A circular hot spring has a diameter of 110 meters. Over time, the diameter
of the spring decreases by 3 meters. By how many square meters does the
area of the hot spring decrease? PLEASEE IM BEGGING ANSWER ;(

The equation of the parabola with the given focus (2, 7) and directrix y = -1 is [tex](x - 2)^2 = 12(y - 3)^2[/tex].Option C.

To find the equation of the parabola with a focus and directrix, we can use the standard form of the equation of a parabola:

For a vertical parabola:

[tex](x - h)^2 = 4p(y - k),[/tex]

where (h, k) is the vertex, and p is the distance from the vertex to the focus and directrix.

In this case, the focus is given as (2, 7), which means the vertex is also (2, 7) since the focus and vertex lie on the axis of symmetry. Additionally, the directrix is given as y = -1, which means the directrix is a horizontal line.

First, let's determine the distance from the vertex to the focus and directrix, which is the value of p. The distance is the absolute difference between the y-coordinate of the focus (7) and the y-coordinate of the directrix (-1):

p = |7 - (-1)| = 8.

Now we can substitute the values of the vertex (h, k) = (2, 7) and p = 8 into the standard form equation:

[tex](x - 2)^2 = 4(8)(y - 7).[/tex]

Simplifying further:

[tex](x - 2)^2 = 32(y - 7).[/tex]

Expanding the equation:

[tex]x^2 - 4x + 4 = 32y - 224.[/tex]

Rearranging the terms:

[tex]x^2 - 4x - 32y + 228 = 0.[/tex]

Therefore, the equation of the parabola with the given focus (2, 7) and directrix y = -1 is  [tex](x - 2)^2 = 12(y - 3)^2.[/tex] SO Option C is correct.

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By using the table of values, a graph of the exponential function is shown in the image below.

In Mathematics and Geometry, an exponential function can be modeled by using this mathematical equation:

[tex]f(x) = a(b)^x[/tex]

Where:

Based on the information provided above, we can logically deduce the following exponential function;

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

Next, we would create a table of values based on the exponential function;

when x = 0, the y-value is given by;

y = 2⁰

y = 1

when x = 1, the y-value is given by;

y = 2¹

y = 2

x                     y____

-2                 0.25

-1                  0.5

0                   1

1                    2

2                   4

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The value of measure of angle 1, angle 2, and angle 3 are,

⇒ ∠1 = 106 degree

⇒ ∠2 = 74°

⇒ ∠3 = 74°

We have to given that;

Angle in figure is,

⇒ 74°

Since, From figure,

angle 1 is,

⇒ ∠1 = 180 - 74

⇒ ∠1 = 106 degree

Hence, We get;

Measure of angle 2 is,

⇒ ∠2 = 180 - 106

⇒ ∠2 = 74°

And, Measure of angle 3 is,

⇒ ∠ 3 = ∠ 2

By definition of alternate interior angle.

⇒ ∠3 = 74°

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The measure of ∠1 is 106 because it is a straight angle with 74° angles.

The measure of ∠2 is 74 because it is a vertical angle with 74° angles.

The measure of ∠3 is 74 because ∠3 and 74° are corresponding angles.

We have,

The angles that are in the same position on a given two parallel lines intersected by a transversal line are called the corresponding angles.

Corresponding angles are always equal.

Now,

∠1 + 74 = 180

∠1 = 180 - 74

∠1 = 106

And,

∠2 and 74 are vertical angles.

So,

∠2 = 74

And,

∠3 and 74 are corresponding angles.

So,

∠3 = 74

Thus,

The measure of ∠1 is 106 because it is a straight angle with 74° angles.

The measure of ∠2 is 74 because it is a vertical angle with 74° angles.

The measure of ∠3 is 74 because ∠3 and 74° are corresponding angles.

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The rule used to determine the term value from its position in the sequence is often referred to as the "nth term" rule.

The nth-term rule involves identifying a pattern or relationship between the position (n) of a term in the sequence and the value of that term.

By analyzing the pattern, such as the common difference or common ratio, the nth-term rule allows us to express the value of any term in the sequence based on its position.

This rule provides a formula or equation that relates the position of a term to its corresponding value in the sequence.

Mathematically, the rule is:

T(n) = a + (n - 1) * d

where:

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Answer:

28 meters Aprox

Step-by-step explanation:

To find the width of the rectangular park, we can use the Pythagorean theorem. The diagonal running from one corner to the opposite corner forms a right triangle with the length and width of the park.

Given:

Length of the park (L) = 45 meters

Diagonal distance (d) = 53 meters

Using the Pythagorean theorem:

d² = L² + W²

(53 meters)² = (45 meters)² + W²

2809 = 2025 + W²

W² = 2809 - 2025

W² = 784

Taking the square root of both sides:

W ≈ √784

W ≈ 28

Therefore, the width of the rectangular park is approximately 28 meters.

Jonah swam a distance of 29 meters from one corner of the pool to the opposite corner.

To determine how far Jonah swam from one corner of the pool to the opposite corner, we can use the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the two sides of the right-angled triangle are the length and width of the swimming pool, which are 21 meters and 20 meters respectively. The distance Jonah swam represents the hypotenuse of the triangle.

Using the Pythagorean theorem:

[tex]Hypotenuse^2 = Length^2 + Width^2[/tex]

[tex]Hypotenuse^2[/tex] =[tex]21^2 + 20^2[/tex]

[tex]Hypotenuse^2[/tex]= 441 + 400

[tex]Hypotenuse^2[/tex]= 841

Taking the square root of both sides, we find:

Hypotenuse =[tex]\sqrt{841[/tex]

Hypotenuse = 29

Jonah swam a distance of 29 meters from one corner of the pool to the opposite corner.

According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

The two sides of the right-angled triangle are the length and width of the swimming pool, which are 21 meters and 20 meters respectively. The distance Jonah swam represents the hypotenuse of the triangle.

Using the Pythagorean theorem:

[tex]Hypotenuse^2 = Length^2 + Width^2\\Hypotenuse^2 = 21^2 + 20^2\\Hypotenuse^2 = 441 + 400\\Hypotenuse^2 = 841[/tex]

Taking the square root of both sides, we find:

Hypotenuse = [tex]\sqrt841[/tex]

Hypotenuse = 29

Jonah swam a distance of 29 meters from one corner of the pool to the opposite corner.

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Answer:

B Mean: 17.5  | W Mean: 20.5 | B Mode: 15 | W Mode: 20.5 | B Mode: 14 & 15 | W Mode: 20 & 21 | B Range: 17 | W Range: 3

Step-by-step explanation:

Answer: A

Step-by-step explanation: 32.07 cm^2

Applying the tangent ratio, the value of x in the image, rounded to the nearest thousandth is: 15.824.

The tangent ratio, commonly referred to as "tangent," is a trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle in a right triangle. It is expressed as:

tan (∅) = opposite/adjacent

We have the following:

Reference angle (∅) = 53 degrees

Length of opposite side = 21

Length of adjacent side = x

Plug in the values:

tan 53 = 21/x

x * tan 53 = 21

x = 21 / tan 53

x = 15.824

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The simplified exponential expression in the context of this problem is given as follows:

[tex](6m^5g^5)^2 = 32m^{10}g^{10}[/tex]

The exponential expression in the context of the problem is defined as follows:

[tex](6m^5g^5)^2[/tex]

The power of a power rule is used when a single base is elevated to multiple exponents, and states that simplified expression is obtained keeping the base, while the exponents are multiplied.

Applying the exponent of 2, we have that:

Hence the simplified exponential expression in the context of this problem is given as follows:

[tex](6m^5g^5)^2 = 32m^{10}g^{10}[/tex]

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Answer:

[tex](y - 4)^2 = -8(x - 2).[/tex]

Step-by-step explanation:

The equation of a parabola with a vertical axis of symmetry, vertex (h, k), and focus (h+a, k) is given by:

[tex](y - k)^2 = 4a(x - h)[/tex]

In this case, the vertex is (2, 4) and the focus is (0, 4).

Comparing this to the general equation, we have h=2, k=4, and h+a=0.

From h+a=0, we can solve for a:

a=-h = -2

Substituting the values of h, k, and p into the equation, we get:

[tex](y - 4)^2 = 4(-2)(x - 2)[/tex]

Simplifying further:

[tex](y - 4)^2 = -8(x - 2)[/tex]

Therefore, the parabola equation is[tex](y - 4)^2 = -8(x - 2).[/tex]

The number in the green box is 31.78° .

Given,

Triangle with sides 11 , 12 , 20.

From Cosine Law.

Let △ABC have sides a, b,  c.

Here,

a = 11

b = 20

c = 12

Length of the side opposite vertex A is called a.

Length of the side opposite vertex B is called c.

Length of the side opposite vertex C is called c.

Let θ = ∠C

Then, the Cosine Law says that

c²=a²+b²−2abcosθ.

Substitute the values of a , b , c .

12²  = 11² + 20² - 2 × 11× 20 cosθ

θ = 31.78°

Hence the angle can be found out from cosine law.

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The solution to the given differential equation is y(x) = (C1 + C2x) * e^(3x), where C1 and C2 are arbitrary constants.

To solve the given differential equation, we need to find the function y(x) that satisfies the equation:

(D^2 - 6D + 9)y(x) = 0,

where D represents the differentiation operator.

Let's break down the solution process step by step:

Characteristic Equation

First, we'll find the characteristic equation associated with the given differential equation. For a second-order linear homogeneous differential equation of the form aD^2y + bDy + cy = 0, the characteristic equation is obtained by replacing D with λ:

λ^2 - 6λ + 9 = 0.

Solving the Characteristic Equation

Now, we solve the characteristic equation to find the values of λ. Factoring the equation, we get:

(λ - 3)^2 = 0.

From this, we see that λ = 3 (with a multiplicity of 2).

General Solution

The general solution of the differential equation is given by:

y(x) = C1e^(λ1x) + C2xe^(λ2*x),

where C1 and C2 are arbitrary constants, and λ1, λ2 are the distinct roots of the characteristic equation.

In our case, since we have repeated roots, the general solution simplifies to:

y(x) = C1e^(3x) + C2xe^(3*x).

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Hello!:

24/12 - 8

= 2 - 8

= -6

The nearest Whole number, the missing dimension (height) of the cylinder is approximately 39 inches.

The missing dimension of the cylinder, we can use the formula for the volume of a cylinder:

Volume = π * r^2 * h

Given that the volume is 10,000π in³ and the diameter of the base is 32 inches, we can determine the radius (r) of the cylinder.

The diameter is twice the radius, so the radius is half of the diameter:

r = 32 inches / 2 = 16 inches

Substituting the known values into the volume formula, we have:

10,000π in³ = π * (16 in)^2 * h

Cancelling out the common factor of π, we get:

10,000 = 16^2 * h

Simplifying further:

10,000 = 256 * h

To isolate h, we divide both sides of the equation by 256:

h = 10,000 / 256

Calculating the value of h:

h ≈ 39.06

Rounded to the nearest whole number, the missing dimension (height) of the cylinder is approximately 39 inches.

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Although the rectangles have the same base lengths, the heights and areas are not directly proportional in this case.

Let's denote the base length of both rectangles as 'b'. If one rectangle has a height that is twice the height of the other rectangle, we can denote the heights as 'h' and '2h', respectively.

The area of a rectangle is calculated by multiplying the base length by the height. Therefore, the area of the first rectangle with height 'h' would be A₁ = b * h, and the area of the second rectangle with height '2h' would be A₂ = b * (2h) = 2b * h.

Comparing the two areas, we have A₁ = b * h and A₂ = 2b * h. It is evident that the areas are not proportional because the area of the second rectangle is twice the area of the first rectangle.

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The maximum revenue, to the nearest whole dollar, is $180,625 (option c).

To find the maximum revenue, we need to determine the vertex of the quadratic function represented by the revenue equation.

The revenue equation is given as:

R(p) = -4p² + 1700p

The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be found using the formula:

x = -b / (2a)

For the given revenue equation, a = -4 and b = 1700. Plugging these values into the formula, we have:

p = -1700 / (2 × -4)

p = -1700 / -8

p = 212.5

The maximum revenue occurs at p = 212.5.

To find the maximum revenue, we substitute this value back into the revenue equation:

R(p) = -4(212.5)² + 1700(212.5)

R(p) = -4(45256.25) + 361250

R(p) = -181025 + 361250

R(p) = 180625

Therefore, the maximum revenue, to the nearest whole dollar, is $180,625 (option c).

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We can deduce here that in the diagram the ratios  AD : DB  and  AE : EC  are equal.

Triangles that resemble one another but may differ in size are said to be similar triangles. They have equal corresponding angles and proportional corresponding sides, in other words.

Two triangles are similar if and only if the following conditions are met:

The corresponding sides of two triangles that are comparable are proportionate. The length of the comparable side in the other triangle can be obtained by multiplying the length of a side in one triangle by the same factor.

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Answer:

The figure is omitted--please sketch it to confirm my answer.

[tex]\pi {(5 \sqrt{2}) }^{2} - {10}^{2} [/tex]

[tex]50\pi - 100 = 50(\pi - 2) = 57.08[/tex]

The area inside the square and outside the circle is about 57.08 cm^2.

It will take approximately 2.32 hours to fill 11 tanks.

First, we need to convert the capacity of the tanks from gallons to liters, as the rate at which we empty the bottle is given in liters. Since 1 gallon is equal to 3.8 liters, the capacity of each tank is 50 gallons * 3.8 liters/gallon = 190 liters.

Now, let's calculate the time it takes to fill one tank:

The bottle is emptied at a rate of 2 liters every 8 seconds. To find the time it takes to fill one tank, we divide the capacity of the tank (190 liters) by the rate at which we empty the bottle (2 liters/8 seconds):

Time for one tank = 190 liters / (2 liters/8 seconds) = 190 liters * (8 seconds/2 liters) = 760 seconds.

Next, we need to find the total time required to fill 11 tanks. Since each tank takes 760 seconds to fill, we multiply this by 11 to account for the 11 tanks:

Total time to fill 11 tanks = 760 seconds/tank * 11 tanks = 8,360 seconds.

Finally, we convert the total time from seconds to hours. There are 60 seconds in a minute and 60 minutes in an hour, so:

Total time to fill 11 tanks = 8,360 seconds * (1 minute/60 seconds) * (1 hour/60 minutes) = 2.32 hours (rounded to two decimal places).

Therefore, it will take approximately 2.32 hours to fill 11 tanks.

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a) The volume of the hemisphere is V₁ = 261.80 km³

b) The volume of the hemisphere is V₂ = 1,526.81 feet³

Given data ,

a)

Let the volume of the hemisphere be represented as V₁

where the radius of the hemisphere is r₁ = 5 km

And , volume of hemisphere is V = ( 2/3 )πr³

On simplifying , we get

V₁ = ( 2/3 )π ( 5 )³

V₁ = 261.80 km³

b)

Let the volume of the hemisphere be represented as V₂

The diameter of the hemisphere = 18 feet

where the radius of the hemisphere is r₁ = 9 feet

And , volume of hemisphere is V = ( 2/3 )πr³

On simplifying , we get

V₂ = ( 2/3 )π ( 9 )³

V₂ = 1,526.81 feet³

Hence , the volume of the hemisphere is solved

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The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller.

To determine the probabilities in question, we need to consider the information provided and make some assumptions based on general knowledge about the population of the United States.

P(B) when you have no other information:

Without any other information, we cannot accurately determine the probability of event B, which represents "The woman is taller than 5 feet 4 inches." We would need additional data on the height distribution of women in the United States to calculate this probability.

P(B) when you know A is true:

If we know that event A is true, meaning "The woman is a professional basketball player," we can make some assumptions based on the nature of professional basketball players.

Generally, professional basketball players tend to be taller than the average population due to the physical requirements of the sport. Therefore, the probability of event B, "The woman is taller than 5 feet 4 inches," would likely be greater when we know event A is true.

P(B) when you know A is false:

If event A is false, meaning "The woman is not a professional basketball player," we cannot make any definitive conclusions about the probability of event B, "The woman is taller than 5 feet 4 inches." The height of an individual is not solely determined by their profession, so without further information, we cannot determine if event B is more or less likely when event A is false.

In summary, based on the given information, we can conclude that the probabilities of event B are not equal under different scenarios. The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller. However, without any other information, we cannot determine the probability of event B or make comparisons when event A is false.

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The value of expression (s - f) (- 7) would be,

⇒ (s - f) (- 7)  = 119

We have to given that,

Functions are defined as,

⇒ s (x) = 2x²

⇒ f (x) = 3x

Now, We can find the value of (s - f) (- 7) is,

⇒ (s - f) (- 7)

⇒ s (- 7) - f (- 7)

⇒ 2 (- 7)² - (3 × - 7)

⇒ 2×49 + 21

⇒ 98 + 21

⇒ 119

Therefore, The value of expression (s - f) (- 7) would be,

⇒ (s - f) (- 7)  = 119

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Answer:

Step-by-step explanation:

For the package of 10 folders: Unit price = $1.20 / 10 = $0.12 For the package of 8 folders: Unit price = $1.12 / 8 = $0.14

Answer:

6

Step-by-step explanation:

The correct answer is C. 6.

If we substitute 6 for x in the equation, we get 4(8-6) = 4(2) = 8.

This is the only value of x that makes the equation true.

Answer:

Slope = -7/3

Step-by-step explanation:

-7x = 6 + 3y is in the standard form of a line, whose general equation is

Ax = C + By (it's sometimes written in terms of C and is Ax + By = C, but in this problem, it's written in terms of Ax).

We can find the slope of the line by converting from standard form to slope-intercept form, whose general equation is y = mx + b, where

Step 1:  Subtract 6 from both sides:

(-7x = 6 + 3y) - 6

-7x - 6 = 3y

Step 2:  Divide both sides by 3 to isolate y:

(-7x - 6 = 3y) / 3

-7/3x - 2 = y

Thus, the slope of the line is -7/3

Answer:  Therefore the slope is [tex]-\frac{7}{3}[/tex].

Step-by-step explanation:

We can rewrite the equation -7x=6+3y in slope-intercept form y = mx + b, where the m is the slope of the line, and b is the y-intercept.

-7x = 6 + 3y

-6     -6        

-7x - 6 = 3y

[tex]\frac{-7x}{3}-\frac{6}{3} =\frac{3y}{3}[/tex]

[tex]\frac{-7}{3}x-2 =y[/tex]

Therefore the slope is [tex]-\frac{7}{3}[/tex].

Considering the definition of an equation and the way to solve it, there are 84 green counters in the box.

An equation is the equality existing between two algebraic expressions connected through the equals sign in which one or more unknown values appear.

The solution of a equation means determining the value that satisfies it. To solve an equation, keep in mind:

Knowing that:

the equation in this case is:

2/9 × total counters on the box =24

Solving:

total counters on the box =24÷ 2/9

The first step in dividing by a fraction is to find the reciprocal (reverse the numerator and denominator) of the second fraction.

Then, the two numerators and the two denominators must be multiplied and, if necessary, the fractions are simplified.

total mountain bikes =24× 9/2= 24/1× 9/2

total mountain bikes =(24×9)/ (1×2)

total mountain bikes =216/2

total mountain bikes =108

Then, there are 108 green and yellow counters in the box.

So, the amount of green counters in the box is calculated as:

Amount of green counters= 7/9× 108

Amount of green counters= 7/9× 108

Amount of green counters= 84

Finally, there are 84 green counters in the box.

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The expression for the total number of packets of crisps that Jane buys is given as follows:

p + c.

The amounts of packets of crisps purchased are given as follows:

The expression for the total number of packets of crisps that Jane buys is given by the addition of these two amounts.

Hence the expression for the total number of packets of crisps that Jane buys is given as follows:

p + c.

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Based on the information, it should be noted that 31.25 subscriptions were left when y=5.

From the information, Saleema was investigating the newspaper subscriptions for her local newspaper. At the beginning of the year, She noted 1000 subscriptions for Subscribed to the newspaper

To find the approximate number of subscriptions left when y=5, we can substitute y=5 into the function N(y) = 1000 * (0.5)^y and evaluate it.

N(5) = 1000 * (0.5)^5

N(5) = 1000 * (0.5)^(5)

N(5) = 1000 * (0.03125)

N(5) ≈ 31.25

Approximately 31.25 subscriptions were left when y=5.

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