graph the line passing through (−4,−1) whose slope is m=-4/5

Answer:

See below

Step-by-step explanation:

Squaring a number means you multiply it by itself, and the exponent is 2, whereas multiplying a number by 2 means you're doubling it.

Answer:There are several ways to calculate this.

The easiest way would be to determine the distance the bird flew.

40km/h x 4h = 160km

30km/h x 2,5h= 75km

Then add up the total distance: 160km + 75km = 235km

Next add up the total time: 4h + 2,5h = 6,5h

So in total the bird flew 235km in 6,5h. Now just divide both numbers by 6,5 as what you want to know is how far the bird got during an average hour.

Step-by-step explanation:

Answer:

195

Step-by-step explanation:

a = 3/2

According to the formula tn= a + (n-1)d

81/2= 3/2 + (13 - 1)d

81/2= 3/2 + 12d

81/3 = 12d

Therefore 27/12 = d

Sn= n/2 [2a + (n-1)d]

[tex]S_{13}[/tex] = 13/2 [2(3/2) + (13-1)(27/12)]

     = 13/2 (3 + 27)

     = 39/2 + 351/2

     = 390/2

     = 195

The probability of writing the first 7 digits of your phone number is 1/604,800.

Given data ,

To determine the probability of writing the first 7 digits of your phone number, we need to consider the total number of possible outcomes and the favorable outcomes.

Total number of possible outcomes:

When picking 7 digits without replacement from the set of 10 digits (0-9), the total number of possible outcomes is given by 10P7 (permutations of 10 objects taken 7 at a time):

10P7 = 10! / (10 - 7)! = 10! / 3! = 10 * 9 * 8 * 7 * 6 * 5 * 4 = 604,800.

Favorable outcomes:

To have written the first 7 digits of your phone number, you need to pick those specific digits in the correct order. Since the phone number has no repeated digits, there is only one specific order that satisfies this condition.

Therefore, the number of favorable outcomes is 1.

Probability:

The probability is given by the ratio of favorable outcomes to total possible outcomes:

Probability = Favorable outcomes / Total possible outcomes

= 1 / 604,800.

Hence, the probability is 1/604,800.

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Whwn the line of sight from the park ranger station, P, to the lifeguard chair, L, the length of PL is 0.76 miles.

It should be noted that since the line of sight from the park ranger station, P, to the lifeguard chair, L, on the beach of a lake is perpendicular to the path joining the campground, C, and the first aid station, F, then the path from the park ranger station to the lifeguard chair is the hypotenuse of a right triangle with legs of 0.35 miles and 0.65 miles.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Therefore, the length of PL is equal to:

= ✓(0.35² + 0.65²)

= 0.76 miles.

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Answer:

Because we are adding 2/5, we would be moving in the positive direction, which is to the right.

Step-by-step explanation:

therefore, the correct option should be A.

There are 14 Geometry problems in Vardan's homework assignment.

The number of geometry problems in Vardan's homework assignment, we need to calculate 58 1/3 percent of the total number of problems.

First, let's convert 58 1/3 percent to a decimal by dividing it by 100:

58 1/3 percent = 58.33/100 = 0.5833

Next, we multiply the decimal by the total number of problems:

Number of geometry problems = 0.5833 * 24

To calculate this, we can multiply 0.5833 by 24:

Number of geometry problems = 0.5833 * 24 = 14

Therefore, there are 14 geometry problems in Vardan's homework assignment.

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The given shape is a quadrilateral and a trapezoid.

A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles.

A trapezoid is a quadrilateral with at least one pair of parallel sides.

In a trapezoid, the parallel sides are called the bases of the trapezoid.

The other two sides are called the legs of the trapezoid.

The given shape is not a parallelogram because only one pair of sides are parallel and other sides are not.

Hence, the given shape is a quadrilateral and a trapezoid.

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Answer:

2.5

Step-by-step explanation:

The explanation is attached below.

The function whose graph is given include the following: D. y = cosx + 1.

In Mathematics and Geometry, the standard form of a cosine function can be represented or modeled by the following mathematical equation (formula):

y = acos(x - h) + k

Where:

By critically observing the graph of the given cosine function, we can reasonably infer and logically deduce that the parent cosine function y = cosx was vertically shifted (translated) upward by 1 unit, in order to produce the transformed cosine function as follows;

y = cosx

g(x) = y + 1

g(x) = cosx + 1

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a. Using plantwide allocation based on direct labor-hours at a rate of $3 per labor-hour:

Strawberry: $1,744 per 1,000 gallons

Vanilla: $1,534 per 1,000 gallons

Chocolate: $1,889 per 1,000 gallons

b. Allocation rates:

Department SV: $3.60 per machine-hour

Department C: $1.50 per labor-hour

c. Using department allocation rates:

Strawberry: $1,780 per 1,000 gallons

Vanilla: $1,573 per 1,000 gallons

Chocolate: $1,886 per 1,000 gallons

a. To compute the total cost of 1,000 gallons of each flavor using plantwide allocation, we need to allocate overhead based on direct labor-hours at a rate of $3 per labor-hour.

Strawberry:

Direct labor cost: $757 per 1,000 gallons

Labor-hours: 60 per 1,000 gallons

Overhead allocation: $3 per labor-hour * 60 labor-hours = $180

Raw materials cost: $807 per 1,000 gallons

Total cost of 1,000 gallons of strawberry flavor:

Direct labor cost + Overhead allocation + Raw materials cost = $757 + $180 + $807 = $1,744

Vanilla:

Direct labor cost: $832 per 1,000 gallons

Labor-hours: 65 per 1,000 gallons

Overhead allocation: $3 per labor-hour * 65 labor-hours = $195

Raw materials cost: $507 per 1,000 gallons

Total cost of 1,000 gallons of vanilla flavor:

Direct labor cost + Overhead allocation + Raw materials cost = $832 + $195 + $507 = $1,534

Chocolate:

Direct labor cost: $1,132 per 1,000 gallons

Labor-hours: 50 per 1,000 gallons

Overhead allocation: $3 per labor-hour * 50 labor-hours = $150

Raw materials cost: $607 per 1,000 gallons

Total cost of 1,000 gallons of chocolate flavor:

Direct labor cost + Overhead allocation + Raw materials cost = $1,132 + $150 + $607 = $1,889

b. To compute the allocation rate for each department, we will divide the overhead cost by the corresponding allocation base.

Allocation rate for Department SV:

Overhead: $90,972

Machine-hours: 25,270

Allocation rate = Overhead / Machine-hours = $90,972 / 25,270 = $3.60 per machine-hour

Allocation rate for Department C:

Overhead: $27,750

Labor-hours: 18,500

Allocation rate = Overhead / Labor-hours = $27,750 / 18,500 = $1.50 per labor-hour

c. To compute the cost of 1,000 gallons of each flavor using the department allocation rates, we will multiply the allocation rate by the corresponding allocation base for each flavor.

Strawberry:

Machine-hours: 60 per 1,000 gallons

Allocation rate for Department SV: $3.60 per machine-hour

Overhead allocation: $3.60 per machine-hour * 60 machine-hours = $216

Direct labor cost: $757 per 1,000 gallons

Raw materials cost: $807 per 1,000 gallons

Total cost of 1,000 gallons of strawberry flavor:

Direct labor cost + Overhead allocation + Raw materials cost = $757 + $216 + $807 = $1,780

Vanilla:

Machine-hours: 65 per 1,000 gallons

Allocation rate for Department SV: $3.60 per machine-hour

Overhead allocation: $3.60 per machine-hour * 65 machine-hours = $234

Direct labor cost: $832 per 1,000 gallons

Raw materials cost: $507 per 1,000 gallons

Total cost of 1,000 gallons of vanilla flavor:

Direct labor cost + Overhead allocation + Raw materials cost = $832 + $234 + $507 = $1,573

Chocolate:

Labor-hours: 157 per 1,000 gallons

Allocation rate for Department C: $1.50 per labor-hour

Overhead allocation: $1.50 per labor-hour * 157 labor-hours = $235

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Answer:

[tex]\ln(14k)=\ln(14)+\ln(k)[/tex]

Step-by-step explanation:

Recall the following property of logarithmic function:

[tex]\ln(ab)=\ln(a)+\ln(b)[/tex]

By using the property above, we expand the given logarithmic expression as follows:

[tex]\ln(14k)=\ln(14)+\ln(k)[/tex]

Answer:

To expand the logarithm as much as possible, we need to use the properties of logarithms that allow us to rewrite the expression as a sum, difference, or product of logs. One such property is the product rule, which states that ln(ab) = ln(a) + ln(b) for any positive numbers a and b. Using this rule, we can rewrite ln(14k) as ln(14) + ln(k). This is the most simplified form of the expression, as we cannot further expand ln(14) or ln(k) using the properties of logarithms. Therefore, the final answer is:

ln(14k) = ln(14) + ln(k)

Note that we enclosed the arguments of the logarithm functions in parentheses and included a multiplication sign between 14 and k, as instructed.

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1. The surface area of the rectangular prism is 872 square inches.

2. The surface area of the cylinder is 378o.56 square millimeters.

The rectangular prism has three pairs of equal faces: the top and bottom faces, the front and back faces, and the left and right faces.

The formula for the surface area of a rectangular prism is:

Surface Area = 2(length × width) + 2(length × height) + 2(width × height)

Surface Area = 2(8 in × 12 in) + 2(8 in × 17 in) + 2(12 in × 17 in)

Surface Area = 192 in² + 272 in² + 408 in²

Surface Area = 872 in².

The surface area of a cylinder consists of two circular bases and the curved surface area.

The formula for the surface area of a cylinder is: 2π(radius × height) + 2π(radius²).

Surface Area = 2π(14 mm × 29 mm) + 2π(14 mm)²

Surface Area = 2π(406 mm²) + 2π(196 mm²)

Surface Area = 812π mm² + 392π mm²

Surface Area = 1204π mm²

As π = 3.14:

Surface Area = 1204 × 3.14 mm²

Surface Area = 3780.56 mm²

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The measure of angle 1 and angle 4 are equal, that is 62°, by vertical angle theorem.

Given that, m∠1=62° and lines t and l intersect.

From the given figure,

m∠1 and m∠4 are vertically opposite angles.

Vertical angles theorem or vertically opposite angles theorem states that two opposite vertical angles formed when two lines intersect each other are always equal (congruent) to each other.

Here, m∠1 =m∠4=62°

Hence, it is proved.

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Answer:-2

Step-by-step explanation:

x(4)+16=8

Answer:

B) [tex]A=0.5(f\cdot g)(x)[/tex]

Step-by-step explanation:

[tex]\displaystyle A=\frac{1}{2}bh\\\\A=\frac{1}{2}(2x)(x^2+3)\\\\A=\frac{1}{2}g(x)f(x)\\\\A=\frac{1}{2}(f\cdot g)(x)[/tex]

Answer:

Option 2 is the correct answer.

Step-by-step explanation:

The equation that can be used to represent the area of the triangle is:

A(x) = 0.5 * (f ⋅ g)(x)

Let's break it down step by step:

Step 1: Understanding the equation components

f(x) represents the height of the triangle, given as x² + 3.

g(x) represents the base of the triangle, given as 2x.

(f ⋅ g)(x) represents the product of f(x) and g(x), or the multiplication of their respective values at a specific x.

Step 2: Evaluating the equation

To find the area of the triangle when x = 2, we substitute x = 2 into the equation.

A(2) = 0.5 * (f ⋅ g)(2)

Now, let's substitute the functions f(x) and g(x) with their corresponding values at x = 2.

f(2) = 2² + 3 = 4 + 3 = 7

g(2) = 2(2) = 4

Substituting these values into the equation:

A(2) = 0.5 * (7 ⋅ 4)

= 0.5 * 28

= 14

Therefore, when x = 2, the area of the triangle is 14.

Step 3: Interpretation and Conclusion

The equation A(x) = 0.5 * (f ⋅ g)(x) correctly represents the area of the triangle. It takes into account the height function f(x) = x² + 3 and the base function g(x) = 2x, multiplied together and multiplied by 0.5 (or divided by 2), which is a common factor in the formula for the area of a triangle.

Hence, the equation A(x) = 0.5 * (f ⋅ g)(x) can be used to represent the area of the triangle for any given value of x.

Answer:

To find the value of the condominium in 2020, we can work backward from the given information. Since we know that the value appreciates by 15% each year, we can calculate the value for the previous year using the formula:

Value in previous year = Value in current year / (1 + appreciation rate)

Let's calculate the value of the condominium in 2020:

Value in 2021 = $899,300 / (1 + 0.15) ≈ $781,521.74

Value in 2020 = Value in 2021 / (1 + 0.15) ≈ $781,521.74 / (1 + 0.15) ≈ $678,309.34

Therefore, the value of the condominium in 2020 is approximately $678,309.34.

Step-by-step explanation:

Consequently, there are 5 elements in A B'.

Given the following definitions:

U = {1, 2, 3, 4, 5, 6, 7}A = {1, 2, 4, 5}B = {1, 3, 5, 7}

The complement of a set B is the set of all elements that belong to the universal set U but not to B.

A’ = {x | x ∈ U and x ∉ A} = {3, 6, 7}B’ = {x | x ∈ U and x ∉ B} = {2, 4, 6}

The union of sets A and B is the set of all elements that belong to set A or set B, or both.

A ∪ B = {x | x ∈ A or x ∈ B}

= {1, 2, 3, 4, 5, 7}A ∪ B'

= {x | x ∈ A or x ∈ B’}

= {1, 2, 4, 5, 6}

Therefore, the number of elements in A ∪ B' is 5.

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The cost of four points is:4 x $1,370 = $5,480Thus, the four points will cost the homebuyers $5,480.

Points can help lower mortgage rates on fixed-rate loans. The concept of points, which are basically prepaid interest, is a little complicated.

Each point is worth one percent of the loan amount, and paying points can lower your interest rate by a certain amount, typically about one-eighth to one-quarter of a percentage point.

The cost of points in the given scenario can be found using the following steps:

The loan amount to purchase a condominium is $137,000. The homebuyers obtained a 15-year fixed-rate loan with a rate of 5.05%.

If the homebuyers opt for four points, their loan rate will decrease to 4.81%.

To figure out how much the points will cost the homebuyers, we must first determine the cost of one point. Since one point is equal to 1% of the loan amount, one point on a $137,000 loan is:1% of $137,000 = $1,370

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The probability of selecting a number that contains 7 and 8 is 1/8

Here, we have,

The tree diagram shows the sample space of​ two-digit numbers that can be created using the digits 9,7,1, and 8.

so, we get,

There are 16 samples given.

now, we have to find that the probability of choosing a number from the sample space that contains both 7 and 8.

here, the number of this event = 2

so, we get,

The probability of selecting a number that contains 7 and 8 is:

2/16 = 1/8

Hence, The probability of selecting a number that contains 7 and 8 is 1/8

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The equation for the trigonometric graph is:

y = 4 cos(x/2)

We have given is a graph with maximum point at 4 and minimum point at -4 with x intercepts are = -π, π, 3π, 5π, 7π, 9π, 11π.....

We need to identify the equation of the trigonometric graph.

To identify the equation of the given graph with maximum point at 4 and minimum point at -4, we can start by analyzing the characteristics of the cosine function.

The cosine function oscillates between 1 and -1 and has a period of 2π. The general form of the cosine function is:

y = A cos(Bx + C) + D

where A represents the amplitude, B represents the frequency (1/period), C represents the phase shift, and D represents the vertical shift.

Given that the maximum point is at 4 and the minimum point is at -4, we can determine the amplitude, A, to be 4.

y = 4 cos(Bx + C) + D

Next, let's consider the x-intercepts given as -π, π, 3π, 5π, 7π, 9π, 11π, and so on. The x-intercepts of the cosine function occur when the angle inside the cosine function, Bx + C, is equal to (2n + 1)π/2, where n is an integer.

We can start by analyzing the first x-intercept, which is -π. Setting Bx + C equal to -π, we have:

B(-π) + C = -π

Simplifying, we get:

B = -1 - C/π

Similarly, for the x-intercepts at π, 3π, 5π, 7π, 9π, 11π, and so on, we can write the following equations:

Bπ + C = π

B(3π) + C = π

B(5π) + C = π

B(7π) + C = π

B(9π) + C = π

B(11π) + C = π

Now, notice that these equations are satisfied when C is an odd multiple of π, and B = 1/2.

Therefore, we can conclude that:

C = (2n + 1)π

and

B = 1/2

Substituting these values into the equation, we have:

y = 4 cos((1/2)x + (2n + 1)π) + D

Finally, considering the vertical shift, we know that the maximum point is at 4 and the minimum point is at -4. Since the amplitude is 4, the vertical shift is the average of the maximum and minimum points, which is 0. Therefore, D = 0.

The final equation for the trigonometric graph is:

y = 4 cos(x/2)

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Answer(s):

[tex]\displaystyle y = 4sin\:(\frac{1}{2}x + \frac{\pi}{2}) \\ y = -4cos\:(\frac{1}{2}x \pm \pi) \\ y = 4cos\:\frac{1}{2}x[/tex]

Step-by-step explanation:

[tex]\displaystyle y = Asin(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-\pi} \hookrightarrow \frac{-\frac{\pi}{2}}{\frac{1}{2}} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4\pi} \hookrightarrow \frac{2}{\frac{1}{2}}\pi \\ Amplitude \hookrightarrow 4[/tex]

OR

[tex]\displaystyle y = Acos(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4\pi} \hookrightarrow \frac{2}{\frac{1}{2}}\pi \\ Amplitude \hookrightarrow 4[/tex]

You will need the above information to help you interpret the graph. First off, keep in mind that although this looks EXACTLY like the cosine graph, if you plan on writing your equation as a function of sine, then there WILL be a horisontal shift, meaning that a C-term will be involved. As you can see, the photograph on the right displays the trigonometric graph of [tex]\displaystyle y = 4sin\:\frac{1}{2}x,[/tex] in which you need to replase “cosine” with “sine”, then figure out the appropriate C-term that will make the graph horisontally shift and map onto the cosine graph [photograph on the left], accourding to the horisontal shift formula above. Also keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY REALLY ARE, so you must be careful with your calculations. So, between the two photographs, we can tell that the sine graph [photograph on the right] is shifted [tex]\displaystyle \pi\:units[/tex] to the right, which means that in order to match the cosine graph [photograph on the left], we need to shift the graph BACKWARD [tex]\displaystyle \pi\:units,[/tex] which means the C-term will be negative; so, by perfourming your calculations, you will arrive at [tex]\displaystyle \boxed{-\pi} = \frac{-\frac{\pi}{2}}{\frac{1}{2}}.[/tex] So, the sine equation of the cosine graph, accourding to the horisontal shift, is [tex]\displaystyle y = 4sin\:(\frac{1}{2}x + \frac{\pi}{2}).[/tex] Now, with all that being said, in this case, sinse you ONLY have a graph to wourk with, you MUST figure the period out by using wavelengths. So, looking at where the graph hits [tex]\displaystyle [-2\pi, -4],[/tex] from there to [tex]\displaystyle [-6\pi, -4],[/tex] they are obviously [tex]\displaystyle 4\pi\:units[/tex]apart, telling you that the period of the graph is [tex]\displaystyle 4\pi.[/tex] Now, the amplitude is obvious to figure out because it is the A-term, but of cource, if you want to be certain it is the amplitude, look at the graph to see how low and high each crest extends beyond the midline. The midline is the centre of your graph, also known as the vertical shift, which in this case the centre is at [tex]\displaystyle y = 0,[/tex] in which each crest is extended four units beyond the midline, hence, your amplitude. So, no matter how far the graph shifts vertically, the midline will ALWAYS follow.

I am delighted to assist you at any time.

Answer:

See below

Step-by-step explanation:

Since the spinner has the numbers 1, 2, and 3 on it, and we want to find the probability of spinning a number less than 2, there is only one possible outcome that satisfies this condition, which is spinning a 1. Therefore, the probability of spinning a number less than 2 is:

P(less than 2) = P(1) = 1/3

So the probability of spinning a number less than 2 is 1/3.

Answer:

Step-by-step explanation:

To find out how many people would have an IQ between 106 and 125, we need to calculate the area under the normal distribution curve between these two IQ values. We can do this by calculating the z-scores corresponding to these IQ values and then using a standard normal distribution table or a calculator.

First, let's calculate the z-score for an IQ of 106 using the formula:

z = (x - μ) / σ

where x is the IQ score (106), μ is the mean (100), and σ is the standard deviation (15).

z = (106 - 100) / 15 = 0.4

Next, let's calculate the z-score for an IQ of 125:

z = (x - μ) / σz = (125 - 100) / 15 = 1.67

Using a standard normal distribution table or a calculator, we can find the corresponding cumulative probabilities for these z-scores.

The cumulative probability for a z-score of 0.4 is approximately 0.6554.The cumulative probability for a z-score of 1.67 is approximately 0.9525.

To find the proportion of people with an IQ between 106 and 125, we subtract the cumulative probability corresponding to the lower z-score from the cumulative probability corresponding to the higher z-score:

0.9525 - 0.6554 = 0.2971

This means that approximately 29.71% of the population falls within the IQ range of 106 to 125.

To find out how many people out of the randomly selected 1450 would have an IQ in this range, we multiply the proportion by the sample size:

0.2971 * 1450 ≈ 431.15

Rounding to the nearest whole number, we find that approximately 431 people would have an IQ between 106 and 125 out of the randomly selected sample of 1450 individuals.

Answer:

To solve this question, we'll need to use the properties of normal distribution. Specifically, we'll use the z-scores, which tell us how many standard deviations away from the mean a given value is. The formula for calculating a z-score is:

Z = (X - μ) / σ

where:

- X is the value we're interested in,

- μ is the mean, and

- σ is the standard deviation.

The z-score corresponding to an IQ of 106 is (106 - 100) / 15 ≈ 0.40, and the z-score corresponding to an IQ of 125 is (125 - 100) / 15 ≈ 1.67.

Next, we need to determine the proportion of individuals in a normal distribution that falls between these z-scores. This is found by looking up these z-scores in a standard normal distribution table, or using a software function that provides the cumulative distribution function of the standard normal distribution.

The approximate values are:

- The cumulative probability associated with a z-score of 0.40 is about 0.6554.

- The cumulative probability associated with a z-score of 1.67 is about 0.9525.

The proportion of individuals with IQs between 106 and 125 is the difference between these probabilities, or about 0.9525 - 0.6554 = 0.2971.

Now, to find the number of people out of 1450 with IQs between 106 and 125, we simply multiply this proportion by 1450 and round to the nearest whole number:

0.2971 * 1450 ≈ 431

So, we expect about 431 people out of the randomly selected 1450 to have an IQ between 106 and 125.

Answer:

theta = n pi and theta = (2n + 1)^pi/2, Where n is an integer.

Step-by-step explanation:

cos(2theta) + (sin(2theta) = 0

2 cos(theta) sin(theta) + 2 sin(theta) cos(theta) = 0

2 sin(theta) cos(theta)(1 + 1) = 0

4 sin(theta) cos(theta) = 0

For sin(theta) = 0.theta = n pi,  Where n is an integer.

For cos(theta) = 0.theta  = (2n + 1)^pi / 2, Where n is an integer

The solution for the original equation is:

theta = n pi, and  theta = (2n + 1)^pi / 2, Where n is an integer.

Hope this helps!

Answer: 2

Step-by-step explanation: 10 divided by 5 equals 2

Answer:

The lengths are equal so the triangle is equilateral

Step-by-step explanation:

We can write the points as follows,

(2,0), (-1,[tex]\sqrt{3}[/tex]) (-1,-[tex]\sqrt{3}[/tex])

now if it is an equilateral triangle, all side lengths must be equal

first we compute the sides(vectors)

(2-(-1),-[tex]\sqrt{3}[/tex]) = (3,-[tex]\sqrt{3}[/tex]) = side 1

(2-(-1),[tex]\sqrt{3}[/tex]) = (3,[tex]\sqrt{3}[/tex]) = side 2

(-1+1,[tex]\sqrt{3}[/tex]+[tex]\sqrt{3}[/tex]) = (0,2[tex]\sqrt{3}[/tex]) = side 3

now we compute the lengths of the sides using pythagoras theorem

(3)^2 + (-[tex]\sqrt{3}[/tex])^2 = (length of side 1)^2 = 9 + 3 = 12

similarly, (3)^2 + ([tex]\sqrt{3}[/tex])^2 = 12 = Length of side 2 squared

and,( 2[tex]\sqrt{3}[/tex])^2 = length of side 3 squared = 12

since the squares are equal, so the lengths must also be equal

so the triangle is equilateral

this is just a quick addition to the superb posting by "hamza0100" above

well, indeed, in the argand or imaginary plane, for those values above we have the coordinates of A(2 , 0) , B(-1 √3) and C(-1 , -√3), let' use the distance formula for those fellows

[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ A(\stackrel{x_1}{2}~,~\stackrel{y_1}{0})\qquad B(\stackrel{x_2}{-1}~,~\stackrel{y_2}{\sqrt{3}}) ~\hfill AB=\sqrt{(~~ -1- 2~~)^2 + (~~ \sqrt{3}- 0~~)^2} \\\\\\ ~\hfill AB=\sqrt{( -3)^2 + ( \sqrt{3})^2} \implies \boxed{AB=\sqrt{ 12 }}[/tex]

[tex]B(\stackrel{x_1}{-1}~,~\stackrel{y_1}{\sqrt{3}})\qquad C(\stackrel{x_2}{-1}~,~\stackrel{y_2}{-\sqrt{3}}) \\\\\\ BC=\sqrt{(~~ -1- (-1)~~)^2 + (~~ -\sqrt{3}- \sqrt{3}~~)^2} \\\\\\ ~\hfill BC=\sqrt{( 0)^2 + ( -2\sqrt{3})^2} \implies \boxed{BC=\sqrt{ 12 }}[/tex]

[tex]C(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-\sqrt{3}})\qquad A(\stackrel{x_2}{2}~,~\stackrel{y_2}{0}) ~\hfill CA=\sqrt{(~~ 2- (-1)~~)^2 + (~~ 0- (-\sqrt{3})~~)^2} \\\\\\ ~\hfill CA=\sqrt{( 3)^2 + (-\sqrt{3})^2} \implies \boxed{CA=\sqrt{ 12 }} \\\\[-0.35em] ~\dotfill\\\\ AB=BC=CA=\sqrt{12}\implies 2\sqrt{3}\hspace{5em}\qquad equilateral\textit{\LARGE \checkmark}[/tex]

The total surface area of the cylindrical brass pipe, including the internal surface area, is approximately 34183.08 cm².

The formula for the surface area of a cylinder includes the curved surface area (CSA) and the two circular base areas.

1. Outer Surface Area:

The curved surface area (CSA) of a cylinder is given by the formula: CSA = 2πrh, where r is the radius and h is the height.

Given:

Outer diameter = 10 cm

Outer radius (R) = 10 cm / 2 = 5 cm

Height (h) = 600 cm

Outer CSA = 2π(5)(600) = 6000π cm²

The circular base areas can be calculated using the formula: Base area = πr², where r is the radius.

Outer base area = π(5)² = 25π cm²

Therefore, the total outer surface area is the sum of the curved surface area and the two circular base areas:

Total outer surface area = Outer CSA + 2 * Outer base area = 6000π + 2 * 25π = 6050π cm²

2. Inner Surface Area:

The inner radius (r) is given as 4 cm. We can use the same formulas to calculate the inner surface area.

Inner CSA = 2π(4)(600) = 4800π cm²

Inner base area = π(4)² = 16π cm²

Total inner surface area = Inner CSA + 2 * Inner base area = 4800π + 2 * 16π = 4832π cm²

Total surface area = Total outer surface area + Total inner surface area = 6050π + 4832π = 10882π cm²

To find the numerical value, we can use the approximation π ≈ 3.14:

Total surface area ≈ 10882 * 3.14 = 34183.08 cm²

Therefore, the total surface area of the cylindrical brass pipe, including the internal surface area, is approximately 34183.08 cm².

Learn more about surface area at https://brainly.com/question/16519513

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