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The equation of the tangent to the circle at point A is 6x + 7y - 93 = 0

The equation of a circle in standard form is (x-h)² + (y-k)² = r²,

(h,k) is the center of the circle

r is the radius.

The radius formula ⇒ √((x₂ - x₁)² + (y₂ - y₁)²).

Here,

x₁ = -1, y₁ = 2 (center of the circle),

x₂ = 5, y₂ = 9 (point A on the circle).

∴

r = √((5 - (-1))² + (9 - 2)²) = √(36 + 49) = √85.

Now, we have the equation of the circle: (x - (-1))² + (y - 2)² = 85, or (x + 1)² + (y - 2)² = 85.

The slope of the radius from the center of the circle to point A ⇒ (y₂ - y₁) / (x₂ - x₁)

= (9 - 2) / (5 - (-1)) = 7/6.

tangent line is the negative reciprocal of the slope of the radius, ∴ -6/7.

The equation of a line in point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.

The slope of the tangent line (m) is -6/7 and it passes through point A(5,9). Substituting these values in, it becomes

y - 9 = -6/7 (x - 5).

Multiplying every term by 7 to clear out the fraction and to have the equation in the ax + by + c = 0 form, we get:

7y - 63 = -6x + 30,

or

6x + 7y - 93 = 0.

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Answer:

B. 3188.5 cubic inches.

Step-by-step explanation:

The volume of a cone is calculated using the following formula:

Volume = (1/3) * π * r² * h

Where:

In this problem, we are given that r = 17 inches and S.h = 20 inches.

First we need to find height h.

py using Pythagorous theorem,we get

c²=a²+b²

here c= slight height and a is radius

20²=17²+b²

20²-17²=b²

111=b²

b=√(111)

Plugging these values into the formula, we get:

Volume = ⅓*π* 17² *√(111) = 3188.5 cubic inches

Therefore, the volume of the cone is 3188.5 cubic inches.

The type of sytudy that we have here is the observational study.

In an observational study, the researcher observes and analyzes data from individuals without actively intervening or manipulating any variables.

In this case, the researcher is observing and comparing the health outcomes of two groups of individuals: those who have eaten a vegan diet for the past five years and those who have not.

The participants are not randomly assigned to the groups, and the researcher does not actively control or manipulate the diet of the individuals.

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To convert from degrees to radians, we use the conversion factor that 180 degrees is equal to π radians (or π/180 radians per degree).

Given that the angle is 45 degrees, we can calculate the radian measure as follows:

Radian measure = (45 degrees) * (π/180 radians per degree)

Radian measure = 45π/180

Simplifying further:

Radian measure = π/4

Therefore, the radian measure of a 45 degree angle is π/4.

A square and a traingle are present in a large rectangle with given dimensions in the figure.

[tex]\qquad\displaystyle \tt \dashrightarrow \: 12 \times 8[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 96 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 4 \times 4[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 16 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 2 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 10 \: \: unit {}^{2} [/tex]

[ area of square / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{16}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{1}{6} [/tex]

[ total area except area of triangle / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{96 - 10}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{86 }{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{43}{48} [/tex]

The sum of the exterior angles in a regular 20-gon is given as follows:

360º.

An exterior angle of a polygon is defined as the angle between a side and its adjacent extended side.

The sum of exterior angles formula states the sum of all exterior angles in any polygon is 360°, no matter the number of sides.

For this problem, we have a 20-gon, that is a polygon of 20 sides, however, as the formula states, the sum of the exterior angles in a regular 20-gon is given as follows:

360º.

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The value of AB is,

⇒ AB = 5.9

(rounded to nearest tenth)

We have to given that,

A right triangle ABC is shown.

Now, By trigonometry formula,

we get;

⇒ cos 33° = Base / Hypotenuse

Substitute all the values, we get;

⇒ cos 33° = AB / 7

⇒ 0.84 = AB / 7

⇒ AB = 0.84 × 7

⇒ AB = 5.88

⇒ AB = 5.9

(rounded to nearest tenth)

Thus, We get;

AB = 5.9

(rounded to nearest tenth)

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The Quadratic equation in standard form that corresponds to the given parabola is (x + 1)^2 = 12y.

The quadratic equation in standard form that corresponds to the graph of the parabola passing through the points (2, 0) and (-4, 0), we can use the vertex form of a parabola equation, which is (x - h)^2 = 4a(y - k). the vertex of the parabola. The vertex is the midpoint of the line segment connecting the two given points.

The x-coordinate of the vertex is the average of the x-coordinates of the two points:

(2 + (-4))/2 = -2/2 = -1

The y-coordinate of the vertex is the same as the y-coordinate of both given points:

y = 0

Therefore, the vertex of the parabola is (-1, 0).

Now, let's find the value of 'a', which represents the coefficient in front of the y-term. We know that the distance from the vertex to either of the given points is equal to 'a'. In this case, the distance from the vertex (-1, 0) to either (2, 0) or (-4, 0) is 3 units.

So, 'a' = 3.

Now, we can write the quadratic equation in standard form:

(x - h)^2 = 4a(y - k)

Plugging in the values we found:

(x - (-1))^2 = 4(3)(y - 0)

Simplifying:

(x + 1)^2 = 12y

Therefore, the quadratic equation in standard form that corresponds to the given parabola is (x + 1)^2 = 12y.

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In the equation [tex]y = -2(0.5)^x[/tex], the initial amount (a) is -2, and the growth/decay factor (b) is 0.5.

In the given equation, [tex]y = -2(0.5)^x[/tex], we can identify the initial amount and the growth/decay factor.

The equation is in the form of exponential decay, as the base (0.5) is between 0 and 1, resulting in the function decreasing as x increases.

The initial amount, denoted as "a," is the coefficient in front of the exponential term. In this case, the initial amount is -2. This means that when x = 0, y = -2.

The growth/decay factor, denoted as "b," is the base of the exponential term. In this equation, the base is 0.5. The value of the base determines how quickly the function decreases or decays.

To summarize:

Initial amount (a) = -2

Growth/decay factor (b) = 0.5

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The measure for NS is 3.5 cm and the value of x is 2.

We have,

JK = 16, MP= 8, LP = 2x + 4, and PS= 3.5.

Since PS is perpendicular to ML then

MP = LP

So, 8 = 2x+ 4

2x = 8-4

2x= 4

x= 4/2

x= 2

Now, ML = MP + PL = 16

and, JK = ML= 16

Then, NS = PS

NS = PS = 3.5 cm

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The relationship between angles 8 and 7 is that they are supplementary. option B is correct.

Given that a quadrilateral, with three parallel lines, we need to find the relation between angles 8 and 7,

We know that the adjacent angles between the parallel lines are supplementary,

We know that,

Supplementary angles are a pair of angles that add up to 180 degrees. In other words, if you have two angles that are supplementary, the sum of their measures will always be 180 degrees.

So,

The relation between the angles is that they are Supplementary angles.

Hence the option B is correct.

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a = 30°

b = 180-135 = 45°

total angle inside triangle = 180°

x = 180-(30+45) = 105°

Applying the angle of intersecting secants theorem, the measure of arc LN in the circle is: 56°.

Given the circle in the image above where the two secants intersect outside the circle, the angle of intersecting secants theorem states that:

external angle formed = 1/2 * (the measure of arc KP - the measure of arc LN)

Plug in the values:

20 = 1/2 * (96 - m(LN))

2 * 20 = 96 - m(LN)

40 = 96 - m(LN)

m(LN) = 96 - 40

m(LN) = 56°

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The total cost of producing the additional units is $767.50.

In order to determine the total cost of producing 10 additional units assuming 2 units are currently being produced, we would have to integrate the marginal cost function for this handbag manufacturer with respect to x, and over the interval [10, 2].

Based on the information provided above, the marginal cost function for this handbag manufacturer is given by this function;

C'(x) = 0.046875x² − x + 100

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x² − x + 100)dx

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x²)dx − ₂∫¹⁰(x)dx + ₂∫¹⁰(100)dx

₂∫¹⁰C'(x)dx = 0.046875x³/3|¹⁰₂ - x²/2|¹⁰₂ + 100x|¹⁰₂

₂∫¹⁰C'(x)dx = 0.015625(10³ - 2³) - 1/2(10² - 2²) + 100(10 - 2)

₂∫¹⁰C'(x)dx = 0.015625(1000 - 8) - 0.5(100 - 4) + 100(8)

₂∫¹⁰C'(x)dx = 0.015625(992) - 0.5(96) + 800

₂∫¹⁰C'(x)dx = 15.5 - 48 + 800

₂∫¹⁰C'(x)dx = $767.50

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Fred will pay $43.05 in sales tax if he buys the new refrigerator.

To calculate the sales tax Fred will pay for the new refrigerator, we first need to find the amount of the sales tax.

The cost of the new refrigerator is $615.

To calculate the sales tax, we multiply the cost by the tax rate of 7%:

So, Sales tax = 7% of $615

Sales tax = (7/100) x $615

Sales tax = $43.05

Therefore, Fred will pay $43.05 in sales tax if he buys the new refrigerator.

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The slope of the line that is perpendicular to the line y = 3x - 8 is -1/3.

The given line has an equation in slope-intercept form, which is y = 3x - 8. In this form, the coefficient of x represents the slope of the line.

Therefore, the slope of the given line is 3.

To find the slope of a line perpendicular to the given line, we need to take the negative reciprocal of the slope.

The negative reciprocal of 3 is -1/3.

Therefore, the slope of the line that is perpendicular to the line y = 3x - 8 is -1/3.

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The length of y in the right triangle is 15 units.

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The sum of angles in a triangle is 180 degrees.

The side y in the triangle XYZ can be found using trigonometric ratios as follows:

Therefore,

sin 45 = opposite / hypotenuse

opposite sides = y

hypotenuse side = 15√2

sin 45 = y / 15√2

cross multiply

y = 15√2 × sin 45

y = 15√2 × 1 / √2

y = 15 units

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Answer:

Infinite solutions (D).

Step-by-step explanation:

Here is how:

To determine the true statement about the given equation, let's simplify it step by step:

-9(x + 3) + 12 = -3(2x + 5) - 3x

Distributing the -9 and -3 on the left and right sides respectively:

-9x - 27 + 12 = -6x - 15 - 3x

Combining like terms:

-9x - 15 = -9x - 15

Now, let's analyze the equation. We have -9x on both sides, and -15 on both sides. By subtracting -9x from both sides and -15 from both sides, we obtain:

0 = 0

This equation is true regardless of the value of x. In other words, it holds for all values of x. Therefore, the equation has infinitely many solutions.

Answer:

The correct answer is: "The equation has one solution, x = 0"

Answer:

$653

Step-by-step explanation:

:]

Let's use x to represent the cost of one refrigerator and y to represent the cost of one fan.

The equations become:

Equation 1: x + 2y = 1219

Equation 2: 2x + 3y = 2155

Using the same substitution method:

From Equation 1, we have:

x = 1219 - 2y

Substitute this expression for x in Equation 2:

2(1219 - 2y) + 3y = 2155

Simplify the equation:

2438 - 4y + 3y = 2155

-y = 2155 - 2438

-y = -283

===> y = 283

Now substitute the value of y back into Equation 1 to find x:

===> x + 2(283) = 1219

===> x + 566 = 1219

===> x = 1219 - 566

===> x = 653

Therefore, the cost of one refrigerator is $653.

Answer: (4x-12) + ( 1/2x y -10) = 6

Step-by-step explanation:

First, input 4 for x and 6 for y into the equation so it looks like this:

(4(4)-12) + (1/2(4)(6)-10)

Now solve inside the parentheses starting with the first one. 4 * 4 = 16 so the inside of the first parentheses should look like (16 - 12) which equals 4.

For the second set of parentheses, 1/2 * 4 * 6 = 12, so the inside of that parentheses would look like (12 - 10), which equals 2.

At this point, the equation should look like this: (4) + (2). If you add those two together, your answer should be 6.

The height of the right rectangular prism is 7.6 cm.

To determine the height of the right rectangular prism, we can use the formula for the volume of a prism:

Volume = Base Area * Height

Given that the volume is 380 cm³ and the base area is 50 cm², we can write the equation as:

380 = 50 * h

Now, let's solve for h by dividing both sides of the equation by 50:

h = 380 / 50

Simplifying the expression:

h = 7.6 cm

Therefore, the height of the right rectangular prism is 7.6 cm.

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The perimeter of shaded figure is 10 unit.

We know,

The perimeter of a figure is the total distance around its boundary. To calculate the perimeter, you need to sum the lengths of all the sides of the figure.

From the figure

length of rectangle = 4 unit

width of rectangle =  1 unit

Now, the perimeter of shaded figure

= 2 (l + w)

= 2 (4 +1 )

= 2 x 5

= 10 unit

Thus, the perimeter of figure is 10 unit.

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The derivative of function f(w) = [tex]2/(w^2 - 4)^5 ~is ~f'(w) = -40~w~(w^2 - 4)^{-6}.[/tex]

We have,

To find the derivative of the function [tex]f(w) = 2/(w^2 - 4)^5[/tex], we can use the chain rule and the power rule for differentiation.

Let's go through the steps:

First, rewrite the function as [tex]f(w) = 2(w^2 - 4)^{-5}.[/tex]

Now, let's differentiate f(w) with respect to w:

[tex]f'(w) = d/dw~ [2(w^2 - 4)^{-5}][/tex]

To apply the chain rule, we need to differentiate the outer function and multiply it by the derivative of the inner function.

Using the power rule, the derivative of (w² - 4) with respect to w is 2w.

Applying the chain rule:

[tex]f'(w) = -10 \times 2(w^2 - 4)^{-6} \times 2w[/tex]

Simplifying further:

[tex]f'(w) = -40w(w^2 - 4)^{-6}[/tex]

Therefore,

The derivative of function f(w) = [tex]2/(w^2 - 4)^5 ~is ~f'(w) = -40~w~(w^2 - 4)^{-6}.[/tex]

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Answer:

The perimeter of the rectangle is P = 4x^2 + 16x - 4.

Step-by-step explanation:

1. L = 3x + 2, W = 2x^2 + 5x - 4

2. P = 2(L + W)

3. P = 2((3x + 2) + (2x^2 + 5x - 4))

4. P = 2(2x^2 + 8x - 2)

5. P = 4x^2 + 16x - 4

Step-by-step explanation:

ln 54.60 = 4         e^x both sides

e ^ (ln 54.60) = e^4

         54.60 = e^4                 Done.

The difference between an observational study and an experiment is that  in an observational study, the researchers do not control treatment, and in an experiment, they do

In an observational study, it should be noted that the participants are measured or surveyed without any attempt to influence them. *

However the controlled experiment, participants or objects are divided into groups, and one group is given a treatment while the other is not.

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The value of x in the given figure is 15.

In the given figure

The length of section of chords are given

We have to find the value of x

In order to find the value of x

Apply the intersecting chord theorem,

The intersecting chords theorem, often known as the chord theorem, is a basic geometry statement that defines a relationship between the four line segments formed by two intersecting chords within a circle. It asserts that the products of the line segment lengths on each chord are equal.

Therefore,

From figure we get,

⇒ 10/x = 8/12

⇒ x = 120/8

⇒  x = 15  

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The area under the curve y = x² + 5x + 4 and the x-axis is [tex]1\frac{2}{3}[/tex] square units.

To find the area under the curve y = x² + 5x + 4 and the x-axis, we need to integrate the given function over a specific interval.

We need to find the definite integral of the function over a suitable interval.

Let's find the definite integral of the function from its roots (where the curve intersects the x-axis).

First, let's find the roots of the function by setting y = 0:

x² + 5x + 4 = 0

Factoring the quadratic equation:

(x + 1)(x + 4) = 0

Setting each factor equal to zero:

x + 1 = 0 => x = -1

x + 4 = 0 => x = -4

The roots of the function are x = -1 and x = -4.

To find the area under the curve, we integrate the function from x = -4 to x = -1:

∫[x=-4 to -1] (x² + 5x + 4) dx

Integrating the function:

∫[x=-4 to -1] (x² + 5x + 4) dx = [1/3x³ + (5/2)x² + 4x] from -4 to -1

Substituting the limits:

[1/3(-1)³ + (5/2)(-1)² + 4(-1)] - [1/3(-4)³ + (5/2)(-4)² + 4(-4)]

Simplifying:

[-1/3 + 5/2 - 4] - [-64/3 + 40/2 - 16]

[tex]1\frac{2}{3}[/tex]

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