jason answered 12 questions correctly on a test worth 150 points. each question was worth 10 points. what percent did jason earn on the test?

Explaining the quadratic formula application in solving an equation.

The quadratic formula:

The answer is:

It will take 20.5 months to the populations to be equal.

Since from the statement we know that both populations are growing, we need to use the formula to calculate the exponential growth.

The exponential growth is defined by the following equation:

[tex]P(t)=StartPopulation*e^{\frac{growthpercent}{100}*t}[/tex]

Now,

Calculating for the rabbits, we have:

[tex]StartPopulation=20\\GrowthPercent=5\\[/tex]

So, writing the equation for the rabbits, we have:

[tex]P(t)=20*e^{\frac{5}{100}*t}[/tex]

[tex]P(t)=20*e{0.05}*t}[/tex]

Calculating for the fox, we have:

[tex]StartPopulation=30\\GrowthPercent=3\\[/tex]

So, writing the equation for the fox, we have:

[tex]P(t)=30*e{\frac{3}{100}*t}[/tex]

[tex]P(t)=30*e^{0.03}*t}[/tex]

Then, if we want to calculate how long does it takes for these populations to be equal, we need to make their equations equal, so:

[tex]20*e^{0.05}*{t}=30*e^{0.03}*{t}\\\\\frac{20}{30}=\frac{e^{0.03}*{t}}{e^{0.05}*t}}\\\\0.66=e^{0.03t-0.05t}=e^{-0.02t}\\\\0.66=e^{-0.02t}\\\\ln(0.66)=ln(e^{-0.02t})\\\\-0.41=-0.02t\\\\t=\frac{-0.41}{-0.02}=20.5[/tex]

Hence, we have that it will take 20.5 months to the populations to be equal.

To find out how long it takes for the populations to be equal, set up and solve an equation using the growth rates of the rabbit and fox populations.

To find out how long it takes for the rabbit population and the fox population to be equal, we can set up and solve an equation. Let's start by setting up an equation for each population growth:

Rabbit population: P(t) = 20 * (1 + 0.05)^t

Fox population: P(t) = 30 * (1 + 0.03)^t

We want to find the value of t when the two populations are equal, so we set the equations equal to each other and solve for t:

20 * (1 + 0.05)^t = 30 * (1 + 0.03)^t

Divide both sides by 20:

(1 + 0.05)^t = 1.5 * (1 + 0.03)^t

Now take the natural logarithm of both sides:

t * ln(1 + 0.05) = ln(1.5 * (1 + 0.03)^t)

Divide both sides by ln(1 + 0.05):

t = ln(1.5 * (1 + 0.03)^t) / ln(1 + 0.05)

Using a calculator, we can approximate the value of t to the nearest tenth of a month.

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Answer: B

Step-by-step explanation: You have no more than $65 which means you will want to spend equal to or less. Since you have $65, you will be able to spend up to that amount but you will not be able to spend any more than that. So, you will need to make sure the final price remains under or equal to your amount.

Answer:

The area of a sector GHJ is [tex]36.3\ cm^{2}[/tex]

Step-by-step explanation:

step 1

we know that

The area of a circle is equal to

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

we have

[tex]r=8\ cm[/tex]

substitute

[tex]A=\pi (8)^{2}[/tex]

[tex]A=64\pi\ cm^{2}[/tex]

step 2

Remember that the area of a complete circle subtends a central angle of 360 degrees

so

by proportion find the area of a sector by a central angle of 65 degrees

[tex]\frac{64\pi}{360}=\frac{x}{65}\\ \\x=64\pi (65)/360[/tex]

Use [tex]\pi =3.14[/tex]

[tex]x=64(3.14)(65)/360=36.3\ cm^{2}[/tex]

Answer: first option.

Step-by-step explanation:

Given the right triangle shown in the figure, to calculate the measure of the angle m∠C, you can use the inverse function of the cosine:

[tex]\alpha=arccos(\frac{adjacent}{hypotenuse})[/tex]

You can identify in the figure, that, for the angle ∠C:

[tex]\alpha=\angle C\\adjacent=7\\hypotenuse=15[/tex]

Then, since you know the lenght of the adjacent side and the lenght of the hypotenuse, you can substitute these values into  [tex]\alpha=arccos(\frac{adjacent}{hypotenuse})[/tex].

Therefore, the measure of the angle ∠C is:

[tex]\angle C=arccos(\frac{7}{15})\\\\\angle C=62.2\°[/tex]

Answer:

[tex]2.1\ in[/tex]

Step-by-step explanation:

we know that

The scale of the world globe is equal to

[tex]\frac{13.9}{10,480}\frac{in}{km}= 0.0013\frac{in}{km}[/tex]

To find the distance between City C and City D on the globe, multiply the actual distance by the scale

[tex]0.0013\frac{in}{km}*(1,590\ km)=2.1\ in[/tex]

Answer: Option 'C' is correct.

Step-by-step explanation:

Since we have given that

The expression is defined as

[tex]f(x)=12(1.035)^x------------------------(1)[/tex]

As we know the general form of exponential function:

[tex]f(x)=a(1+r)^x--------------------------(2)[/tex]

Here, a denotes the initial amount.

r denotes the growth rate.

On comparing, we get that

[tex]1+r=1.035\\\\r=1.035-1\\\\r=0.035\\\\r=0.035\times 100\%\\\\r=3.5\%[/tex]

Hence, option 'C' is correct.

Answer:

C) 3.5%

Step-by-step explanation:

For this case we have the following expression:

[tex]-5 (3x + 7) -4 (x-4) +11[/tex]

For simplicity we follow the steps:

We apply distributive property to the terms within the parenthesis, bearing in mind that:

[tex]- * + = -\\- * - = +\\-15x-35-4x + 16 + 11 =[/tex]

We add similar terms taking into account that equal signs are added and the same sign is placed, while different signs are subtracted and the sign of the major is placed.

[tex]-15x-4x-35 + 16 + 11 =\\-19x-8[/tex]

ANswer:

[tex]-19x-8[/tex]

Answer: 19x - 8

-5(3x + 7)

Apply the distributive law: a( b + c ) = ab + ac

a = -5, b = 3x, c = 7

= -5 * 3x + (-5) x 7

Apply minus - plus rules

+(-a) = -a

= -5 * 3x - 5 x 7

Simplify

-5 * 3x - 5 x 7

Multiply the numbers: 5 x 3 = 15

= -15x - 5 x 7

Multiply the numbers: 5 x 7 = 35

= -15x - 35

Plug in the values

-15x - 35 - 4(x - 4) + 11

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

-4(x - 4)

Apply the distributive law: a( b - c ) = ab - ac

a = -4, b = x, c = 4

= -4x - (-4) * 4

Apply minus - plus rules

-(-a) = a

= -4x + 4 * 4

Multiply the numbers: 4 * 4 = 16

= -4x + 16

Plug in the values

-15x - 35 - 4x + 16 + 11

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

Simplify

-15x - 35 - 4x + 16 + 11

Group like terms

= -15x - 4x - 35 + 16 + 11

Add similar elements: -15x - 4x = -19x

= -19x - 35 + 16 + 11

Add/Subtract the numbers: -35 + 16 + 11 = -8

= 19x - 8

Final answer:

To find the number of oranges left for Jenny to peel in the evening, subtract the number of oranges she already peeled from the total number of oranges.

Explanation:

To find the number of oranges left for Jenny to peel in the evening, we need to subtract the number of oranges she already peeled in the morning and afternoon from the total number of oranges.

Jenny peeled 30% of the oranges in the morning, which is (30/100) * 300 = 90 oranges.

Jenny peeled 45% of the oranges in the afternoon, which is (45/100) * 300 = 135 oranges.

Therefore, the number of oranges left for Jenny to peel in the evening is 300 - 90 - 135 = 75 oranges.

Answer:

It's answers 1 and 3

Step-by-step explanation:

The meeting is not mandatory and only volunteers are participating in the task he wanted, this shows bias and doesn't correctly represent the whole team.

Answer:

The ages at present are:

Explanation:

Translate the word language to algebraic expressions.

1) A father who is 42 years old has a son and a daughter.

2) The daughter is three times as old as the son.

3) In 10 years the sum of all their ages will be 100 years

            ↑                          ↑                                  ↑                             ↑

age of the father     age of the son          age of the daughter      sum

4) How old are the two siblings at present:

Solve the equation

        4x = 100 - 72

        4x = 28

        x = 28 / 4

        x = 7

5) Answers:

6) Verification:

        age of the son: 7 + 10 = 17

        age of the daughter: 21 + 10 = 31

        age of the father: 42 + 10 = 52

        sum of the ages: 17 + 31 + 52 = 100 ⇒ correct.

Answer: Son’s age:7       Daughter’s age:21

Step-by-step explanation:

1. The daughter is 3 times older than the son. Son will be x. Then the daughter will be 3x

2. In ten years the sum of their ages will be 100. Father+Son+Daughter=(42+10)+(x+10)+(3x+10)=100

3. Solve the equation:

  72+4x=100

  x=7

4. You have found the age of the son, 7. Now find the daughter. 3*7=21. The daughter is 21.

     Success.!

Answer:

y+3= -6(x+9) is the answer

Step-by-step explanation:

Answer:

[tex]y+3=-6(x+9)[/tex]

Step-by-step explanation:

We are given that

Slope of a line=-6

Given point =(-9,-3)

We have to find the equation which represents the line.

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

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

Substitute the values then we get

The equation of line passing through the point (-9,-3) with slope -6 is given by

[tex]y-(-3)=-6(x-(-9))[/tex]

[tex]y+3=-6(x+9)[/tex]

Hence, the equation of line that passes through (-9,-3) and has  a slope -6 is given by

[tex]y+3=-6(x+9)[/tex]

Answer:

The relationship between the circumference of a circle and its diameter represent  a direct variation

The constant of proportionality is equal to [tex]\pi[/tex]

Step-by-step explanation:

we know that

A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]

The circumference of a circle is equal to

[tex]C=\pi D[/tex]

Let

C=y

D=x

substitute

[tex]y=\pi x[/tex]

therefore

The relationship between the circumference of a circle and its diameter represent  a direct variation

The constant of proportionality is equal to [tex]\pi[/tex]

Answer:

D. y = 10000 - 750x

Step-by-step explanation:

The answer is D. y = 10000 - 750x, where:

y = the current value of the car,

10000 is the initial value of the car

750 is the depreciation it has every year

x is the number of years.

The 10000 has to be fixed and not multiplied by anything (unlike answer A or C) because that's the initial value of the car.  Then it has to be reduced (meaning we take value of out it, so a subtraction), so that excludes A and B.  The devaluation occurs every year, so it has to be multiplied by the number of years (excluding answers A and C again).  So, only answer D remains.

Final answer:

The equation that best models the depreciation of the car is: y = 10000 - 750x. This equation represents the value of the car decreasing by $750 each year.

Explanation:

The equation that best models the depreciation of the car is: y = 10000 - 750x.

This equation is derived from the given information that the car depreciates by $750 each year, which is a constant amount. The equation represents the value of the car, denoted by 'y', decreasing by $750 for each year, denoted by 'x'.

For example, if we plug in x = 1 into the equation, we get y = 10000 - 750(1) = 9250, which means the car is worth $9250 after the first year.

Move the 5 to the other side:

[tex]2\sec(x)=1-5=-4[/tex]

Divide both sides by 2:

[tex]\sec(x) = -2[/tex]

Recall the definition:

[tex]\sec(x)=-2 \iff \dfrac{1}{\cos(x)}=-2[/tex]

Invert both sides

[tex]\cos(x) = -\dfrac{1}{2}[/tex]

This is true when

[tex]x=\pm \dfrac{\pi}{3}[/tex]

If you need both angles to be in [0,2pi], you can recall

[tex]\cos\left(-\dfrac{\pi}{3}\right) = \cos\left(-\dfrac{\pi}{3}+2\pi\right) = \cos\left(\dfrac{5\pi}{3}\right)[/tex]

So, the solutions are

[tex]x=\dfrac{\pi}{3},\quad x=\dfrac{5\pi}{3}[/tex]

Answer:

2pi/3 and 4pi/3

Step-by-step explanation:

this is the answer according to apex

V= 3,840 cm

Volume is L x W x H

and W is the same as Depth

So you would multiply those and multiply it again by four to make up for the other pieces of wood.

Final answer:

The total volume of wood used in making the square frame is 3840 cubic centimeters, which is calculated by finding the volume of one piece (960 cm³) and multiplying it by the number of pieces (4).

Explanation:

The question concerns the calculation of the total volume of wood used in making a square frame. Each piece of wood is a rectangular prism with given dimensions. To find the total volume, one must multiply the length, height, and depth of a single piece and then multiply the result by the number of pieces.

To calculate the volume of a single piece of wood, we use the formula for the volume of a rectangular prism, which is length × width × height. For one piece of wood:

Length = 40 cm

Width (depth) = 6 cm

Height = 4 cm

The volume of one piece of wood is therefore 40 cm × 6 cm × 4 cm = 960 cm³.

Since there are four pieces that make up the frame, the total volume of wood used is:

960 cm³/piece × 4 pieces = 3840 cm³

Thus, the total volume of the wood used in the frame is 3840 cubic centimeters.

Answer:

49.2 miles

Step-by-step explanation:

The distance remains the same.                        d

The time spent on the outbound trip is t1 = -------------

                                                                         35 mph

and that on the inbound (return) trip is

                                                                              d

                                                                t2 = -------------

                                                                         45 mph

We combine these two fractions and set the sum = to 2.5 hours:

d(1/35 + 1/45) = 5/2

We wish to solve for d, the distance between home and work.

The LCD of 35, 45 and 2 is 630.

1/35 becomes 18/630; 1/45 becomes 14/630, and 5/2 becomes 1575/630.  Then we have the simpler equation   d(18 + 14) = 1575, or

d(32) = 1575, and d is then

d = 1575/32 = 49.2 miles

Answer:

Step-by-step explanation:

If your function is

[tex]x+y^2=ln(\frac{x}{y} )[/tex], that is definitely not the answer you should get after taking the derivative implicitely.  Rewrite your function to simplify a bit:

[tex]x+y^2=ln(x)-ln(y)[/tex]

Take the derivative of x terms like "normal", but taking the derivative of y with respect to x has to be offset by dy/dx.  Doing that gives you:

[tex]1+2y\frac{dy}{dx}=\frac{1}{x}-\frac{1}{y}\frac{dy}{dx}[/tex]

Collect the terms with dy/dx on one side and everything else on the other side:

[tex]2y\frac{dy}{dx}+\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}-1[/tex]

Now factor out the common dy/dx term, leaving this:

[tex]\frac{dy}{dx}(2y+\frac{1}{y})=\frac{1}{x}-1[/tex]Now divide on the left to get dy/dx alone:

[tex]\frac{dy}{dx}=\frac{\frac{1}{x}-1 }{2y+\frac{1}{y} }[/tex]

Simplify each set of fractions to get:

[tex]\frac{dy}{dx}=\frac{\frac{1-x}{x} }{\frac{2y^2+1}{y} }[/tex]

Bring the lower fraction up next to the top one and flip it upside down to multiply:

[tex]\frac{dy}{dx}=\frac{1-x}{x}[/tex]×[tex]\frac{y}{2y^2+1}[/tex]

Simplifying that gives you the final result:

[tex]\frac{dy}{dx}=\frac{y-xy}{x(2y^2+1)}[/tex]

or you could multiply in the x on the bottom, as well.  Same difference as far as the solution goes.  You'd use this formula to find the slope of a function at a point by subbing in both the x and the y coordinates so it doesn't matter if you do the distribution at the very end or not.  You'll still get the same value for the slope.

Answer:

B. [tex]LA=2\pi rh[/tex]

Step-by-step explanation:

The lateral area of the right cylinder refers to the curved surface area.

The lateral area of the right cylinder does not include the two circular bases.

The lateral area is given by the formula;

[tex]LA=2\pi rh[/tex]

The correct choice is B.

Answer:

thats the correct answer =B

Step-by-step explanation:

Answer:

FALSE

Correlations tell us the strength and the direction of the relationship between two variables. The main result of a correlation is called the correlation coefficient (or "r").

Hope this helps and have a great day!!

[tex]Sofia[/tex]

Answer:false

Step-by-step explanation:

Answer:

Conjugate radicals.

Step-by-step explanation:

The two roots a+√b and a-√b are called Conjugate radicals.

Just like in the complex number system where we have complex conjugates such as of 4+7i and 4 - 7i, the radicals also have their conjugate radicals. The conjugate radical of a+√b is simply obtained by changing the sign of the radical part of the expression to obtain a-√b. Therefore, the two expressions given are conjugate radicals

Answer:

Conjugate.

Step-by-step explanation:

The difference is in the signs. They are conjugate radicals.

Answer:

  264 miles

Step-by-step explanation:

Using the relation ...

  distance = speed · time

we can rearrange to get ...

  speed = distance/time

We can choose to let d represent the distance we want to find. Then Jina's speed going to the mountains is d/6. Her speed coming home is then d/6+22. It takes Jina 4 hours at that speed to cover the same distance, so we have ...

  d = 4(d/6 +22)

  d = 2/3d +88 . . . . eliminate parentheses

  1/3d = 88 . . . . . . . subtract 2/3d

  d = 264 . . . . . . . . . multiply by 3

Jina lives 264 miles from the mountains.

I need a clear understanding....

then jeans will cost $38.25

Answer:

the answer would be $35.70

Step-by-step explanation:

0.70 x $60 = $42

0.85 x $42 = $35.70

This is using multipliers so the amount off will be taken away from 1 and that answer times amount needed to found from will give the answer you are looking for

The answer is:

The equation D  will produce the shown circle.

[tex]6x^{2}+6y^{2}=144[/tex]

Since the graph is showing a circle, we need to find the equation of a circle that has a radius which is between 0 and 5 units, and has a center located at the origen (0,0).

Also, we need to remember the standard form of a circle:

[tex](x+h)^{2} +(y+k)^{2}=r^{2}[/tex]

Where,

x, is the x-coordinate of the x-intercept point

y, is the y-coordinate of the y-intercept point

h, is the x-coordinate of the center.

k, is the y-coordinate of the center.

r, is the radius of the circle.

So, discarding each of the given options, we have:

First option:

A.

[tex]\frac{x^{2} }{20}+ \frac{y^{2} }{20}=1\\\\\frac{1}{20}(x^{2}+y^{2})=1\\\\x^{2}+y^{2}=20*1\\\\x^{2}+y^{2}=20[/tex]

Where,

[tex]radius=\sqrt{20}=4.47=4.5[/tex]

Now, can see that even the center is located at the point (0,0), the radius of the circle is equal to 4.5 units and from the graph we can see that the radius of the circle is more than 4.5 units but less than 5 units, the option A is not the equation that produces the shown circle.

Second option:

B.

[tex]20x^{2} -20y^{2}=400\\\\\frac{1}{20}(x^{2} -y{2})=400\\\\x^{2} -y{2}=400*20[/tex]

Where,

[tex]radius=\sqrt{8000}=89.44units[/tex]

We can see that even the center is located at the point (0,0), the radius of the circle is 89.44 units, so, the option B is not the equation that produces the shown circle.

Third option:

C.

[tex]x^{2}+y^{2}=16[/tex]

Where,

[tex]radius=\sqrt{16}=4units[/tex]

We  can see that even the center is located at the point (0,0), the radius of the circle is 4 units, which is less than the radius of the circle shown in the graph, so, the option C is not the equation that produces the shown circle.

D.

[tex]6x^{2}+6y^{2}=144\\\\6(x^{2} +y^{2})=144\\\\x^{2} +y^{2}=\frac{144}{6}=24\\\\[/tex]

Where,

[tex]radius=\sqrt{24}=4.89units[/tex]

Now, we have that the radius of the circle is 4.89 units, which is approximated equal to 0, also, the center of the circle is located at (0,0) so, the equation D will produce the shown circle.

[tex]6x^{2}+6y^{2}=144[/tex]

Have a nice day!

The equation that represents the given graph is:

              [tex]6x^2+6y^2=144[/tex]

By looking at the given graph we observe that the graph is a circle with center at (0,0) and the radius is close to 5.

Now, we know that:

The general equation of a circle with center (h,k) and radius r is given by:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

Here (h,k)=(0,0)

Hence, the equation of the circle is:

            [tex]x^2+y^2=r^2[/tex]

A)

[tex]\dfrac{x^2}{20}+\dfrac{y^2}{20}=1\\\\i.e.\\\\x^2+y^2=20[/tex]

i.e.

[tex]x^2+y^2=(2\sqrt{5})^2[/tex]

This equation is a equation of a circle with center at (0,0)

and radius is: [tex]2\sqrt{5}\ units[/tex]

i.e. the radius is approximately equal to 4.5 units.

But the radius is close to 5.

Hence, option: A is incorrect.

B)

[tex]20x^2-20y^2=400\\\\i.e.\\\\x^2-y^2=20[/tex]

This is not a equation of a circle.

This equation represents a hyperbola.

Hence, option: B is incorrect.

C)

[tex]x^2+y^2=16[/tex]

which could be represented by:

[tex]x^2+y^2=4^2[/tex]

i.e. the radius of circle is: 4 units

which is not close to 5.

Hence,option: C is incorrect.

D)

[tex]6x^2+6y^2=144[/tex]

On dividing both side of the equation by 6 we get:

[tex]x^2+y^2=24[/tex]

i.e.

[tex]x^2+y^2=(\sqrt{24})^2[/tex]

i.e.

Radius is: [tex]\sqrt{24}\ units[/tex]

which is approximately equal to 4.9 units which is close to 5 units.

Answer:

  C. Draw a diagram. Draw 4 groups of dots to show the 3 types of exercise and the weeks. Write 30, 45, 25, and 8 in each circle. Add the four numbers.

Step-by-step explanation:

You might be able to get there starting with strategy C, but it is incomplete as written and will not solve the problem.

Answer:

m < EJB = half of m < OBE.

m < BDE = 180 minus m < BOE.

m < OED = m<OBD.

Step-by-step explanation:

First part :  Because  angled subtended by an arc at the circumference = half of angle at the center.

Second: Because  The 2 angles OBD and OED = 90 degrees.

Third: DB and DE are both tangents to the circle, and OE and OB are both radii. So m < OED = m<OBD = 90 degrees.

Answer:

1. B. one-half

2. D. 180 minus

3. A. equal to

Answer:

(6, -70) is not on the graph

Step-by-step explanation:

we have

[tex]y=-3x^{2}+6x-4[/tex]

we know that

If a ordered pair is on the graph of the quadratic equation, then the ordered pair must satisfy the quadratic equation

Verify each case

Substitute the x-coordinate of the ordered pair in the quadratic equation to find the value of y and then compare the results

case 1) (6, -70)

For x=-6

[tex]y=-3(-6)^{2}+6(-6)-4=-148[/tex]

[tex]-148\neq-70[/tex]

therefore

the ordered pair is not on the graph

case 2) (4, -28)

For x=4

[tex]y=-3(4)^{2}+6(4)-4=-28[/tex]

[tex]-28=-28[/tex]

therefore

the ordered pair is on the graph

case 3) (-8,-244)

For x=-8

[tex]y=-3(-8)^{2}+6(-8)-4=-148[/tex]

[tex]-244=-244[/tex]

therefore

the ordered pair is on the graph

case 4) (12,-364)

For x=12

[tex]y=-3(12)^{2}+6(12)-4=-148[/tex]

[tex]-364=-364[/tex]

therefore

the ordered pair is on the graph

Answer:

Part 1) [tex]9x-7y=-25[/tex]

Part 2) [tex]2x-y=2[/tex]

Part 3) [tex]x+8y=22[/tex]  

Part 4) [tex]x+8y=35[/tex]

Part 5) [tex]3x-4y=2[/tex]

Part 6) [tex]10x+6y=39[/tex]

Part 7) [tex]x-5y=-6[/tex]

Part 8)

case A) The equation of the diagonal AC is [tex]x+y=0[/tex]

case B) The equation of the diagonal BD is [tex]x-y=0[/tex]

Step-by-step explanation:

Part 1)

step 1

Find the midpoint

The formula to calculate the midpoint between two points is equal to

[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]

substitute the values

[tex]M=(\frac{2-6}{2},\frac{-3+5}{2})[/tex]

[tex]M=(-2,1)[/tex]

step 2

The equation of the line into point slope form is equal to

[tex]y-1=\frac{9}{7}(x+2)\\ \\y=\frac{9}{7}x+\frac{18}{7}+1\\ \\y=\frac{9}{7}x+\frac{25}{7}[/tex]

step 3

Convert to standard form

Remember that the equation of the line into standard form is equal to

[tex]Ax+By=C[/tex]

where

A is a positive integer, and B, and C are integers

[tex]y=\frac{9}{7}x+\frac{25}{7}[/tex]

Multiply by 7 both sides

[tex]7y=9x+25[/tex]

[tex]9x-7y=-25[/tex]

Part 2)

step 1

Find the midpoint

The formula to calculate the midpoint between two points is equal to

[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]

substitute the values

[tex]M=(\frac{1+5}{2},\frac{0-2}{2})[/tex]

[tex]M=(3,-1)[/tex]

step 2

Find the slope

The slope between two points is equal to

[tex]m=\frac{-2-0}{5-1}=-\frac{1}{2}[/tex]

step 3

we know that

If two lines are perpendicular, then the product of their slopes is equal to -1

Find the slope of the line perpendicular to the segment joining the given points

[tex]m1=-\frac{1}{2}[/tex]

[tex]m1*m2=-1[/tex]

therefore

[tex]m2=2[/tex]

step 4

The equation of the line into point slope form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=2[/tex] and point [tex](1,0)[/tex]

[tex]y-0=2(x-1)\\ \\y=2x-2[/tex]

step 5

Convert to standard form

Remember that the equation of the line into standard form is equal to

[tex]Ax+By=C[/tex]

where

A is a positive integer, and B, and C are integers

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

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

Part 3)

In this problem AB and BC are the legs of the right triangle (plot the figure)

step 1

Find the midpoint AB

[tex]M1=(\frac{-5+1}{2},\frac{5+1}{2})[/tex]

[tex]M1=(-2,3)[/tex]

step 2

Find the midpoint BC

[tex]M2=(\frac{1+3}{2},\frac{1+4}{2})[/tex]

[tex]M2=(2,2.5)[/tex]

step 3

Find the slope M1M2

The slope between two points is equal to

[tex]m=\frac{2.5-3}{2+2}=-\frac{1}{8}[/tex]

step 4

The equation of the line into point slope form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=-\frac{1}{8}[/tex] and point [tex](-2,3)[/tex]

[tex]y-3=-\frac{1}{8}(x+2)\\ \\y=-\frac{1}{8}x-\frac{1}{4}+3\\ \\y=-\frac{1}{8}x+\frac{11}{4}[/tex]

step 5

Convert to standard form

Remember that the equation of the line into standard form is equal to

[tex]Ax+By=C[/tex]

where

A is a positive integer, and B, and C are integers

[tex]y=-\frac{1}{8}x+\frac{11}{4}[/tex]

Multiply by 8 both sides

[tex]8y=-x+22[/tex]

[tex]x+8y=22[/tex]  

Part 4)

In this problem the hypotenuse is AC (plot the figure)

step 1

Find the slope AC

The slope between two points is equal to

[tex]m=\frac{4-5}{3+5}=-\frac{1}{8}[/tex]

step 2

The equation of the line into point slope form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=-\frac{1}{8}[/tex] and point [tex](3,4)[/tex]

[tex]y-4=-\frac{1}{8}(x-3)[/tex]

[tex]y=-\frac{1}{8}x+\frac{3}{8}+4[/tex]

[tex]y=-\frac{1}{8}x+\frac{35}{8}[/tex]

step 3

Convert to standard form

Remember that the equation of the line into standard form is equal to

[tex]Ax+By=C[/tex]

where

A is a positive integer, and B, and C are integers

[tex]y=-\frac{1}{8}x+\frac{35}{8}[/tex]

Multiply by 8 both sides

[tex]8y=-x+35[/tex]

[tex]x+8y=35[/tex]

Part 5)  

The longer diagonal is the segment BD (plot the figure)  

step 1

Find the slope BD

The slope between two points is equal to

[tex]m=\frac{4+2}{6+2}=\frac{3}{4}[/tex]

step 2

The equation of the line into point slope form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=\frac{3}{4}[/tex] and point [tex](-2,-2)[/tex]

[tex]y+2=\frac{3}{4}(x+2)[/tex]

[tex]y=\frac{3}{4}x+\frac{6}{4}-2[/tex]

[tex]y=\frac{3}{4}x-\frac{2}{4}[/tex]

step 3

Convert to standard form

Remember that the equation of the line into standard form is equal to

[tex]Ax+By=C[/tex]

where

A is a positive integer, and B, and C are integers

[tex]y=\frac{3}{4}x-\frac{2}{4}[/tex]

Multiply by 4 both sides

[tex]4y=3x-2[/tex]

[tex]3x-4y=2[/tex]

Note The complete answers in the attached file

Answer:

1

Step-by-step explanation:

y-intercept is defined as the point where the graph crosses the y-axis. The value of x coordinate at this point is zero, as along entire y-axis, the value of x coordinate is always zero. So substituting x = 0 in the function will give us the y-coordinate of the y-intercept of the given function.

[tex]f(x)=x^{3}-x^{2} -x+1[/tex]

Substituting x = 0 in this function, we get:

[tex]f(0)=0^{3}-0^{2}-0+1=1[/tex]

Thus, the y-coordinate of the y-intercept is 1. Therefore the y-intercept of the function in ordered pair will be: (0, 1)

Answer:

the length of the conjugate axis  is 16

Step-by-step explanation:

We know that the general equation of a hyperbola with transverse horizontal axis has the form:

[tex]\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1[/tex]

Where the point (h, k) are the coordinates of the center of the ellipse

2a is the length of the transverse horizontal axis

2b is the length of the conjugate axis

In this case the equation of the ellipse is:

[tex]\frac{(x-2)^2}{36} + \frac{(y+1)^2}{64} = 1[/tex]

Then

[tex]b^2 = 64\\\\b = \sqrt{64}\\\\b = 8\\\\2b = 16[/tex]

Finally the length of the conjugate axis is 16

Final answer:

The length of the conjugate axis of the given hyperbola \(\frac{(x-2)²}{36} - \frac{(y+1)²}{64} =1\) is 16 units.

Explanation:

The question asks to find the length of the conjugate axis of a hyperbola given by the equation:

\(\frac{(x-2)²}{36} - \frac{(y+1)²}{64} =1\)

In the equation of a hyperbola of the form \(\frac{(x-h)²}{a²} - \frac{(y-k)²}{b²} = 1\), where \((h,k)\) is the center of the hyperbola, \(a²\) is the denominator under the \(x\)-term, and \(b^2\) is the denominator under the \(y\)-term, the length of the transverse axis is \(2a\) and the length of the conjugate axis is \(2b\).

For the given hyperbola:

a² = 36, so \(a = 6\)

b² = 64, so \(b = 8\)

Therefore, the length of the conjugate axis is \(2 \times 8 = 16\) units.