Find the values of x in this equation: x – 15 / x = 2.


A) -7, 3

B) -5, 2

C) -7, 5

D) -2, 5

E) -3, 5

Answer:

  26 minutes on the bicycle

Step-by-step explanation:

Let t and b represent the number of minutes on the treadmill and bicycle, respectively. The problem statement tells us two ways to relate these values.

  t + b = 60 . . . . . Amy exercises a total of 60 minutes (1 hour)

  9t +6.5b = 475 . . . . . she burns a total of 475 calories

Since we want to find the value of "b", we can use the first equation to write an expression that will substitute for t.

  t = 60 -b

  9(60 -b) +6.5b = 475

  540 -2.5b = 475 . . . . . . simpify

  -2.5b = -65 . . . . . . . . . . subtract 540

  b = 26 . . . . . . . . . . . . . . . divide by -2.5 (equivalently, multiply by -2/5)

Amy spends 26 minutes exercising on the bicycle.

Setting up a system of linear equations can help us find the time spent on each exercise. Form two equations based on the given information: total time spent exercising and total calories burned. Solve these equations to find the time spent on each activity.

To solve these kinds of problems, it is helpful to use a system of linear equations. We need to find two expressions for the total calories burned, based on the different exercises Amy does, and then solve for the time she spends on each exercise.

Step 1: Identify the variables. Let's denote the time Amy spends jogging as J (in minutes) and the time she spends bicycling as B (also in minutes).

Step 2: Write two equations based on the information given. We know:

Step 3: Solve for B in the first equation (B = 60 - J) and substitute this into the second equation. Simplifying will give J, the number of minutes Amy spends jogging. Substituting J back into the first equation will give B, the number of minutes spent bicycling.

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Answer:

  C) All integers where n ≥ 1

Step-by-step explanation:

N is the term number, so is a "counting number," or "natural number." It is a positive integer.

Answer:

64 oz

Step-by-step explanation:

y = k/x

We want to find k first.

4 = k/8

4(8) = (k/8)(8)

32 = k

We now need to find y knowing that k = 32 and x = 2.

y = k/x

y = 32/2

y = 16

Did you follow?

Answer:

16

Step-by-step explanation:

Inverse variation follows this equation:

y = k/x

We know one point, so we input x and y, and solve for k.

4 = k/8

k = 32

The inverse variation equation is

y = 32/x

Now we let x = 2, and use our equation to find y.

y = 32/2

y = 16

Answer:

Option 1:(x-4)^2+y^2=100

Step-by-step explanation:

Given center = (h,k) = (4,0)

The point (-2,8) lies on circle which means the distance between the point and center will be equal to the radius.

So,

The distance formula will be used:

[tex]d = \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}} \\=\sqrt{(4+2)^{2}+(0-8)^{2}}\\=\sqrt{(6)^{2}+(-8)^{2}}\\=\sqrt{36+64}\\ =\sqrt{100}\\ =10\ units[/tex]

Hence radius is 10.

The standard form of equation of circle is:

(x-h)^2+(y-k)^2 = r^2

Putting the values

(x-4)^2+(y-0)^2=10^2

(x-4)^2+y^2=100

Hence option 1 is correct ..

Answer:

A

Step-by-step explanation:

Answer:

∠A ≅ ∠A; reflexive property

Step-by-step explanation:

Statement and reason are missing in the proof is "∠A ≅ ∠A reflexive property". Option 3 is the right choice.

The image shows a proof of the congruence of two isosceles triangles, ΔABC and ΔAYX. The proof uses the reflexive property of congruence to show that ∠A ≅ ∠A, and then the SAS similarity theorem to show that ΔABC ≅ ΔAYX.

The reflexive property of congruence is a fundamental property of congruence, and it is used in many proofs in geometry. It is important to understand the reflexive property of congruence and how it can be used in proofs.

Option 3 is the right choice.

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Answer:

  D.)  (14, 0); the time it takes for the bird to reach the ground

Step-by-step explanation:

The attached graph shows a plot of the table values and the two offered solution options.

The dependent variable in this scenario is the bird's height above the ground. When that is zero, the bird is on the ground. This fact makes choices B and C seem ridiculous.

We note from the table and graph that the bird is on a path that decreases in height by 3 feet every 2 seconds. If the bird continues that rate of descent, it will reach the ground after 14 seconds.

That is, its (time, height) pair will be (14, 0), matching choice D.

_____

Choosing any answer to this question requires making assumptions that are inconsistent with real-world bird behavior. At least, the problem statement should say what assumptions are applicable.

Answer: D.) (14, 0); the time it takes for the bird to reach the ground

Step-by-step explanation:

Answer:

A) open up if [tex]a[/tex] is positive .

B) open down if [tex]a[/tex] is negative.

Step-by-step explanation:

If you are given a quadratic in vextex form, [tex]y=a(x-h)^2+k[/tex], then the parabola is:

A) open up if [tex]a[/tex] is positive .

B) open down if [tex]a[/tex] is negative.

Answer:

See below.

Step-by-step explanation:

The coefficient of x^2 will tell you this. If it is positive the graph opens upwards , negative it will open downwards.

For example

-(x - 4)^2 + 1  will open downwards and (x - 4)^2 + 1 will open upwards.

Answer:

8.4 gallons

Step-by-step explanation:

We have a boater traveled 532 miles and we want to know how much gas he used for this trip.

We are also given if he travels 63 miles then has used 1 gallon.

532 miles ->x gallons

63 miles   ->1  gallon

The information here is lined up for you already.

We have from the line, this proportion:

[tex]\frac{532}{63}=\frac{x}{1}[/tex]

x/1=x so we have:

[tex]\frac{532}{63}=x[/tex]

[tex]8.\overline{4}=x[/tex]

So approximately 8.4 gallons was used on this trip of 532 miles.

Answer:

8.4 gallons

Step-by-step explanation:

A boater travels 532 miles. Assuming the boat averages 63 miles per gallon, there are 8.4 gallons of gasoline that were used.

Answer:

Descartes' rule states that the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients of the terms or less than the sign changes by a multiple of 2.

The Fundamental Theorem of Algebra states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution, furthermore any polynomial of degree n has n roots.  

Remember that the complex numbers include the real numbers.

Suppose we are given the polynomial x^3+3x^2-x-x^4-2, we arrange the terms of the polynomial in the descending order of exponents:

-x^4+x^3+3x^2-x-2, count the number of sign changes, there are 2 sign changes in the polynomial, so the possible number of positive roots of the polynomial is 2 or 0.

returning to our polynomial above, -x^4+x^3+3x^2-x-2, it has degree 4 and so has n roots. Note that complex roots always come in pairs, so here is what can be said from these two rules:  

degree 1 has 1 real root

degree 2 has 2 real roots or 2 complex roots

degree 3 has 3 real roots or 1 real root and 2 complex roots

degree 4 has 4 real roots or 2 real roots and 2 complex roots  

note that if the degree is odd, there will be at least 1 real root

Step-by-step explanation:

The number of complex roots and the possible positive and negative real roots of a polynomial can be predicted using the Fundamental Theorem of Algebra and Descartes' Rule of Signs. The Fundamental Theorem of Algebra ensures at least one complex root for every polynomial, while Descartes' Rule predicts possible real roots based on the number of sign changes in the polynomial equation.

To predict the number of complex roots and find the number of possible positive and negative real roots to a polynomial, you can use both Descartes' Rule and the Fundamental Theorem of Algebra. These two concepts in mathematics can give us interesting insights into the roots of polynomial equations.

First, the Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root. This theorem can ensure us that we have a starting point, knowing that every polynomial equation will have at least one solution, even if it's complex.

Second, you can use Descartes' Rule of Signs to predict possible positive and negative roots. This rule uses the number of sign changes in the polynomial to give possible number of positive real roots. You can get the possible number of negative real roots by replacing x with -x and count the sign changes again.

For example, for the equation f(x) = x5 – 4x4 + 2x3 + 8x2 -12x + 6, there are two sign changes in the original equation, so there can be two or zero positive real roots. If we replace x with -x, we get three sign changes, suggesting three or one negative real roots.

An important aspect to remember is that Descartes' Rule of Signs gives us possible quantities of roots, but not the exact amount or their values, and it can't predict complex roots. However, by utilizing the Fundamental Theorem of Algebra in conjunction with Descartes' Rule, we can get a fuller picture of the roots of a polynomial equation.

Answer:

5 or B

Step-by-step explanation:

Answer:

(d):  0.7734

Step-by-step explanation:

Using the normcdf( statistical function built into my old TI 83-Plus calculator, I find that the area under the standard normal curve from the extreme left to the value 69 is 0.2266.  Thus, about 0.2266 of students would fail and

1.000-0.2266, or 0.7734 would pass.  This corresponds to Answer (d).

Answer:

40,000 workers

Step-by-step explanation:

This question deals with simple rates proportion.

If it takes 20 years for X workers to complete a job,

if we double the amount of workers, we will take half the time.

hence to halve the time from 20 years to 10 years, we will need to double the amount of workers.

In this case we will need 20,000 workers x 2 = 40,000 workers

Answer: 40,000

Step-by-step explanation:

(20÷10)×20,000

715 calculators will be defective, from a sample of 13200.

Calculation using the unitary method:-

In a sample of 240 = 13 calculators are found defective.

so, in a sample of 1 =13/240 calculator is defective.

 ∴in a sample of 13200 = 13/240*13200 calculators are defective.

                                        715 calculators are defective (answer)

Problem-solving is the act of defining a problem; figuring out the purpose of the trouble; identifying, prioritizing, and selecting alternatives for an answer; and imposing an answer.

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Answer:

Step-by-step explanation:

When Riko left his house, Yuto was 5.25 miles along the path.

Average speed of Riko = 0.35 miles per minute

Average speed of Yuto = 0.25 miles per minute

First we will calculate the time in which Riko will catch Yuto on the track.

Relative velocity of Riko as compared to Yuto will be = velocity of Riko - velocity of Yuto

= 0.35 - 0.25

= 0.10 miles per minute

Now we this relative velocity tells that Riko is moving and Yuto is in static position.

By the formula,

Average speed = [tex]\frac{Distance}{Time}[/tex]

0.10 = [tex]\frac{5.25}{t}[/tex]

t = [tex]\frac{5.25}{0.1}[/tex]

t = 52.5 minutes

Now we know that Rico will catch Yuto in 52.5 minutes. Before this time he will be behind Yuto.

So duration of time in which Rico is behind Yuto will be 0 ≤ t ≤ 52.5

Here time can not be less than zero because time can not be with negative notation. It will always start from 0.

Answer:

Sample Response:

The solution means that Riko will be behind Yuto from the time she leaves the house, which corresponds to t = 0, until the time she catches up to Yuko after 52.5 minutes, which corresponds to t = 52.5. The reason that t cannot be less than zero is because it represents time, and time cannot be negative.

The 3 known points of the parabola are the origin (0,0) and the bottom left and right points ( -5,-50) (5,-50)

The general form of a parabola that goes through the origin is:  

y = ax^2  

We need to solve for a:

-50 = a(5)^2  

-50 = 25a  

a = -50/25

a = -2

Now the equation of the parabola is y = (-2)x^2.

To solve for the vertical clearance (y) 2 meters from the edge

Subtract the distance from the edge 2 meters from the original edge which is 5 meters, 5-2 = 3.

Now replace x with 3 and solve for y:

y = (-2)3^2

y = -2(9)

y = -18

The clearance would then 50 - 18 = 32 meters.

Explanation:

Tommy calculated the area of the base by squaring the diameter, not the radius. If he's going to use the diameter in the area formula, he needs to divide the result by 4.

  A = πr² = πd²/4

Answer:

Sample response: Tommy used the diameter instead of the radius in the formula to find the area of the base. The diameter needs to be divided by 2 to find the radius. The correct area of the base is 1 pi square inches. The correct total volume is 3 pi cubic inches.Step-by-step explanation:

Answer: 6 hours

Step-by-step explanation:

Given : Time taken by Type JQ pump to fill the tank = 72 hours

Then , Time taken by Type JQ pump to fill half of the tank = 36 hours

Time taken by Type JT pump to fill the tank = 18 hours

Then, Time taken by Type JT pump to fill the tank = 9 hours

Now, if he tank starts at half full then the time taken by two Type JQ pumps and a Type JT pump all three pumps working together to fill the tank :-

[tex]\dfrac{1}{t}=\dfrac{1}{36}+\dfrac{1}{36}+\dfrac{1}{9}\\\\\Rightarrow\dfrac{1}{t}=\dfrac{1+1+4}{36}=\dfrac{1}{6}\\\\\Rightarrow t=6[/tex]

Hence, it will take 6 hours to fill the tank.

Answer:

  TRUE

Step-by-step explanation:

They are called independent and consistent.

_____

If they describe the same line, they are dependent. If they describe parallel lines, they are inconsistent.

Answer:

Width 13, Length = 22

Step-by-step explanation:

Let L = length and W = width

Given, length is 9 more than width, ie.

L = W + 9  ----> eq 1

Also given that area = 286 square inches.

Area of rectangle = length x width , or

286 = L x W   -----> eq 2

2 equations, 2 unknowns => we can solve this system of equations by substitution.

Substitute eq 1 into eq 2,

286 = (W + 9) x W

286 = W² + 9W

W² + 9W - 286 = 0 (solve quadratic equation using your favorite method)

w = 13 or w = -22 (impossible since width cannot be negative)

hence W = 13 (substitute this back into equation 1)

L = 13 + 9 = 22

The width of the rectangle is 13 inches, and the length is 22 inches, obtained by solving the quadratic equation derived from the provided area and the relationship between the length and the width.

To find the length and width of a rectangle when given the area and the relationship between the length and the width, we can set up an algebraic equation. Let w represent the width and l represent the length of the rectangle. According to the problem, l = w + 9. The area of the rectangle, which is width times length, is given as 286 square inches, so our equation is w x (w + 9) = 286.

Solving this quadratic equation step by step:

Therefore, the width of the rectangle is 13 inches and the length is 22 inches.

Answer:

$0.60

Step-by-step explanation:

The costs are:

$0.15 + $0.25 = $0.40

He sells it for $1.25.

The value added is:

$1.25 - $0.40 = $0.85

The farmer adds a value of $0.60 to each pineapple by growing them and selling them to the convenience store.

The farmer obtains a revenue of $1.25 per pineapple, while the total production cost per pineapple is the sum of the seed, fertilizer, and forgone output costs, which amounts to $0.15 + $0.25 + $0.25 = $0.65. Therefore, the value added by the farmer to each pineapple is the revenue obtained from selling each pineapple minus the cost of producing each pineapple, which equals $1.25 - $0.65 = $0.60.

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Collecting data from a sample that does not represent the population is known as sampling bias. It leads to inaccurate results as it may not correctly represent the diversity or behaviors of the entire population. Researchers use random sampling methods to reduce bias, but may also use non-probability sampling acknowledging its limitations.

Collecting data from a sample that does not represent the population is called sampling bias. Sampling bias occurs when the elements selected for inclusion in a study do not accurately reflect the larger population from which they were drawn. This can lead to inaccurate results and conclusions because the sample may not capture the diversity or behaviors of the entire population. For instance, an online poll conducted by a newspaper may only reach individuals with internet access, thereby excluding a significant portion of the population. To mitigate sampling bias, researchers aim to utilize random sampling methods, such as simple random sampling, stratified sampling, cluster sampling, and systematic sampling, which give each individual in the population an equal chance of being selected.

However, due to constraints such as cost and feasibility, researchers may sometimes opt for non-probability sampling, like convenience sampling, acknowledging its limitations in representativeness. This highlights the importance of carefully choosing a sampling method that is as representative of the population as possible to ensure more accurate and generalizable results.

Answer:

Proofs contained within the explanation.

Step-by-step explanation:

These induction proofs will consist of a base case, assumption of the equation holding for a certain unknown natural number, and then proving it is true for the next natural number.

a)

Proof

Base case:

We want to shown the given equation is true for n=1:

The first term on left is 2 so when n=1 the sum of the left is 2.

Now what do we get for the right when n=1:

[tex]\frac{1}{2}(1)(3(1)+1)[/tex]

[tex]\frac{1}{2}(3+1)[/tex]

[tex]\frac{1}{2}(4)[/tex]

[tex]2[/tex]

So the equation holds for n=1 since this leads to the true equation 2=2:

We are going to assume the following equation holds for some integer k greater than or equal to 1:

[tex]2+5+8+\cdots+(3k-1)=\frac{1}{2}k(3k+1)[/tex]

Given this assumption we want to show the following:

[tex]2+5+8+\cdots+(3(k+1)-1)=\frac{1}{2}(k+1)(3(k+1)+1)[/tex]

Let's start with the left hand side:

[tex]2+5+8+\cdots+(3(k+1)-1)[/tex]

[tex]2+5+8+\cdots+(3k-1)+(3(k+1)-1)[/tex]

The first k terms we know have a sum of .5k(3k+1) by our assumption.

[tex]\frac{1}{2}k(3k+1)+(3(k+1)-1)[/tex]

Distribute for the second term:

[tex]\frac{1}{2}k(3k+1)+(3k+3-1)[/tex]

Combine terms in second term:

[tex]\frac{1}{2}k(3k+1)+(3k+2)[/tex]

Factor out a half from both terms:

[tex]\frac{1}{2}[k(3k+1)+2(3k+2][/tex]

Distribute for both first and second term in the [ ].

[tex]\frac{1}{2}[3k^2+k+6k+4][/tex]

Combine like terms in the [ ].

[tex]\frac{1}{2}[3k^2+7k+4[/tex]

The thing inside the [ ] is called a quadratic expression.  It has a coefficient of 3 so we need to find two numbers that multiply to be ac (3*4) and add up to be b (7).

Those numbers would be 3 and 4 since

3(4)=12 and 3+4=7.

So we are going to factor by grouping now after substituting 7k for 3k+4k:

[tex]\frac{1}{2}[3k^2+3k+4k+4][/tex]

[tex]\frac{1}{2}[3k(k+1)+4(k+1)][/tex]

[tex]\frac{1}{2}[(k+1)(3k+4)][/tex]

[tex]\frac{1}{2}(k+1)(3k+4)[/tex]

[tex]\frac{1}{2}(k+1)(3(k+1)+1)[/tex].

Therefore for all integers n equal or greater than 1 the following equation holds:

[tex]2+5+8+\cdots+(3n-1)=\frac{1}{2}n(3n+1)[/tex]

//

b)

Proof:

Base case: When n=1, the left hand side is 1.

The right hand at n=1 gives us:

[tex]\frac{1}{4}(5^1-1)[/tex]

[tex]\frac{1}{4}(5-1)[/tex]

[tex]\frac{1}{4}(4)[/tex]

[tex]1[/tex]

So both sides are 1 for n=1, therefore the equation holds for the base case, n=1.

We want to assume the following equation holds for some natural k:

[tex]1+5+5^2+\cdots+5^{k-1}=\frac{1}{4}(5^k-1)[/tex].

We are going to use this assumption to show the following:

[tex]1+5+5^2+\cdots+5^{(k+1)-1}=\frac{1}{4}(5^{k+1}-1)[/tex]

Let's start with the left side:

[tex]1+5+5^2+\cdots+5^{(k+1)-1}[/tex]

[tex]1+5+5^2+\cdots+5^{k-1}+5^{(k+1)-1}[/tex]

We know the sum of the first k terms is 1/4(5^k-1) given by our assumption:

[tex]\frac{1}{4}(5^k-1)+5^{(k+1)-1}[/tex]

[tex]\frac{1}{4}(5^k-1)+5^k[/tex]

Factor out the 1/4 from both of the two terms:

[tex]\frac{1}{4}[(5^k-1)+4(5^k)][/tex]

[tex]\frac{1}{4}[5^k-1+4\cdot5^k][/tex]

Combine the like terms inside the [ ]:

[tex]\frac{1}{4}(5 \cdot 5^k-1)[/tex]

Apply law of exponents:

[tex]\frac{1}{4}(5^{k+1}-1)[/tex]

Therefore the following equation holds for all natural n:

[tex]1+5+5^2+\cdots+5^{n-1}=\frac{1}{4}(5^n-1)[/tex].

//

Answer:

If you meant [tex]9^x+4=11[/tex], then the answer is approximately 0.866.

If you meant [tex]9^{x+4}=11[/tex], then the answer is approximately -2.909 which looks like what you meant based on the choices.

Step-by-step explanation:

[tex]9^x+4=11[/tex]

First step is to get the exponential part by itself. The part that has the variable exponent which is the [tex]9^x[/tex] term.

To do this we need to subtract 4 on both sides:

[tex]9^x=11-4[/tex]

Simplify:

[tex]9^x=7[/tex]

The equivalent logarithmic form is:

[tex]\log_9(7)=x[/tex]

I always say to myself the logarithm is the exponent that is how I know what to put opposite the side containing the log.

Anyways if you don't have options for computing [tex]\log_b(a)[/tex] in your calculator you need to use the change of base formula.

[tex]\frac{\log(7)}{\log(9)}=x[/tex]

So [tex]x \approx 0.8856[/tex]

I don't see this as a choice so maybe you actually meant the following equation:

[tex]9^{x+4}=11[/tex]

Let's see if this is the correct interpretation.

So the exponential part is already isolated.

So we just need to put in the equivalent logarithmic form:

[tex]\log_9(11)=x+4[/tex]

Now we subtract 4 on both sides:

[tex]\log_9(11)-4=x[/tex]

Again if you don't have the option for computing [tex]\log_b(a)[/tex] in your calculator, you will have to use the change of base formula:

[tex]\frac{\log(11)}{\log(9)}-4=x[/tex]

[tex]x \approx -2.909[/tex]

i know im late, but the answer is b. -2.909 :)

Answer:

5.25

Step-by-step explanation:

If CD is perpendicular to AB, a right angle is formed by the perpendicular lines at the base of the triangle. If we know that AB is 3 we can divided that by two to use half the triangle and create a right trianle with a base of 1.5 (half of 3) a height of rad 3 now all we have to do is use P-Thags to find AC which is the hypotenuse. After doing 1.5 squared + rad 3 squared = ac squared you will get the answer of AC = 5.25

Answer:

√(21) / 2

Step-by-step explanation:

The vertical line divides triangle ABC into two right angles which are congruent.

DB has length AB/2, which comes out to 3/2.  Regard DB as the base of one of the two right triangles.  If the height, CD, is √3, then by the Pythagorean Theorem |AC|² = (3/2)² + (√3)².

Thus, |AC| = √( (9/4) + 3 ), or  √( (9 + 12)/4 ), or √(21) / 2

[tex]\bf \textit{ellipse, horizontal major axis} \\\\ \cfrac{(x- h)^2}{ a^2}+\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2- b ^2} \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\bf 2x^2+8y^2=16\implies \cfrac{2x^2+8y^2}{16}=1\implies \cfrac{2x^2}{16}+\cfrac{8y^2}{16}=1\implies \cfrac{x^2}{8}+\cfrac{y^2}{2}=1 \\\\\\ \cfrac{(x-0)^2}{(\sqrt{8})^2}+\cfrac{(y-0)^2}{(\sqrt{2})^2}=1\qquad \begin{cases} a=\sqrt{8}\\ b=\sqrt{2}\\ c=\sqrt{a^2-b^2}\\ \qquad \sqrt{6} \end{cases}~\hfill \begin{array}{llll} \stackrel{\textit{vertices}}{(\pm \sqrt{8},0)}\qquad \stackrel{\textit{foci}}{(\pm \sqrt{6},0)}\\\\ ~\hfill \stackrel{\textit{center}}{(0,0)}~\hfill \end{array}[/tex]

Answer:

Step-by-step explanation:

2x² + 8y² = 16

divide both sides of equation by the constant

2x²/16 + 8y²/16 = 16/16

x²/8 + y²/2 = 1

x² has a larger denominator than y², so the ellipse is horizontal.

General equation for a horizontal ellipse:

  (x-h)²/a² + (y-k)²/b² = 1

with

  a² ≥ b²

  center (h,k)

  vertices (h±a, k)

  co-vertices (h, k±b)

  foci (h±c, k), c² = a²-b²

Plug in your equation, x²/8 + y²/2 = 1.

(x-0)²/(2√2)² + (y-0)²/(√2)² = 1

h = k = 0

a = 2√2

b = √2

c² = a²-b² = 6

c = √6

center (0,0)

vertices (0±2√2,0) = (-2√2, 0) and (2√2, 0)

co-vertices (0, 0±√2) = (0, -√2) and (0, √2)

foci (0±√6, 0) = (-√6, 0) and (√6, 0)

Answer:

  B.  a circle inscribed inside the triangle

Step-by-step explanation:

The incenter is the center of the inscribed circle. The incenter is the intersection of angle bisectors.

___

The center of mass and balance is the centroid, the intersection of medians.

Answer:

B.

Step-by-step explanation:

That is obtained when you split the angle of each vertex into two equal parts and trace the bisector. The intersection point is the incenter or the center of the circle inscribed inside the triangle.

Answer:

A

Step-by-step explanation:

To find the coordinates of the vertices , treat the inequalities as equations and solve them in pairs, simultaneously, that is

x + y = 9 → (1)

y - 2x = - 21 → (2) ← add 2x to both sides

y = - 21 + 2x → (3)

Substitute y = - 21 + 2x into (1)

x - 21 + 2x = 9

3x - 21 = 9 ( add 21 to both sides )

3x = 30 ( divide both sides by 3 )

x = 10

Substitute x = 10 into (3)

y = - 21 + 20 = - 1

Coordinates of vertex = (10, - 1 )

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y - 2x = - 21 → (1)

y = - 4x + 15 → (2)

Substitute y = - 4x + 15 into (1)

- 4x + 15 - 2x = - 21

- 6x + 15 = - 21 ( subtract 15 from both sides )

- 6x = - 36 ( divide both sides by - 6 )

x = 6

Substitute x = 6 into (2)

y = - 24 + 15 = - 9

Coordinates of vertex = (6, - 9 )

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x + y = 9 → (1)

y = - 4x + 15 → (2)

Substitute y = - 4x + 15 into (1)

x - 4x + 15 = 9

- 3x + 15 = 9 ( subtract 15 from both sides )

- 3x = - 6 ( divide both sides by - 3 )

x = 2

Substitute x = 2 into (2)

y = - 8 + 15 = 7

Coordinates of vertex = (2, 7 )