Determine whether the following equation defines y as a function of x: x^2 + y^2 = 169. How was this solved? Where does the 2 (pictured) come from?

The volume generated by rotating the region bounded by the curves [tex]$x = \sqrt{y}$[/tex], [tex]$x = 0$[/tex], and [tex]$y = 4$[/tex] about the x-axis using the method of cylindrical shells is [tex]$4\pi$[/tex] cubic units.

The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.

To find the volume generated by rotating the region bounded by the curves [tex]$x = \sqrt{y}$[/tex], [tex]$x = 0$[/tex], and [tex]$y = 4$[/tex] about the x-axis using the method of cylindrical shells, we need to follow these steps:

Sketch the region and the axis of rotation. The region bounded by the curves [tex]$x = \sqrt{y}$[/tex], [tex]$x = 0$[/tex], and [tex]$y = 4$[/tex] is a quarter-circle with radius 2 centered at the origin. The axis of rotation is the x-axis.

Choose a vertical strip with width $\Delta x$ that runs parallel to the y-axis and intersects the region. The height of this strip is given by the equation $y = 4 - x^2$.

Imagine rotating this strip around the x-axis to form a cylindrical shell with thickness [tex]$\Delta x$[/tex], height [tex]$4 - x^2$[/tex], and radius [tex]$x$[/tex].

The volume of this cylindrical shell is given by the formula [tex]$V = 2\pi x(4-x^2)\Delta x$[/tex].

To find the total volume of the solid, we need to add up the volumes of all the cylindrical shells. This can be done by taking the limit as the width of the strips $\Delta x$ approaches zero and summing up the volumes of the resulting shells. This gives us the integral:

[tex]$V = \int_{0}^{2} 2\pi x(4-x^2) dx$[/tex]

We can simplify this integral by expanding the expression inside the parentheses, which gives us:

[tex]$V = \int_{0}^{2} 8\pi x - 2\pi x^3 dx$[/tex]

We can then integrate each term separately to get:

[tex]$V = [4\pi x^2 - \frac{1}{2}\pi x^4]_{0}^{2}$[/tex]

[tex]$V = (4\pi \cdot 2^2 - \frac{1}{2}\pi \cdot 2^4) - (4\pi \cdot 0^2 - \frac{1}{2}\pi \cdot 0^4)$[/tex]

[tex]$V = 8\pi - 4\pi = 4\pi$[/tex]

Therefore, the volume generated by rotating the region bounded by the curves [tex]$x = \sqrt{y}$[/tex], [tex]$x = 0$[/tex], and [tex]$y = 4$[/tex] about the x-axis using the method of cylindrical shells is [tex]$4\pi$[/tex] cubic units.

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The system of equations modeling the number of imports and the number of exports is given as follows:

The solution f(x) = g(x) represents the month at which the number of imports and the number of exports was the same.

The system of equations is modeled by linear functions in slope-intercept form, as follows:

y = mx + b.

In which:

Hence the equations are given as follows:

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Answer:

The number of additional streetlights installed every week is 2, and this will continue for 52 weeks. Therefore, the total number of streetlights added during this period will be:

2 streetlights/week x 52 weeks = 104 streetlights

If the city currently has 138 streetlights, then after 30 weeks, the total number of streetlights will be:

138 streetlights + 2 streetlights/week x 30 weeks = 198 streetlights

Therefore, the city will have 198 streetlights at the end of 30 weeks.

Step-by-step explanation:

1. The problem states that the city currently has 138 streetlights.

2. The city council has decided to install 2 additional streetlights at the end of each week for the next 52 weeks. This means that every week, the city will add 2 streetlights to its total.

3. To find out how many streetlights will be added in total over the 52-week period, we can multiply the number of additional streetlights per week (2) by the number of weeks (52):

2 streetlights/week x 52 weeks = 104 streetlights

Therefore, over the 52-week period, the city will add 104 streetlights.

4. To find out how many streetlights the city will have at the end of 30 weeks, we need to calculate how many additional streetlights will be added during this time period. Since the city is adding 2 streetlights per week, we can multiply 2 by the number of weeks (30):

2 streetlights/week x 30 weeks = 60 streetlights

This means that over the course of 30 weeks, the city will add 60 streetlights.

5. To find out the total number of streetlights at the end of 30 weeks, we need to add the current number of streetlights to the number of streetlights added during the 30-week period. Therefore, we can add 138 (the current number of streetlights) and 60 (the number of streetlights added during the 30-week period):

138 streetlights + 60 streetlights = 198 streetlights

Therefore, the city will have 198 streetlights at the end of 30 weeks.

Since the ratio of Y to X is not constant for different values of X, the relationship between Y and X is not proportional, and there is no constant of proportionality.

Proportion is a mathematical concept that expresses the relationship between two or more quantities that are related to each other in a consistent way. A proportion can be expressed as an equation in which two ratios are set equal to each other, and it states that the ratios of two quantities are always the same. In other words, if we know that two quantities are proportional, we can use this relationship to find one quantity given the other.

Here,

Based on the given table, we can see that the number of cookies (Y) is directly proportional to the number of packages (X). Thus, we can write:

Y = kX

where k is the constant of proportionality.

a) When k = 6, we can find the value of Y for different values of X as follows:

When X = 12, Y = kX = 6(12) = 72

When X = 10, Y = kX = 6(10) = 60

When X = 8, Y = kX = 6(8) = 48

When X = 6, Y = kX = 6(6) = 36

When X = 4, Y = kX = 6(4) = 24

When X = 2, Y = kX = 6(2) = 12

Thus, the constant of proportionality is k = 6.

b) When k = 12, we can find the value of Y for different values of X as follows:

When X = 12, Y = kX = 12(12) = 144

When X = 10, Y = kX = 12(10) = 120

When X = 8, Y = kX = 12(8) = 96

When X = 6, Y = kX = 12(6) = 72

When X = 4, Y = kX = 12(4) = 48

When X = 2, Y = kX = 12(2) = 24

Thus, the constant of proportionality is k = 12.

c) When k = 1/6, we can find the value of Y for different values of X as follows:

When X = 12, Y = kX = (1/6)(12) = 2

When X = 10, Y = kX = (1/6)(10) = 5/3

When X = 8, Y = kX = (1/6)(8) = 4/3

When X = 6, Y = kX = (1/6)(6) = 1

When X = 4, Y = kX = (1/6)(4) = 2/3

When X = 2, Y = kX = (1/6)(2) = 1/3

Thus, the constant of proportionality is k = 1/6.

e) Since the ratio of Y to X is not constant for different values of X, the relationship between Y and X is not proportional, and there is no constant of proportionality.

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Answer: 0

Step-by-step explanation:

1. Bring the Variables to one side and the constants to the other (3n + 2n + 3n = 7 - 7)

2. Solve ( 8n = 0, n = 0/8, n = 0)

Answer:

459..

hope you understand

A line of best fit, also known as a regression line, is a straight line that best represents the relationship between two variables in a scatter plot. The association between the variables is strong.

To determine whether there is a strong association between the two variables, you need to examine the pattern of the data points in relation to the line of best fit.

If the data points are tightly clustered around the line of best fit, this indicates a strong association between the two variables. On the other hand, if the data points are more spread out and do not follow a clear pattern, this suggests a weak association.

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The transformation that converts the graph of f(x) = x - 1 into the graph of g(x) = 9x - 9 is a vertical stretch by a factor of 9, followed by a vertical shift downwards by 8 units.

What are some instances of graphs?

Some instances of graphs include line graphs, bar graphs, scatter plots, pie charts, and histograms, which are visual representations of data or mathematical functions.

To see this, we can rewrite g(x) as:

g(x) = 9(x - 1) = 9x - 9

This shows that g(x) is a vertical stretch of f(x) by a factor of 9, which means that every y-value of g(x) is 9 times the corresponding y-value of f(x). The horizontal axis remains the same.

Next, we can see that g(x) is shifted downward by 8 units compared to f(x). This is because g(x) has a y-intercept of -9, while f(x) has a y-intercept of -1. This means that every y-value of g(x) is 8 units less than the corresponding y-value of f(x).

Therefore, the transformation that converts the graph of f(x) = x - 1 into the graph of g(x) = 9x - 9 is a vertical stretch by a factor of 9, followed by a vertical shift downwards by 8 units.
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Answer:

  (c)  42.9 ft²

Step-by-step explanation:

You want the area of the right triangle with hypotenuse 17.3 ft and long side 16.5 ft.

The Pythagorean theorem tells you the relationship between the side lengths is ...

  c² = a² +b²

The missing side, b, can be found as ...

  b² = c² -a²

  b = √(c² -a²) = √(17.3² -16.5²) = √27.04 = 5.2

The area of the triangle is ...

  A = 1/2bh

  A = 1/2(16.5 ft)(5.2 ft) = 42.9 ft²

The area of the triangle is 42.9 square feet.

Answer: B

V=957.7 cubic inches

Step-by-step explanation:

V=Hxpixr^2

V=12.2x3.14x5^2           r=5 because it is half of the diamter, which is 10.

V=12.2x3.14x25

V=957.7 cubic inches

Answer:

First, we need to evaluate g(-1):

g(-1) = (-1)^2 - 7(-1) - 9

= 1 + 7 - 9

= -1

Now, we can plug in g(-1) into f(x):

f(g(-1)) = f(-1)

= -1 + 8

= 7

Therefore, f(g(-1)) = 7.

Hypotenuse equals opposite/tan(45°) = 300/1 = 300 feet. Boat A is therefore 300 feet horizontally away from the lighthouse.

The numerical measurement of distance is the separation between two objects or points. Typically, it is expressed in terms of metres, kilometres, miles, or feet. Distances can be calculated either along a simple path (the Euclidean distance) or along a more complicated one (such as a curved surface or a circuitous route). Distance is a crucial notion that is used to define the separation between things, the length of a route or path, and the displacement of an object through time in many disciplines, including physics, mathematics, geography, and engineering.

given

We may utilise trigonometry and create some equations based on the provided angles and distances to solve this problem.

a) We can use the tangent function to calculate the separation between boat A and the lighthouse:

opposite/hypotenuse = tan(45°)

where the hypotenuse is the distance between boat A and the lighthouse, the other side is the lighthouse's height (i.e., the horizontal distance we want to find). By calculating the hypotenuse, we obtain:

Hypotenuse equals opposite/tan(45°) = 300/1 = 300 feet. Boat A is therefore 300 feet horizontally away from the lighthouse.

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The dοtted line is the line y=13x3, and the shaded area is abοve it.

As is well knοwn, the equatiοn fοr a straight line is y = mx + c. Where 'c' is the y-axis intercept and 'm' is the slοpe οn the y-axis. The fοrmula fοr a hοrizοntal line is y = 3.

In οrder tο graph the inequality y≥13x−3, we can first plοt the line y=13x3

(which has a slοpe οf 13 and a y-intercept οf -3) as a dοtted line.

Since the inequality is y≥13x−3, we must shade the area abοve the line.

The dοtted line is the line y=13x3, and the shaded area is abοve it.

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The scatter graph is used to show correlation between variables

A scatter graph, also known as a scatter plot or scatter chart, is a type of data visualization tool that uses coordinates to display the values of two variables for a set of data.

In a scatter graph, each point on the graph represents a single data point, with one variable plotted on the x-axis and the other variable plotted on the y-axis.

The position of each point is determined by the values of the two variables for that data point.

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The value of the expressions are 1. Distributive property: -3 √3 (2 + √6) = -6 √3 - 9 √2. [2] (-2 √3 + 2)(√3 - 5) = 12 √3 - 16. [3] (-2 -3 √5)(5 - √5) = -15 √5 + 5.

The distributive property in algebra says that the sum or difference of the products of the number and each term in the sum or difference is equal to the product or difference of the number and each phrase in the sum or difference. To put it another way, a(b + c) = ab + ac and a(b - c) = ab - ac are true for any value of a, b, and c. The distributive property is frequently employed in parenthetical expansion and algebraic simplification.

1. Distributive property:

-3 √3 (2 + √6)

= -6 √3 - 3 √3 √6

= -6 √3 - 3 √18

= -6 √3 - 9 √2

2. Multiplying:

(-2 √3 + 2)(√3 - 5)

= -2 √3 (√3) + 2 (√3) - 10 + 10 √3

= -2 (3) - 10 + 12 √3

= 12 √3 - 16

3. Multiplying:

(-2 -3 √5)(5 - √5)

= -2 (5) - 3 √5 (5) + 2 √5 + 15

= -10 - 15 √5 + 2 √5 + 15

= -15 √5 + 5

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Therefore, the number of animals at the start of year 3 is approximately 408.04.

An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.

Here,

We are given that the population at the start of year 1 (P1) is 400. Using the recurrence relation Pt + 1 = 1.01Pt, we can find the population at the start of year 2 (P2) and year 3 (P3) as follows:

P2 = 1.01 * P1 = 1.01 * 400 = 404

P3 = 1.01 * P2 = 1.01 * 404 = 408.04

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a) The probability that it will have 2 or fewer defects is: 0.55 = 55%.

b) The probability it will have 4 or more defects is: 0.25 = 25%.

c) The probability it will have between 1 and 3 defects is: 0.63 = 63%.

d) The expected number of defects is: 2.38.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

For this problem, we are given the frequencies, hence we must identify the outcomes and then add their frequencies.

The probability that it will have 2 or fewer defects is obtained as follows:

P(X = 0) + P(X = 1) + P(X = 2) = 0.12 + 0.18 + 0.25 = 0.55.

The probability it will have 4 or more defects is obtained as follows:

P(X = 4) + P(X = 5) = 0.15 + 0.10 = 0.25.

The probability it will have between 1 and 3 defects is obtained as follows:

P(X = 1) + P(X = 2) + P(X = 3) = 0.18 + 0.25 + 0.20 = 0.63.

The expected value is given by the sum of each value multiplied by it's respective probability, hence:

E(X) = 0 x 0.12 + 1 x 0.18 + 2 x 0.25 + 3 x 0.20 + 4 x 0.15 + 5 x 0.1

E(X) = 2.38.

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1 )Use the algorithm method.

            5 0

×         2 1 2

          1

          100

         500

    1

     10000

     10600

2) Therefore,50×212=10600

              10600

Answer:

a

Step-by-step explanation:

16÷4 is 4 and 12÷4 is 3. I matched up the angles.

After calc, the lenght of AC segment is 3 cm. Alternative A

The similarity of triangles property has a way of calculating how much an unknown segment of the triangle is worth through measures of available equivalent segments of the two triangles and matching angles.

[tex]\begin{array}{l}\raisebox{8pt}{$\sf \dfrac{AC}{BC}=\dfrac{DF}{EF}$}\\\raisebox{8pt}{$\sf \dfrac{AC}{4}=\dfrac{\red{\diagup\!\!\!\!\!\!12}^{\div4}}{\red{\diagup\!\!\!\!\!\!16}^{\div4}}$}\\\raisebox{8pt}{$\sf \dfrac{AC}{\red{\diagup\!\!\!\!4}}=\dfrac{3}{\red{\diagup\!\!\!\!4}}$}\\\bf\therefore AC=3\end{array}[/tex]

Found the length of segment AC, which is 3cm

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The equation of the lines parallel or perpendicular to another line and that pass through a point are listed below:

Case 7: y = (7 / 4) · x + 3

Case 8: y = 2 · x + 1

Case 9: y = (3 / 2) · x + 2

Case 10: y = (6 / 5) · x - 1

Case 11: y = - (2 / 5) · x

Case 12: y = - (5 / 2) · x - 19 / 2

Case 13: y = - x + 3

Case 14: y = - (5 / 4) · x - 5

In this problem we have eight cases of line equations parallel or perpendicular to a line and that pass through a point. The equation of a line is an expression of the form:

y = m · x + b

Where:

There are the following relationships between two lines:

Now we proceed to determine the line equations:

Case 7

Slope

m = 7 / 4

m' = 7 / 4

Intercept

b = y - m · x

b = - 4 - (7 / 4) · (- 4)

b = - 4 + 7

b = 3

Equation of the line

y = (7 / 4) · x + 3

Case 8

Slope

m = 2

m' = 2

Intercept

b = 1 - 2 · 0

b = 1

Equation of the line

y = 2 · x + 1

Case 9

Slope

m = 3 / 2

m' = 3 / 2

Intercept

b = - 4 - (3 / 2) · (- 4)

b = - 4 + 6

b = 2

Equation of the line

y = (3 / 2) · x + 2

Case 10

Slope

m = - 5 / 6

m' = 6 / 5

Intercept

b = 5 - (6 / 5) · 5

b = 5 - 6

b = - 1

Equation of the line

y = (6 / 5) · x - 1

Case 11

Slope

m = 5 / 2

m' = - 2 / 5

Intercept

b = - 2 - (- 2 / 5) · 5

b = 0

Equation of the line

y = - (2 / 5) · x

Case 12

Slope

m = 2 / 5

m' = - 5 / 2

Intercept

b = - 2 + (- 5 / 2) · 3

b = - 2 - 15 / 2

b = - 4 / 2 - 15 / 2

b = - 19 / 2

Equation of the line

y = - (5 / 2) · x - 19 / 2

Case 13

Slope

m = 1

m' = - 1

Intercept

b = 5 - (- 1) · (- 2)

b = 3

Equation of the line

y = - x + 3

Case 14

Slope

m = 4 / 5

m' = - 5 / 4

Intercept

b = 0 - (- 5 / 4) · (- 4)

b = 0 - 5

b = - 5

Equation of the line

y = - (5 / 4) · x - 5

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Answer:

From the information given, we can draw the following diagram:

   S ------- P

       / \       / \

      /   \ QR /   \

     /     \ / /     \

    /       Q       \

   R-----------------P

           MN

Here, SP || QR, so we have ∠QSP = ∠PQR and ∠SPQ = ∠QRP.

Since m∠QRS = 120, we have m∠QRP = 360 - m∠QRS = 360 - 120 = 240.

Since m∠QPS = 135 and ∠QSP = ∠PQR, we have m∠PQR = 360 - m∠QPS = 360 - 135 = 225.

Let x = MN. Since MN is the median of the trapezoid, we have MP = NR = x.

Now, consider the triangles SPQ and RPQ. We have:

tan(∠SPQ) = x/12 (using the tangent ratio in triangle SPQ)

tan(∠RPQ) = x/12 (using the tangent ratio in triangle RPQ)

Since ∠SPQ = ∠RPQ (as they are corresponding angles), we have:

x/12 = x/12

x = 12

Therefore, MN = x = 12.

The inequalities are factorized to;

1. x > 5 and x > -1          

x > 2 and x > -6

2. x < 2

x < -4 and x < 2

3. x > 7

x> 1 and x> 3

From the information given, we have the inequalities;

(x - 5)(x + 1)> 0

expand the bracket, we get;

x² + x - 5x - 5> 0

x(x + 1) - 5(x + 1) > 0

x > 5 and x > -1

2. x-2/x + 3 < 0

cross multiply the values

x - 2< 0

make 'x' the subject

x < 2

3. x -7/x + 4 > 0

cross multiply the values

x - 7> 0

make 'x' the subject

x > 7

x² + 4x - 12 > 0

x² + 6x - 2x - 12> 0

Factorize

x(x + 6) - 2(x + 6)> 0

x > 2 and x > -6

x² - 2x - 8< 0

x² -2x + 4x - 8 < 0

Factorize

x(x - 2) + 4(x -2 )< 0

x < -4 and x < 2

x² - 4x + 3 > 0

x² - 3x - x + 3> 0

factorize

x(x - 3) - 1(x - 3)> 0

x> 1 and x> 3

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The mapping of the functions h(x) = |x|/2, f(x) = 2x + 8, and f(x) = x² - 5x + 2 is defined at;

1). -2, f(-2) = 1

2). 6, f(6) = 20

3). 2, f(2) = -4

A function is a rule that defines a relationship between one variable. It is the mapping whose codomain is the set of real numbers

For x = -2;

h(-2) = |-2|/2

h(x) = 2/2 {absolute value of -2 is 2 units}

h(x) = 1

for x = 6;

f(6) = 2(6) + 8

f(6) = 12 + 8

f(6) = 20

for x = 2;

f(2) = (2)² - 5(2) + 2

f(2) = 4 - 10 + 2

f(2) = -4

Therefore, the mapping of the functions h(x) = |x|/2, f(x) = 2x + 8, and f(x) = x² - 5x + 2 is defined at;

1). -2, f(-2) = 1

2). 6, f(6) = 20

3). 2, f(2) = -4

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The interval notation for the inequality statement "5 less than m is at most 70" is (-∞, 75].

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

The statement given is - 5 less than m is at most 70.

The given sentence can be translated into a mathematical inequality as -

m - 5 ≤ 70

To solve for m, we can add 5 to both sides of the inequality -

m - 5 + 5 ≤ 70 + 5

m ≤ 75

Therefore, m is less than or equal to 75.

In interval notation, we can represent this solution as -

(-∞, 75]

Therefore, the interval value is (-∞, 75].

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The area οf the hοt spring decreases by apprοximately 540.24 square meters.

The area οf a circle is fοund οut using the fοrmula. Area = πr²

Given that, r = d/2 = 110/2 = 55 meters

The οriginal area οf the hοt spring can be calculated as:

[tex]A = \pi r^2 = \pi(55)^2 = 9,525.69[/tex] square meters (rοunded tο twο decimal places)

After the diameter οf the hοt spring decreases by 3 meters, its new diameter is 110 - 3 = 107 meters. Therefοre, its new radius is:

r' = d'/2 = 107/2 = 53.5 meters

The new area οf the hοt spring can be calculated as:

[tex]A' = \pi r'² = \pi(53.5)² = 8,985.45[/tex] square meters (rοunded tο twο decimal places)

The decrease in the area οf the hοt spring is the difference between the οriginal area and the new area:

ΔA = A - A' = 9,525.69 - 8,985.45 = 540.24 square meters (rοunded tο twο decimal places)

Therefοre, the area οf the hοt spring decreases by apprοximately 540.24 square meters.

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