitute the known values in the above equation, so, we have the following representation

cost = 3 * 1 * 185

Evaluate

cost = 555

hence, the cost is $555

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The required equation is y=3x-24, where x is the number of elementary school students and y is the number of middle school students.

An equation is a formula in mathematics that expresses the equality of two expressions by connecting them with the equals sign =. In its most basic form, an equation is a mathematical statement that shows that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical equation that depicts the relationship between two expressions on opposite sides of the sign. It mostly consists of one variable and one equal to symbol. 2x - 4 = 2 is an example.

Here,

Let x be the number of students in elementary school and y be the number of students in middle school.

y=3x-24

The required equation is y=3x-24 where x be the number of students in elementary school and y be the number of students in middle school.

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Answer:

milk

Step-by-step explanation:

milk store is where your dad went

Answer:

4 2/9

Step-by-step explanation:

7 5/9 -3 1/3

68/9 -10/3

(68-30)/9

38/9

4 2/9

Kieran can ride at 235 yd per min if he continues at this rate.

Speed is a scalar quantity that refers to "how fast an object is moving."

Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a short amount of time.

Given that, Kieran is horseback riding. His horse can trot 470 yards in 2 minutes and 1,410 yards in 6 minutes when Kieran is riding.

Here, the question is asking about how fast Kieran riding if he continues at the given rate,

That mean we have to find the speed,

Speed = distance / time

Therefore,

Speed of Kieran = 470 / 2

= 235 yd per min

Speed of Kieran = 1410 / 6

= 235 yd per min

Therefore, we can say kieran us riding for 235 yd in a minute.

Hence, Kieran can ride at 235 yd per min if he continues at this rate.

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The image of the triangle DEF are formed by the following coordinates: D'(x, y) = (1, - 2), E'(x, y) = (4, - 2) and F'(x, y) = (- 0.5, - 5).

In this problem we need to apply a rigid transformation known as dilation, whose definition is introduced below:

P'(x, y) = O(x, y) + k · [P(x, y) - O(x, y)]

Where:

We need to determine the image of triangle DEF. If we know that D(x, y) = (2, - 2), E(x, y) = (4, - 2), F(x, y) = (1, - 4) and k = 1.5, then the images of the vertices of the triangle:

D'(x, y) = (4, - 2) + 1.5 · [(2, - 2) - (4, - 2)]

D'(x, y) = (4, - 2) + 1.5 · (- 2, 0)

D'(x, y) = (4, - 2) + (- 3, 0)

D'(x, y) = (1, - 2)

E'(x, y) = (4, - 2) + 1.5 · [(4, - 2) - (4, - 2)]

E'(x, y) = (4, - 2)

F'(x, y) = (4, - 2) + 1.5 · [(1, - 4) - (4, - 2)]

F'(x, y) = (4, - 2) + 1.5 · (- 3, - 2)

F'(x, y) = (4, - 2) + (- 4.5, - 3)

F'(x, y) = (- 0.5, - 5)

The coordinates of the image of triangle DEF are D'(x, y) = (1, - 2), E'(x, y) = (4, - 2) and F'(x, y) = (- 0.5, - 5)

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Answer:

13+15+17+19+21 = 85

Step-by-step explanation:

Hope you got it !

Answer:

The five consecutive odd integers whose sum will be 85 are 13, 15, 17, 19, and 21.

Step-by-step explanation:

To solve this problem, let's assume the first number is n.

Then the consecutive odd integers will be n+2, n+4, n+6, n+8

Thus we can write,

n+(n+2)+(n+4)+(n+6)+(n+8) = 85

In the next step add all the n's,

5n+20 = 85

Then transfer '20' from LHS to RHS,

5n = 85 - 20

5n = 65

Then in the next step '5' needs to be transferred to RHS,  

n = 65÷5

n = 13

Therefore, n = 13

n+2 = 15

n+4 = 17

n+6 = 19

n+8 = 21

Thus the five consecutive odd integers are 13, 15, 17, 19, and 21.

Answer: The percent of putts that he missed is 60%.

Step-by-step explanation: You divide 27 by 45, and that equals 0.6. After that you multiply 0.6 by 100, which gets it to 60/60%.

The equation y+2 = 4(x-1) is already in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In this equation, the slope is 4 and the y-intercept is -2.

For a line that is parallel to y+2 = 4(x-1), the slope (m) will be the same, which is 4.

To find the equation of a line that passes through a specific point (1,-3), we can use the point-slope form of a linear equation: y - y1 = m(x - x1)

So, using the point (1,-3) and the slope (4), we can write the equation as:

y - (-3) = 4(x - 1)

Simplifying this equation gives:

y + 3 = 4x - 4

Finally, we can write the equation in slope-intercept form by adding 4 to both sides and dividing both sides by 4:

y = 4x - 1

So the equation for the line that passes through (1,-3) and is parallel to y+2 = 4(x-1) is y = 4x - 1

The parent function and transformed function will be y = ln x and y = ln (x + 8) + 12, respectively.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

The function is given below.

y = ln x

The function is transformed by shifting 8 units left and 12 units up. Then replace x with (x + 8) and y with (y - 12). Then we have

y - 12 = ln (x + 8)

y = ln (x + 8) + 12

The parent function and transformed function will be y = ln x and y = ln (x + 8) + 12, respectively. The graph of the function is given below.

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Answer:

∞

Step-by-step explanation:

I don’t know just think it out. That is the explaination