what is the resulting expression when (7x - 4/3) is subtracted from -3/5x + 5/3?

Answer:

Here we try the method of trial and error to find out if the equations have a common answer as zero

Step-by-step explanation:

Now,

(a) x²-4=0

(b) x²=-4

(c) 3x²+12=0

(d) 4x²=16

(e) 2(x-2)2=0

Here if we check the first equation i.e x²-4=0

Equating - 2²-4=4-4

=0

2²-4= 4-4

=0

So option (a) is true

Now x²=-4

Substituting, - 2²=4 & 2²=4

So here we get 4≠-4

Therefore, (b) is not true

Now 3x²+12=0

3(-2)²+12= 3(4)+12

=12+12= 24

3(2)²+12= 12+12

= 24

4x²=16

Substituting, 4(-2)²= 4(4)= 16

4(2)²= 4(4)= 16

So option (d) is also true

Now, 2(x-2)2=0

Substituting, 2(-2-2)2= 2(-4)2

4(-4)= - 16

2(2-2)2= 2(0)2

=4(0)=0

Here when we put x=-2, we get - 16

when we put x=2, we get 0

So the following equation is true only for x=2 and not x=-2

I hope this helps ;)

The total surface area of the television is 21,584 square centimeters.

To find the total surface area of the television, we need to calculate the area of all six sides and add them up.

The front and back sides have the same dimensions and area, so we can find the area of one and multiply it by two. The same goes for the left and right sides.

The area of the front/back sides is:

120 cm x 68 cm = 8160 sq cm

Multiplying by 2 gives us the total area of both front/back sides:

2 x 8160 sq cm = 16,320 sq cm

The area of the left/right sides is:

68 cm x 14 cm = 952 sq cm

Multiplying by 2 gives us the total area of both left/right sides:

2 x 952 sq cm = 1904 sq cm

The area of the top and bottom sides is:

120 cm x 14 cm = 1680 sq cm

Multiplying by 2 gives us the total area of both top/bottom sides:

2 x 1680 sq cm = 3360 sq cm

Adding up all six sides, we get:

16,320 sq cm + 1904 sq cm + 3360 sq cm = 21,584 sq cm

Therefore, the total surface area of the television is 21,584 square centimeters.

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I believe credit cards tend to be the entry point for establishing credit for so many consumers because they provide an easy and accessible way for individuals to begin building their credit history. Credit card companies report to credit bureaus on a regular basis, which helps establish a credit score and credit history.

Additionally, credit cards offer a convenient way for individuals to make purchases and build their credit at the same time. However, it is important for individuals to use their credit cards responsibly and make timely payments in order to maintain good credit standing.

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The percentage of the family's income that is spent on food is 25%.

Firstly, noting down the family's monthly spend on food ($550) and their total monthly income ($2,200).

Next, dividing the amount spent on food by the total income to find the ratio of the spend to income: $550 / $2,200.

Now, calculating this division: 550 ÷ 2,200 = 0.25.

Finally, finding the percentage, multiply the ratio (0.25) by 100: 0.25 x 100 = 25%.

So, the family spends 25% of their monthly income on food.

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The ordered pair that describes the location of E'' is (6, -1).

The initial location of Point EE is (-6, 1). To reflect Point EE over the yy-axis, we will change the sign of the x-coordinate while keeping the y-coordinate the same.

Step 1: Reflect Point EE over the yy-axis to create Point E'.
E' = (6, 1)

Next, to reflect Point E' over the xx-axis, you'll change the sign of the y-coordinate while keeping the x-coordinate the same.

Step 2: Reflect Point E' over the xx-axis to create Point E''.
E'' = (6, -1)

Therefore, the ordered pair that describes the location of E'' is (6, -1).

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The decimal fraction 3.003 can be expressed as the sum of fractions 3,003/1,000.

To express the decimal fraction 3,003 as a sum of fractions with a denominator that is a power of 10, we first need to determine the number of decimal places in the fraction. In this case, there are three decimal places, so we can write:

3,003 = 3 + 0.0 0 3

To express 0.003 as a fraction, we can write it as:

0.003 = 3/1000

So, we can write:

3,003 = 3 + 3/1000

To express this as a fraction with a denominator that is a power of 10, we can write:

3,003 = 3,000/1,000 + 3/1,000

Simplifying this expression, we get:

3,003 = 3,000/1,000 + 3/1,000 = (3,000 + 3)/1,000 = 3,003/1,000

Therefore, the decimal fraction 3,003 can be expressed as a sum of fractions with a denominator that is a power of 10 as 3,003/1,000.

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Kaden will earn $729.38 in interest in 30 months.

First, we need to calculate the annual interest rate equivalent to 2.5% for 1 year:

[tex]1 + r = (1 + 0.025)^1[/tex]

1 + r = 1.025

r = 0.025

So, Kaden's account earns 0.025 or 2.5% interest per year.

Next, we can calculate the amount of interest Kaden will earn in 30 months, which is 2.5 years:

n = 2.5 (number of years)

P = $5,000 (principal)

r = 0.025 (annual interest rate)

We can use the compound interest formula to calculate the final amount A:

[tex]A = P(1 + r)^n[/tex]

[tex]A = 5000(1 + 0.025)^2^.^5[/tex]

A = $5,729.38

The interest earned is the difference between the final amount and the principal:

Interest = A - P

Interest = $5,729.38 - $5,000

Interest = $729.38

Therefore, Kaden will earn $729.38 in interest in 30 months.

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Answer:

To find the number of smaller pieces that can be obtained, we need to divide the total length of the wood by the length of each smaller piece:

Number of pieces = Total length ÷ Length of each piece

Number of pieces = 7.2 m ÷ 0.12 m

Number of pieces = 60

Therefore, 60 smaller pieces can be obtained from the 7.2 m long piece of wood.

Answer:

60

Step-by-step explanation:

To determine how many smaller pieces can be obtained from a 7.2 m long piece of wood, we need to divide the total length of the wood by the length of each smaller piece.

Total length of wood = 7.2 m

Length of each smaller piece = 0.12 m

Number of smaller pieces = Total length of wood / Length of each smaller piece

Number of smaller pieces = 7.2 m / 0.12 m

Number of smaller pieces = 60

Therefore, 60 smaller pieces can be obtained from a 7.2 m long piece of wood, with each smaller piece being 0.12 m in length.

The score that marks the 11% of all scores is approximately 56.875 .

To find the score that marks the 11% of all scores, we need to use the standard normal distribution table, also known as the Z-table, since the given distribution is a normal distribution.

The first step is to find the Z-score that corresponds to the 11th percentile, which is given by: Z = invNorm(0.11) ≈ -1.225

Here, "invNorm" represents the inverse of the standard normal cumulative distribution function, which can be computed using statistical software or a calculator.

The second step is to use the Z-score formula to find the raw score that corresponds to this Z-score:Z = (X - μ) / σ

where X is the raw score we want to find, μ is the mean of the distribution, and σ is the standard deviation. Plugging in the values we have:

-1.225 = (X - 63) / 5

Solving for X, we get:

X = -1.225 * 5 + 63 = 56.875

Therefore, the score that marks the 11% of all scores is approximately 56.875

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The value of x is 5/3.

A fraction is a part of a whole. In arithmetic, the number is expressed as a quotient, in which the numerator is divided by the denominator.

We have equation in fraction are:

[tex]\frac{3x-2}{3x+1} = \frac{1}{2}[/tex]

To solve the value of x

In the above equation, Solve by cross multiplication:

2(3x - 2) = 3x + 1

Open the bracket and multiply by 2 :

6x - 4 = 3x +1

Combine the like terms:

6x - 3x = 1 + 4

Add and subtract the terms:

3x = 5

x = 5/3

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The location of the point Kristin will add to represent the 13 bottles of water sold at 39 degrees Fahrenheit is the bottom left of the scatter plot.

A scatter plot represents the relationship between two variables. In this case, the temperature (independent variable) is plotted along the x-axis, while the number of bottles of water sold (dependent variable) is plotted along the y-axis. As the temperature increases, it is expected that more bottles of water would be sold.

The bottom left area of the scatter plot is where lower values of both temperature and the number of bottles sold would be found. Since 39 degrees Fahrenheit is relatively low and 13 bottles of water is a lower quantity, the point representing this data will be in the bottom left quadrant of the scatter plot.

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Complete question:

Kristin made a scatter plot to represent the relationship between the temperature and the number of bottles of water sold at her concession stand at a soccer tournament. Which is a description of the location of the point Kristin will add to represent the 13 bottles of water that were sold when the temperature was 39 degrees Fahrenheit? Select one:

O  Top left of the scatter plot

O  Bottom right of the scatter plot

O  Top right of the scatter plot

O  Bottom left of the scatter plotâ

The equation of the line passing through the given points is y = 3x-1.

Given that is a line passing through two points (0, 2) and (-1, -1) we need to find the equation of the line using them,

We know that the equation of a line passing through points (x₁, y₁) and (x₂, y₂) is =

y-y₁ = y₂-y₁ / x₂-x₁ (x-x₁)

Here (x₁, y₁) and (x₂, y₂) are (0, 2) and (-1, -1),

Therefore, the required equation is =

y+1 = -1-2/-1 (x-0)

y+1 = 3x

y = 3x-1

Hence, the equation of the line passing through the given points is y = 3x-1.

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A function of x that performs the operations  y = f(x) = 6x^9 + 4, the inverse to the function found in part (a). x = g (y) =  ((y - 4) / 6)^(1/9)

The function that performs the operations of raising x to the ninth power, multiplying by 6, and adding 4 is

f(x) = 6x^9 + 4

To find the inverse function, we need to solve for x in terms of y

y = 6x^9 + 4

Subtract 4 from both sides

y - 4 = 6x^9

Divide both sides by 6

(x^9) = (y - 4) / 6

Take the ninth root of both sides

x = ((y - 4) / 6)^(1/9)

Therefore, the inverse function is

g(y) = ((y - 4) / 6)^(1/9)

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The missing values can be found by setting up proportions for each of the ratios whose values are given. The completed table is shown below:

x              17       1/3       11

y               5.67     3.67     1.21

Ratio y/x   3.67          1/3      0.11

In order to find the missing values of y, we can use the given ratios to set up proportions:

For the first ratio:

y/x = 5.67/17

y = (5.67/17) * x

y = (5.67/17) * 11

y = 3.67

So the first missing value of y is 3.67.

For the second ratio:

y/x = 1/3

y = (1/3) * x

y = (1/3) * 1/3

y = 0.11

Therefore, the second missing value of y is  found as 0.11.

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The area of one garden using each side of the lounging area is 15 feet long is equal to 56.25 square feet.

Shape of the garden landscape is square.

If the lounging area is a square with sides of length 15 feet,

Area of lounging area

= (15 feet) × (15 feet)

= 225 square feet

Four congruent sections of the landscape plan .

Three indoor gardens and one walkway.

Divide the lounging area into four equal square sections.

Each of the congruent sections has an area equal to,

Area of lounging area = 4 × area of one garden

Let's call the area of one garden be x.

⇒225 = 4x

Solving for x, we divide both sides by 4

⇒x = 225/4

⇒x  = 56.25 square feet

Therefore, the area of one garden is 56.25 square feet.

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The value of 'a' remains the same, 'h' is equal to -b/(2a), and 'h' and 'k' cannot both equal zero.

When rewriting a quadratic expression of the form y = ax^2 + bx + c into the vertex form y = a(x - h)^2 + k, the following must be true:

1. The value of 'a' remains the same in both expressions, as it represents the parabola's vertical stretch or compression.

2. 'h' is equal to -b/(2a), which is derived from completing the square to transform the standard form into the vertex form.

3. 'k' and 'c' do not necessarily have the same value. 'k' is the value of the quadratic function when 'x' equals 'h', which can be found by substituting 'h' back into the original equation and solving for 'y'.

4. 'h' and 'k' cannot both equal zero, unless the vertex of the parabola is at the origin (0,0).

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The per capita birth rate of the given population of manned wolves 246 with number of birth as 83 is equals to 0.34  births per individual per year.

Number of births = 83

Initial population = 246

Time period = 1 year

Number of deaths = 42

The per capita birth rate is calculated as the number of births per individual in the population.

Typically expressed as a rate per unit time.

Per capita birth rate as follows,

Per capita birth rate

= (Number of births / Initial population) × (Time period / 1 year)

Substituting these values into the formula, we get,

Per capita birth rate

= (83 / 246) × (1 / 1)

= 0.3374

Rounding this to the hundredths place, we get,

Per capita birth rate = 0.34

Therefore, the per capita birth rate for this population of maned wolves is 0.34 births per individual per year.

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To find dx dy, we need to take the derivative of y with respect to x. Using the chain rule, we have:

dy/dx = d(Vx)/dx = V * d(x)/dx + x * d(V)/dx

Since we are given y = Vx, we can substitute and simplify:

dy/dx = y/x * d(V)/dx + V

Now we can plug in the given values: y = 8, dt/dx = 1.75. We also need to find V:

y = Vx, so V = y/x = 8/x

Now we can substitute and simplify again:

dy/dx = 8/x * d(V)/dx + 8/x

We need to find d(V)/dx. We know that t = f(x,V), so we can use the chain rule again:

dt/dx = df/dx + df/dV * dV/dx

Since t and V are independent, df/dV = 0. So we have:

dt/dx = df/dx + 0 * dV/dx

dt/dx = df/dx

We also know that dt/dx = 1.75. Therefore:

1.75 = df/dx

Now we can find d(V)/dx:

d(V)/dx = d/dx (y/x) = (dy/dx * x - y * dx/dx) / x^2

Since y = 8, we have:

d(V)/dx = (dy/dx * x - 8) / x^2

Substituting what we know, we get:

d(V)/dx = (8/x * 1.75 - 8) / x^2 = 8(1.75 - x) / x^3

Now we can substitute everything into the formula we derived earlier:

dy/dx = 8/x * d(V)/dx + 8/x

dy/dx = 8/x * (8(1.75 - x) / x^3) + 8/x

Simplifying, we get:

dy/dx = 14/x^2 - 1.75

Therefore, when y = 8 and dt/dx = 1.75, dx/dy = 1/(dy/dx) is:

dx/dy = 1 / (14/x^2 - 1.75) = x^2 / (14 - 1.75x^2)
Given y = √x, we first need to find dy/dx, the derivative of y with respect to x. Using the power rule, we can rewrite y = x^(1/2), and the derivative will be:

dy/dx = (1/2)x^(-1/2)

Now, we are given that y = 8, so we need to find the corresponding value of x:

8 = √x
64 = x

Next, we are given dt/dx = 1.75. We need to find dt/dy, which can be calculated by taking the reciprocal of dy/dx:

dt/dy = 1 / (dy/dx)

Now, we substitute x = 64 into the derivative:

dy/dx = (1/2)(64)^(-1/2) = (1/2)(8)^(-1) = 1/16

Finally, we can find dt/dy by taking the reciprocal of dy/dx:

dt/dy = 1 / (1/16) = 16

So, the value of dt/dy when y = 8 and dt/dx = 1.75 is 16.

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Answer:

(-13)^3

Step-by-step explanation:

Exponents can be used for repeated multiplication.

In this case, the number "negative 13" is repeated several times, all connected with multiplication.

There are a total of three "negative 13"s being multiplied together ("negative 13" appears three times on the page).

To rewrite using exponents, we would write one of the following:

(-13)^3

[tex](-13)^3[/tex]

Answer:

Step-by-step explanation:

3x+y = -6

y=-3x-6

The Y-intercept is -6, therefore there is one point (0,-6)

Now go 3 down and one right, so there is your second point (1,-9)

The wood flooring for the floor will cost $14,175.

To calculate the cost of replacing the wooden floor at Greg's gym, we first need to find the area of the rectangular floor. The area of a rectangle can be found using the formula: area = length × width. In this case, the length is 45 feet and the width is 35 feet.

Area = 45 feet × 35 feet = 1575 square feet

Since the cost of wood flooring is $9 per square foot, we can now calculate the total cost:

Total cost = area × cost per square foot = 1575 square feet × $9/square foot = $14,175

So, the wood flooring will cost Greg $14,175 to replace the floor at his gym.

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When we convert the given 54 square feet into square yards we get 6 square yards by using two unit multipliers to convert.

A fraction that equals 1 and is used to convert one set of units to another one is a unit multiplier. The fraction numerator and denominator contain equivalent measurements in different units.

We need to convert the 54 square feet to square yards by using two unit multipliers. by using the two-unit multipliers

given standards :

1 yard = 3 feet

1 square yard = 9 square feet

To convert the 54 square feet to square yards we need to multiply 54 square feet by two unit multipliers which are (1 yard / 3 feet) and (1 yard / 3 feet). Then the equation can be written as:

= 54 square feet × (1 yard / 3 feet) × (1 yard / 3 feet)

= 6 square yards

Therefore, 54 square feet is equal to 6 square yards.

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To convert to polar coordinates, we need to express x and y in terms of r and θ. We have:

x = r cos θ

y = r sin θ

Also, we need to change the limits of integration. The region of integration is the circle centered at the origin with radius 8, so we have:

-π/2 ≤ θ ≤ π/2 (for the upper half of the circle)

0 ≤ r ≤ 8

Now we can express the integrand in terms of r and θ:

[tex]x^2 + y^2 = r^2[/tex] (by Pythagoras)

[tex]20(x^2 + y^2) = 20r^2[/tex]

So the integral becomes:

∫-π/2π/2∫[tex]08r^3 cos^2 θ sin θ dr dθ[/tex]

We can simplify cos^2 θ sin θ using the identity cos^2 θ sin θ = (1/3)sin^3 θ, so we get:

∫-π/2π/2∫[tex]08r^3 (1/3)sin^3 θ dr dθ[/tex]

The integral with respect to r is easy to evaluate:

∫0^8r^3 dr = (1/4)8^4 = 2048

The integral with respect to θ is also easy to evaluate using the fact that sin^3 θ is an odd function:

∫-π/2π/2(1/3)[tex]sin^3[/tex] θ dθ = 0

Therefore, the value of the iterated integral is:

2048(0) = 0

The volume of the solid is zero. This makes sense because the integrand is an odd function of y (or sin θ) and the region of integration is symmetric with respect to the x-axis.

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The work done by the force field F along the semicircle and the line segment is 32/3.

The work done by a force field F along a curve C from point A to point B is given by the line integral:

W = ∫ F dot dr

where dot represents the dot product and dr is the differential displacement vector along the curve C.

Let's divide the curve C into two parts: the semicircle from P to Q, denoted by C1, and the line segment from Q to P, denoted by C2.

For C1, the curve can be parameterized as x = 2cos(t) and y = 2sin(t) for t in [0, pi]. The differential displacement vector is then given by:

dr = (-2sin(t) dt)i + (2cos(t) dt)j

The force field F(x,y) = x^2i - ryj, so we have:

F(x,y) = (2cos^2(t))i - (2rsin(t))j

The dot product F dot dr is then:

F dot dr = (2cos^2(t))(-2sin(t) dt) + (2rsin(t))(2cos(t) dt)

= -4cos^2(t)sin(t) dt + 4rcos(t)sin(t) dt

= 4sin(t)cos(t)(r - cos(t)) dt

Therefore, the work done along C1 is:

W1 = ∫ C1 F dot dr

= ∫[0, pi] 4sin(t)cos(t)(r - cos(t)) dt

This integral can be evaluated using the substitution u = cos(t), du = -sin(t) dt:

W1 = -∫[1, -1] 4u(r - u) du

= 4r∫[1, -1] u du - 4∫[1, -1] u^2 du

= 0

Hence, the work done along C1 is 0.

For C2, the curve is simply the line segment from Q(-2,0) to P(2,0), which is parallel to the x-axis. Therefore, the differential displacement vector is given by:

dr = dx i

where i is the unit vector in the x-direction. The force field is the same as before, F(x,y) = x^2i - ryj. Along C2, y = 0, so the force field reduces to:

F(x,y) = x^2i

The dot product F dot dr is then:

F dot dr = x^2 dx

Therefore, the work done along C2 is:

W2 = ∫ C2 F dot dr

= ∫[-2, 2] x^2 dx

= 32/3

Hence, the work done along C2 is 32/3.

The total work done along the curve C is the sum of the work done along C1 and C2:

W = W1 + W2 = 0 + 32/3 = 32/3

Therefore, the work done by the force field F along the semicircle and the line segment is 32/3.

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From the given graph, the roots of the quadratic equation, 0 = x² - 6x + 5, is 1 and 5. The correct option is the last option x= 1, 5

From the question, we are to determine the roots of the quadratic equation from the provided graph.

From the given information,

The given quadratic equation is

0 = x² - 6x + 5

The roots of a quadratic function are the values of x where the function equals zero. On a graph, this corresponds to the points where the graph intersects the x-axis.

From the graph, we will read the x-coordinates of the points where the graph intersects the x-axis.

From the given graph, the x-coordinates of the points where the graph intersects the x-axis are 1 and 5

Hence, the roots of the quadratic equation is 1 and 5

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First, using the Pythagorean theorem, we get the hypotenuse = 12.

sin = opposite/hypotenuse = [tex]\frac{6\sqrt{3} }{12} = \frac{\sqrt{3} }{2}[/tex]

cos = adjacent/hypotenuse = [tex]\frac{6}{12} =\frac{1}{2}[/tex]

tan = opposite/adjacent = [tex]\frac{6\sqrt{3} }{6} = \sqrt{3}[/tex]

csc = hypotenuse/opposite = [tex]\frac{12}{6\sqrt{3} } =\frac{2}{\sqrt{3} }[/tex]

sec = hypotenuse/adjacent = [tex]\frac{12}{6} =2[/tex]

cot = adjacent/opposite = [tex]\frac{6}{6\sqrt{3} } = \frac{1}{\sqrt{3} }[/tex]

If measure of two "vertically-opposite-angles" are 115° and (2x + 5)°, then the equation to find value of "x" is 115° = (2x + 5)°,and value of "x" is 55.

The Vertically opposite angles are defined as a pair of non-adjacent angles formed by the intersection of two lines. and if the two angles are vertically opposite then their measures are equal, so, to find the value of "x", we equate the measure of both the angles,

The measure of the two angles are 115° and (2x + 5)°,

So, on equating,

We get,

⇒ 115° = (2x + 5)°,

⇒ 110° = 2x,

⇒ x = 55,

Therefore, the value of x is 55.

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The given question is incomplete, the complete question is

The measure of the two vertically opposite angles are 115° and (2x + 5)°, Write an equation that can be used to find the value of x. What is the value of x?

The area of the sidewalk in square meters is 69π.

To find the area of the sidewalk, we need to subtract the area of the flower bed from the area of the outer circle formed by the sidewalk.

First, we need to find the area of the flower bed. We know that the diameter of the flower bed is 20 m, so the radius is half of that, which is 10 m. We can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.

Area of flower bed = π(10m)^2 = 100π square meters

Next, we need to find the area of the outer circle formed by the sidewalk. Since the sidewalk is 3 m wide, the radius of the outer circle will be 10 + 3 = 13 m (10 m for the flower bed radius plus 3 m for the width of the sidewalk).

Area of outer circle = π(13m)^2 = 169π square meters

Finally, we can find the area of the sidewalk by subtracting the area of the flower bed from the area of the outer circle:

Area of sidewalk = Area of outer circle - Area of flower bed
Area of sidewalk = (169π) - (100π)
Area of sidewalk = 69π square meters

Therefore, the area of the sidewalk in square meters is 69π, or approximately 216.6 square meters (if we use 3.14 for π).

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Answer:

Step-by-step explanation:

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2500 = a1 * r^4

a8 = a1 * r^(8-1)

312500 = a1 * r^7

We can divide the second equation by the first equation to eliminate a1:

312500 / 2500 = (a1 * r^7) / (a1 * r^4)

125 = r^3

Taking the cube root of both sides gives us:

r = 5

Now that we know the common ratio, we can use either of the two original equations to find the first term, a1. Using the first equation:

250

The function has a horizontal asymptote at y = 3. Other types of discontinuities can also occur in rational functions

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

A rational function can have a discontinuity at any point where the denominator of the function becomes zero since division by zero is undefined. These points are called "vertical asymptotes."

For example, consider the rational function f(x) = (x² - 1) / (x - 1). The denominator becomes zero when x = 1, which causes a vertical asymptote at x = 1. At x = 1, the function approaches positive infinity from the left-hand side and negative infinity from the right-hand side. This creates a "hole" or a "removable discontinuity" in the graph of the function.

Another type of discontinuity that can occur in a rational function is a "horizontal asymptote." This occurs when the degree of the numerator is less than the degree of the denominator. In this case, the function approaches a horizontal line (the horizontal asymptote) as x approaches infinity or negative infinity.

For example, consider the rational function f(x) = (3x² - 2x + 1) / (x² + 1). As x approaches infinity or negative infinity, the function approaches the horizontal line y = 3.

Therefore, the function has a horizontal asymptote at y = 3.

Other types of discontinuities can also occur in rational functions, such as "slant asymptotes" or "oscillating behavior," but these are less common and typically require more advanced techniques to identify.

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