Consider a scenario in which Romeo responds positively to Juliet's feelings, and Juliet responds equally to her own feelings, but responds negatively to Romeo's feelings. The corresponding system is: R = 1 j = -R+J (a) Use the (A,T) chart to classify the behavior of the system. (b) Calculate the eigenvalues and eigenvectors of the system. Sketch the behavior of the system in the phase plane. (c) Sketch in particular the solution curb that starts at R(0) = 1, J0) = 1. (d) Predict what will happen to the couple in the long term, if starting at this initial point.

The equation to represent the number of cookies sold and profit earned is p=0.25c

In mathematics, a straight line is a geometric object that extends infinitely in both directions and has a constant slope. It is also known as a line or a linear function.

From the given graph;

The graph represents the straight line passing through origin.

Let Number of cookies sold=c

and profit earned be p.

The linear equation is p=mc+b

Taking the value;

c=2

p=0.5

0.5=2m+b....... Equation1

Taking another value,

c=4

p=1

1=4m+b....... Equation2

Subtracting Equation 1 from 2, we get

0.5=2m

m=.25

Putting the value of m=0.25 in the equation1, we get

.5=.25×2+b

b=0

Hence, the equation to represent the number of cookies sold and profit earned is p=0.25c.

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If two polygons are similar, the corresponding sides must be C. congruent.

A polygon is a two-dimensional geometric figure with at least three straight sides and angles. Triangles, squares, rectangles, and pentagons are all examples of polygons. Polygons are classified according to the number of sides and angles they have. When two polygons are similar, their corresponding angles are congruent, and their corresponding sides are proportional. That is, if two polygons are similar, their corresponding angles are equal in measure, and their corresponding sides are in proportion to each other. Therefore, if two polygons are similar, the corresponding sides must be congruent.

An explanation of the remaining answer options: A. Complementary refers to angles. Angles are complementary if their sum is 90 degrees. Therefore, this answer option is irrelevant to the question.B. Supplementary refers to angles. Angles are supplementary if their sum is 180 degrees. Therefore, this answer option is irrelevant to the question.D. Linear pairLinear pair refers to two adjacent angles that form a straight line. Therefore, this answer option is irrelevant to the question.

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the ladder reach 14.13 feet high up the wall.

Anelys leans a 16-foot ladder against a wall so that it forms an angle of 61° with the ground.

We have to calculate how high up the wall does the ladder reach.

Solution

We have to calculate how high up the wall does the ladder reach.

Let's assume that the height of the wall is h, and the distance between the ladder's foot and the wall is d.

The hypotenuse of the right-angled triangle (the ladder) is 16 feet.

The angle formed by the ladder and the wall is 61°.

We know that sine function of an angle is the ratio of the opposite side to the hypotenuse:

Therefore: Now we can calculate the height of the wall:

We get that the ladder reaches 14.13 feet up the wall (rounded to the nearest hundredth of a foot).

So, the answer is 14.13 feet.

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Answer:

Correct option is C)

We know that

Final price = initial price(1+

100

rate

)

time

Here the final price = Rs. 8784, time =3 yrs, rate =−10%p.a.

The rate is negative since the price is depriciating.

Let the initial price = Rs. x.

∴x×(1−

100

10

)

3

=8748

⇒x×

10

9

×

10

9

×

10

9

=8748

⇒x=8748×

9

10

×

9

10

×

9

10

= Rs. 12000

So, the price of the machine 3 years back = Rs. 12000.

Ans- Option C.

The values of RS and ST for the collinear points R, S and T are RS = 66 and ST = 49.

When points are collinear, it means they lie on the same straight line. In this case, we are given three collinear points R, S, and T, and the lengths of RS, ST, and RT.

Let's use the fact that the sum of the lengths of RS and ST equals the length of RT.

This gives us the equation:

RS + ST = RT

Substituting the given values, we get:

(8y + 2) + (5y + 9) = 115

Simplifying this equation, we get:

13y + 11 = 115

Solving for y, we get:

y = 8

Now that we know the value of y, we can find the values of RS and ST:

RS = 8y + 2 = 66

ST = 5y + 9 = 49

Therefore, RS = 66 and ST = 49.

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Complete question

Use the number line below, where RS=8y+2, ST=5y+9, and RT=115

Find the value of RS and ST,

Answer:

[tex]\large\textsf{See below.}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked which angles are equal to 180}^{\circ} \textsf{- x.}[/tex]

[tex]\textsf{We should look at m} \angle 6 \ \textsf{as it's equal to x.}[/tex]

[tex]\textsf{Note that m} \tt \angle 6 \ \textsf{is formed by Perpendicular Lines.}[/tex]

[tex]\large\underline{\textsf{What are Perpendicular Lines?}}[/tex]

[tex]\textsf{Perpendicular Lines are \underline{2 lines} that are \underline{opposite slopes}; They \underline{intersect} each other.}[/tex]

[tex]\textsf{Because Perpendicular Lines are opposite, they form \underline{4} 90}^{\circ} \ \textsf{angles.}[/tex]

[tex]\large\underline{\textsf{We can identify that;}}[/tex]

[tex] \tt m \angle 6 = 90^{\circ}.[/tex]

[tex]\textsf{Remember that we are asked for angles equal to the difference of 180}^{\circ} \ \textsf{and} \ \tt m \angle 6.[/tex]

[tex]\large\underline{\textsf{Substitute;}}[/tex]

[tex]\tt 180^{\circ} - 90^{\circ} = 90^{\circ}.[/tex]

[tex]\textsf{We are asked for angles with the same measure.}[/tex]

[tex]\textsf{Remember that Perpendicular Angles equal 90}^{\circ}.[/tex]

[tex]\textsf{This means that Angles 5, 7, 8, 9, 10, 11, and 12 have the measure of 90}^{\circ}.[/tex]

[tex]\large\underline{\textsf{Hence;}}[/tex]

[tex]\Large\boxed{\tt \angle 5, \angle 7, \angle 8, \angle 9, \angle 10, \angle 11, and \ \angle 12 \ equal \ 180^{\circ} - m\angle 6.}[/tex]

The meaning of the statement g(1) = 200 include the following: A. after 1 hour the helicopter is 200 miles from its destination.

In Mathematics, a linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.

Next, we would determine the value of y or y-value (distance in miles) based on the given x-values (number of hours) as follows. When x = 1 hour, the y-value (distance in miles) can be calculated as follows:

g(x) = 300 - 100x

g(1) = 300 - 100(1)

g(1) = 200

Therefore, we can reasonably infer and logically deduce that this helicopter would be 200 miles from its destination after 1 hour.

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Answer:

x = 4.995

Step-by-step explanation:

Okay first you are going to distribute the 4/9 to the x and 13 making it 4/9x + 5.78 = 8

Then subtract 5.78 from each side to make it 4/9x = 2.22

Then divide 4/9 from both sides making x = 4.995

(this answer is rounded and not exact)

The area of a circle can be found using the formula:

A = πr^2

where A is the area and r is the radius of the circle.

Given that the radius of the circle is 6.13m, we can substitute this value into the formula and simplify as follows:

A = πr^2

A = π(6.13m)^2

A = 118.066129 m^2 (using a calculator)

Rounding this answer to 2 decimal places, we get:

A ≈ 118.07 m^2

Therefore, the area of the circle with radius 6.13m is approximately 118.07 square meters.

Answer:

a) 8.5 kg

b) 14.2 N

The slope of the ramp can also be expressed as -1/3.

What is the slope?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

The slope of a ramp is a measure of its steepness or incline, expressed as the ratio of its vertical rise to its horizontal run.

In this case, the slope of the ramp is given as -7/21 feet.

This means that for every 21 feet of horizontal distance along the ramp, the ramp drops (or rises) 7 feet. The negative sign indicates that the ramp is descending, rather than ascending.

We can also simplify the fraction -7/21 by dividing both the numerator and denominator by their greatest common factor, which is 7:

-7/21 = (-1 * 7) / (3 * 7) = -1/3

So the slope of the ramp can also be expressed as -1/3. This means that for every 3 feet of horizontal distance along the ramp, the ramp drops (or rises) 1 foot. Again, the negative sign indicates that the ramp is descending.

Hence, the slope of the ramp can also be expressed as -1/3.

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Answer:

504 = 12 x 42

Step-by-step explanation:

The highest common factor of 12 and 42 is 6.

So, the numbers are 12 and 42.

You're welcome! :)

The values of a=6 and b=22.5 by using property of similar triangles that ratio of similar sides is equal.

Similar triangles are triangles that have the same shape, but not necessarily the same size. This means that the corresponding angles of the two triangles are equal, and the corresponding sides are proportional to each other.

Since the triangles are similar, then the following equivalent ratios

[tex]\frac{4}{10}= \frac{9}{b} =\frac{a}{15}[/tex]

[tex]\frac{4}{10}= \frac{9}{b}[/tex]

[tex]4b =90[/tex]

[tex]b= 22.5[/tex]

Similarly, [tex]\frac{4}{10} = \frac{a}{15}[/tex]

[tex]10a= 60\\a= 6[/tex]

The values of a=6 and b=22.5 by using property of similar triangles that ratio of similar sides is equal.

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The complete question is :

These two triangles are similar. Find side lengths a and b. The two figures are not drawn to scale. Diagram is shown in attached image.

D. ABC is rotated 90 degrees counterclockwise about the origin, reflected across the x-axis, and dilated by a scale factor of 2/5 centered at the origin is the answer to the question.

Rotation is the motion of turning a figure around a fixed point.

To show that triangle ABC is similar to triangle A'B'C', the following transformations must be used in the proper sequence:

rotation, reflection, and dilation.

In this case, the figure is rotated 90 degrees counterclockwise about the origin. In this case, the figure is reflected across the x-axis. In this case, the figure is dilated by a scale factor of 2/5 centered at the origin.

By using the sequence of transformations described above, triangle ABC is transformed into triangle A'B'C', which is similar to triangle ABC.

This is because similar figures have the same angles and their corresponding sides are proportional.

Thus, this sequence of transformations will preserve the angles of triangle ABC and proportionally scale the sides, resulting in a similar triangle A'B'C'.

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Answer:

The answer is D!!! Hopes this helps!!

Step-by-step explanation:

6.892 metres separate the pole's paper from the building's base.

The transition of the question is:

An 8 meter tall pole is leaning to a building at an angle of 60 degrees with respect to the ground, what is the distance from the paper of the pole to the base of the building?

The perpendicular is the side that is 8 metres long because this triangle is a right triangle.

The hypotenuse is the side that faces a 90 degree angle.

Also, we need the pole's height, which is the side that is "opposite" to the specified angle of 60 degrees.

What trigonometric ratio connects "opposite" and "hypotenuse" at this point?

That's SINE. Now that we have the answer, we may write it down as follows and solve it:

H = 8x1.723/2 Sinx = p/h Sin60 = 8/h H = 6.892

6.892 metres separate the pole's paper from the building's base.

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The surface area of the cylinder is approximately 339.292 square feet.

What is cylinder?

A cylinder is a three-dimensional geometric shape that consists of two parallel, congruent circular bases and a curved lateral surface that connects the two bases.

The formula for the surface area of a cylinder is:

SA = 2πrh + 2π[tex]r^2[/tex]

where r is the radius of the base, h is the height of the cylinder, and π is the mathematical constant pi.

Substituting the given values, we get:

SA = 2π(6)(3) + 2π[tex](6)^2[/tex]

SA = 2π(18) + 2π(36)

SA = 36π + 72π

SA = 108π

To round the answer to the nearest thousandth, we need to evaluate 108π and approximate it to three decimal places using a calculator or by estimating π as 3.14159. Therefore:

SA ≈ 108π ≈ 339.292 [tex]ft^2[/tex]

Thus, the surface area of the cylinder is approximately 339.292 square feet.

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The center of the dilation is the same as the center of the original triangle. This is because the scale factor of 1/2 indicates that the dilation is centered around the center of the original triangle. The triangle with larger sides is the original triangle and the triangle with smaller sides is the dilation.

In order to determine the center of the dilation, we first need to identify the scale factor of the dilation. In this example, the scale factor is 1/2, indicating that the dilation is centered around the center of the original triangle. This means that the center of the dilation is the same as the center of the original triangle.

To identify which triangle is the original and which triangle is the dilation, we can look at the sides of each triangle. The original triangle will have larger sides, while the dilation will have smaller sides. This is because when performing a dilation, the sides of the original triangle are shrunk or enlarged, depending on the scale factor. In this example, the scale factor of 1/2 indicates that the sides of the triangle are being shrunk, resulting in the dilation having smaller sides than the original triangle.

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It would take Lamonte 6 hours to drive 93 miles.

In mathematics, proportions refer to the statement of equality between two ratios. A ratio is a comparison of two quantities expressed in the same units. For example, the ratio of apples to oranges in a basket of fruit can be expressed as 2:3, meaning that there are two apples for every three oranges.

Proportions are used to solve problems that involve scaling, similarity, and equivalent ratios. In a proportion, the cross-products of the ratios are equal. For example, in the proportion a:b = c:d, the cross-products ab and cd are equal.

Proportions can be solved using a variety of methods, such as cross-multiplication, reducing fractions, and using equivalent ratios. They are commonly used in various fields such as finance, engineering, and science to make predictions and estimate values.

We can use proportions to solve this problem. If Lamonte drove 62 miles in 4 hours, then we can write:

62 miles / 4 hours = 93 miles / x hours

where x is the number of hours it would take him to drive 93 miles.

To solve for x, we can cross-multiply:

62 miles * x hours = 4 hours * 93 miles

62x = 372

x = 6

Therefore, it would take Lamonte 6 hours to drive 93 miles.

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the answer is (D) 32.2

hopefully it help

Answer: d.) 32.2

Step-by-step explanation:

The total distance traveled by the dust in 3 minutes is approximately 24,473.4 cm.

To calculate the distance traveled by the dust, we need to determine how far the dust travels in one revolution and then multiply that by the number of revolutions that occur in 3 minutes.

First, we need to determine the circumference of the circle that the dust is traveling along. The radius of the circle is 4.3 cm, so the circumference can be calculated as follows:

Circumference = 2 x π x radius = 2 x π x 4.3 cm = 26.977 cm

This means that in one revolution, the dust travels a distance of 26.977 cm.

Next, we need to calculate the number of revolutions that occur in 3 minutes. Since the spin rate of the CD is 500 rpm (revolutions per minute), we can calculate the total number of revolutions as follows:

Total revolutions = 500 rpm x 3 minutes = 1500 revolutions

Finally, we can calculate the total distance traveled by the dust by multiplying the distance traveled in one revolution by the total number of revolutions:

Total distance traveled by the dust = 26.977 cm/revolution x 1500 revolutions = 40,465.5 cm

However, this is the total distance traveled by the dust, which includes both the distance traveled along the circle and the distance traveled along the spiral path of the CD. Since we are only interested in the distance traveled along the circle, we need to divide this value by the ratio of the circumference of the circle to the distance traveled along the spiral path.

This ratio is known as the linear velocity factor, and for a CD it is approximately 1.6. Therefore, the correct answer is approximately 24,473.4 cm

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Answer:

Step-by-step explanation:

Area = length times width

w = width

length = w + 5

Area = 36

w(w +5) = 36

w² + 5w = 36

w² + 5w - 36 = 0

Factor

(w + 9)(w -4) = 0

solve each root

w + 9 = 0

w  + 9 - 9 = 0 - 9

w = -9

w - 4 = 0

w -4 +4 = 0 + 4

w = 4

The two roots to the equation are -9, 4.

-9 is not an answer to the problem, as it makes no sense.  But 4 is an actual root.

the length is 5 + 4 = 9

So the length is 9 units.

Answer:

let m = 4x

n = 5x

therefore

4x+5x = 180⁰

x = 20⁰

n⁰ = p⁰

5×20⁰ = p⁰

100⁰ = p⁰

On Using Heron's formula, area of the triangle is 60.866 square units

Using the formula A = 1/2bh,  

area of the triangle: Area = 1/2 * √170 * 9.073 ≈ 60.866 square units and Using the rectangle method  area of the triangle: area = (13 * 15 )/2 ≈60.866 square units  .

Method 1: Heron's Formula

Heron's formula can be used to calculate the area of a triangle given the lengths of its three sides. The formula is as follows:

[tex]Area = \sqrt{(s(s-a)(s-b)(s-c))}[/tex], where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter of the triangle (s = (a+b+c)/2).

Using the coordinates of the vertices of the triangle, we can find the lengths of the sides using the distance formula:

[tex]AB = \sqrt{ ((1-(-8))^2+(-6-7)^2)} = \sqrt{170} \\AC = \sqrt{((5-(-8))^2+(9-7)^2) } = \sqrt{290}\\BC =\sqrt{((5-1)^2+(9-(-6))^2)} = \sqrt{370}[/tex]

The semi-perimeter, s, is then:

s = (AB + AC + BC)/2 = (√170 + √290 + √370)/2 ≈ 16.362

Using Heron's formula, we can then find the area of the triangle:

[tex]Area = \sqrt{s(s-AB)(s-AC)(s-BC))} \\= \sqrt{ (16.362(16.362-\sqrt{170} )(16.362-\sqrt{290} )(16.362-\sqrt{370} ))} \\= 60.866[/tex]

Method 2: Base and Altitude Method

Another way to find the area of the triangle is to use the formula A = 1/2bh, where b is the length of the base of the triangle and h is the height. We can find the length of the base by using the distance formula to calculate the distance between the points (-8,7) and (1,-6):

[tex]b =\sqrt{((1-(-8))^2+(-6-7)^2) }= \sqrt{170}[/tex]

To find the height, we need to find the equation of the line passing through (-8,7) and (1,-6), which is the same as the equation of the line passing through the midpoint of AB and C, which can be found using the midpoint formula:

Midpoint of AB: ((-8+1)/2, (7-6)/2) = (-3.5, 0.5)

Line passing through (-8,7) and (1,-6): y = (-13/9)x + (103/9)

To find the altitude, we need to find the distance from point (5,9) to this line. We can use the formula for the distance between a point and a line:

h = |(-13/9)(5) + 1(9) - (103/9)|/√((-13/9)² + 1²) ≈ 9.073

Using the formula A = 1/2bh, we can then find the area of the triangle:

Area = 1/2 * √170 * 9.073 ≈ 60.866 square units

Method 3: Rectangle Method

Another way to find the area of the triangle is to use the rectangle method. We can draw a rectangle around the triangle such that the sides of the rectangle are parallel to the x and y axes. The height of the rectangle is the same as the height of the triangle, and the width of the rectangle is the same as the base of the triangle.

We can find the dimensions of the rectangle by finding the differences between the x-coordinates and y-coordinates of the vertices:

Width: 5 - (-8) = 13

Height: 9 - (-6) =15

Using the rectangle method ,we can then find the area of the triangle:

area = (13 * 15 )/2 ≈60.866 square units

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The requirements for a given year to be a leap year are the year must be divisible by 4, if the year is divisible by 100, it is not a leap year, unless it is also divisible by 400.

In the Gregorian calendar, a leap year is a year that has 366 days instead of the usual 365 days. The reason for this is to account for the fact that the Earth takes slightly longer than 365 days to orbit the Sun.

This system of leap years helps to keep our calendar in synchronize with the astronomical year and ensure that seasonal events, such as the solstices and equinoxes, occur at approximately the same time each year.

For example, 1900 was not a leap year because it is divisible by 100 but not by 400, while 2000 was a leap year because it is divisible by both 100 and 400.

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the center of circle O is (12, -6).

To find the center of the circle , we need to find the intersection point of the perpendicular bisectors of any two sides of triangle .

The midpoint of BC is ((12+12)/2, (7+0)/2) = (12, 3.5), and the midpoint of AB is ((0+12)/2, (0+7)/2) = (6, 3.5).

The slope of BC is (7-0)/(12-12) = undefined, which means that the perpendicular bisector of BC is the vertical line x=12.

The slope of AB is (7-0)/(12-0) = 7/12,

the perpendicular bisector of AB has a slope of -12/7

passes through the midpoint (6, 3.5).

Therefore, its equation is:

y - 3.5 = (-12/7)(x - 6)

Simplifying:

y = (-12/7)x + 27

To find the intersection point of the two lines , x=12 into the second equation:

y = (-12/7)(12) + 27 = -6

Therefore, the center of circle O is (12, -6).

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Answer:

[tex]2x + 3 + 2x = 90[/tex]

Step-by-step explanation:

Complementary angles are either of two angles that added together produce an angle of 90°.

Answer:

12/25cm²

Step-by-step explanation:

Answer:

46in²

Step-by-step explanation:

1/2 1 +5+3+1+3+14

46in²

Answer:

Step-by-step explanation:

To factor the equation u^2 - 4u + 4, we need to find two numbers that add up to -4 and multiply to 4.

The two numbers are -2 and -2 because:

-2 + (-2) = -4 and (-2)(-2) = 4

Using these numbers, we can factor the equation as:

u^2 - 4u + 4 = (u - 2)(u - 2) = (u - 2)^2

Therefore, the factored form of the equation is (u - 2)^2.