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Angles Formed by Chords, Tangents, Secants

We may use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ to get 970299, 1061208, and 992016008 for (i), (ii), and (iii), respectively.

Using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ allows us to expand and simplify the expressions by distributing and collecting like terms. Any integer's cubes can be calculated using this method.

(i) Using the identity, we can write 99 as 100 - 1 and get:

= (99)³

= (100 - 1)³

= 100³ - 3(100²)(1) + 3(100)(1²) - 1³

= 970299

(ii) We can denote 102 as 100 + 2 and use the identity to obtain:

= (102)³

= (100 + 2)³

= 100³ + 3(100²)(2) + 3(100)(2²) + 2³

= 1061208

(iii) Using the identity, we may write 998 as 1000 - 2 and get:

= (998)³

= (1000 - 2)³

= 1000³ - 3(1000²)(2) + 3(1000)(2²) - 2³

= 992016008

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Answer:

Top

Step-by-step explanation:

The closer the data points come to forming a straight line when plotted, the higher the correlation between the two variables, or the stronger the relationship. Therefore, the top option has the most closest lines to a linear function.

(Ex. 1) Y=4x-2

This example shows the corresponding possibilities of a linear regression because of the way the line is represented as graphed, making a straight line as similar to the top option

(Ex. 2) f(x) = x^2-5+15

This example dialates to a similar option like the third option, which isnt linear regression because of it being a quadratic function.

In a nut shell, all data plotted on the graph that are formed closer together and corresponds to a linear equation is a linear regression

The problem involves finding the marginal cost of producing a certain number of DVD players, given the total cost function. Specifically, we are given the total cost function C(x) = 120 + 3x^2-3, where x is the number of DVD players produced, and we need to find the marginal cost when x = 3.

To find the marginal cost, we need to take the derivative of the total cost function with respect to x and evaluate it at x = 3. The marginal cost represents the cost of producing one additional unit of the product, and is an important concept in economics and business.

Understanding the relationship between marginal cost and production volume is crucial for optimizing production and pricing decisions for companies. Marginal cost analysis is used extensively in many fields, including manufacturing, finance, and marketing.

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Therefore, the perimeter of the rectangle is 38 units that is option C.

A coordinate is a set of numbers or values that specifies the position or location of a point or object in a space. In mathematics, coordinates are used to describe the position of a point in a plane, a space or a higher-dimensional object. Coordinates can be represented by ordered pairs or tuples, where the first value corresponds to the position on the horizontal axis, and the second value corresponds to the position on the vertical axis.

Here,

First, we need to find the distance between points A and B to determine the length of the rectangle:

Distance AB = |yB - yA|

= |8 - 2|

= 6

Next, we need to find the distance between points B and C to determine the width of the rectangle:

Distance BC = |xC - xB|

= |6 - (-7)|

= 13

Since opposite sides of a rectangle are equal in length, we know that the distance between points A and D is also 13, and the distance between points C and D is also 6.

Therefore, the perimeter of the rectangle is:

Perimeter = 2 * (length + width)

Perimeter = 2 * (13 + 6)

Perimeter = 2 * 19

Perimeter = 38

So the answer is (C) 38.

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John can afford approximately 6 magazines.

A) To write an equation that models the number of magazines John can afford, let's denote the number of magazines as 'm'. Since each magazine costs $2.95 and John has a total of $23.65 to spend, the equation can be expressed as:

2.95m + 5.95 = 23.65

B) To solve the equation, we can isolate the variable 'm' by subtracting 5.95 from both sides:

2.95m = 23.65 - 5.95

2.95m = 17.70

Then, divide both sides by 2.95:

m = 17.70 / 2.95

m ≈ 6

Therefore, John can afford approximately 6 magazines.

In conclusion, the equation 2.95m + 5.95 = 23.65 models the number of magazines John can afford, and by solving it, we find that he can purchase approximately 6 magazines with the given amount of money.

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Answer:

List all the factors

6: 3, 2, 1 (6)

9:3,(9),1

5:(5),1

2:(2),1

Step-by-step explanation:

The total inventory costs consist of two parts: the cost of storing the shoes and the cost of reordering the shoes. The cost of storing the shoes is given by the formula:

Cost of storing = $6 per shoe pair per year x number of shoe pairs

Since the shoes are stored for a year, this cost is incurred annually. The cost of reordering the shoes is given by the formula:

Cost of reordering = $14 per order + $7 per shoe pair x number of shoe pairs

This cost is incurred each time the store places an order with the manufacturer.

Let x be the number of shoe pairs in each order from the manufacturer. The number of orders needed to sell 1750 shoe pairs each year is given by:

Number of orders = 1750 shoe pairs / x shoe pairs per order

The total inventory costs can be expressed as:

Total cost = Cost of storing + Cost of reordering

Substituting the formulas for the two costs and the expression for the number of orders, we get:

Total cost = $6 per shoe pair per year x 1750 shoe pairs + ($14 per order + $7 per shoe pair x x shoe pairs) x (1750 shoe pairs / x shoe pairs per order)

Simplifying this expression, we get:

Total cost = $10,500 + $14(1750/x) + $7(1750)

Total cost = $10,500 + $24,500/x

Therefore, the function that models the total inventory costs as a function of x is:

Total cost(x) = $10,500 + $24,500/x

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Answer:

55 degrees

Step-by-step explanation:

sin Φ = cos 35

sin Φ = .81915

Φ = arc sin (.81915) = 55 degrees

Or you could just know that the cos of an angle is equal to the sin of its comeplement

To express the volume of D as an iterated triple integral in (a) spherical, (b) cylindrical coordinates, we can use the following formulas:

(a) Spherical Coordinates:

We know that the equation of a sphere with radius 6 units is given by:

x^2 + y^2 + z^2 = 6^2

The equation of the plane 3 units from the center of the sphere is given by:

z = 3

To find the equation of the smaller cap cut from the sphere by this plane, we need to find the upper limit of the radial coordinate. This can be found by solving the equation of the sphere for z, and substituting z = 3:

z = sqrt(6^2 - x^2 - y^2)
3 = sqrt(6^2 - x^2 - y^2)
x^2 + y^2 = 27

Thus, the smaller cap D is the region of the sphere bounded by the plane z = 3 and the surface x^2 + y^2 + z^2 = 6^2, with x^2 + y^2 <= 27.

To express the volume of D as an iterated triple integral in spherical coordinates, we can use the following limits:

0 <= r <= sqrt(27)
0 <= θ <= 2π
arcsin(3/6) <= φ <= π

The volume of D can be expressed as the triple integral:

∫∫∫ D  r^2 sin φ dr dφ dθ

(b) Cylindrical Coordinates:

To express the volume of D as an iterated triple integral in cylindrical coordinates, we can use the following limits:

0 <= r <= sqrt(27)
0 <= θ <= 2π
3 <= z <= sqrt(6^2 - r^2)

The volume of D can be expressed as the triple integral:

∫∫∫ D  r dz dr dθ
(a) In spherical coordinates, the volume element is given by dV = ρ²sin(φ)dρdθdφ. To find the limits of integration, note that the radius of the smaller cap ranges from 0 to 3 units, the angle θ ranges from 0 to 2π, and the angle φ ranges from 0 to φ₀, where φ₀ is the angle between the plane and the line from the sphere's center to the plane's intersection with the sphere.

Using the cosine rule, we can find φ₀ as follows:
cos(φ₀) = (6² + 3² - 3²) / (2 × 6 × 3) = 1/2
φ₀ = π/3

Now, we can express the volume of D as an iterated triple integral in spherical coordinates:
∫(ρ=0 to 3) ∫(θ=0 to 2π) ∫(φ=0 to π/3) ρ²sin(φ)dρdθdφ

(b) In cylindrical coordinates, the volume element is given by dV = rdzdrdθ. To find the limits of integration, note that the height z ranges from 3 to 6 units, the radius r ranges from 0 to √((6 - z)²), and the angle θ ranges from 0 to 2π.

Now, we can express the volume of D as an iterated triple integral in cylindrical coordinates:
∫(z=3 to 6) ∫(θ=0 to 2π) ∫(r=0 to √((6 - z)²)) rdzdrdθ

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We can use an inequality to determine the number of biking outfits Shandra can purchase while staying within her budget.

Let o represent the number of outfits she can purchase. Here are the given terms and costs:
- Initial budget: $760
- Bicycle cost: $433.54
- 2 reflectors cost: 2 * $18.41 = $36.82
- Bike gloves cost: $10.76
- Outfit cost: $66.40 each

Now, we can set up the inequality:
760 >= 433.54 + 36.82 + 10.76 + 66.40 * o

First, combine the constants:
760 >= 481.12 + 66.40 * o

Now, subtract 481.12 from both sides:
278.88 >= 66.40 * o

Finally, divide both sides by 66.40:
o <= 4.2

Since Shandra can only purchase whole outfits, the maximum number of outfits she can buy is 4. So the inequality representing this situation is o <= 4.

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Answer:

-5z/v³ + 8 + 6v²z

Do you have to simplify it further or?

Step-by-step explanation:

[tex] \frac{ - 5z + 8v {}^{3} + 6v {}^{5} z}{v {}^{3} } = \frac{ - 5z}{v {}^{3} } + \frac{8v {}^{3} }{ {v}^{3} } + \frac{6v {}^{5} }{v {}^{3} } = \frac{ - 5z}{v {}^{3} } + 8 + 6v {}^{2}z[/tex]

The function table that matches the given rule output = input - 3 is

Input = -2, 1, 6 and output = -5, -2, 3

A) first function table

Output = Input - 3

Value of input:- -2

Putting the value of the input

Output = -2 -3

The value of output we get

Output = -5

Value of input:- 1

Putting the value of the input

Output = 1 -3

The value of output we get

Output = -2

Value of input:- 6

Putting the value of the input

Output = 6 -3

The value of output we get

Output = 3

Hence function table A is correct match

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The correct option is the last one, the surface is 3425 m²

Remember that for a sphere of radius R, the volume is:

V = (4/3)pi*R³

S = 4pi*R²

Where pi = 3.14

Here the volume is  6,000π m³, then the radius will be:

R =∛( (3/4)*6,000m³)

R = 16.51 m

Then the surface area is:

[tex]S = 4*3.14*( 16.51 m)^2 = 3,424 m^2[/tex]

The option that is closser to it is the fourth one, so that is the correct option.

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On the basic of square peg sydney smith written point the probability that it is im-possible to fit a square peg in a round hole is equals to the zero.

Probability is defined as the number of chances of occurrence of an event. It is the ratio of the number of favorable outcomes to the total outcomes in that sample space. Mathematical formula is written as, probability of an event, P(E) = (Number of favorable outcomes)/(Total possible outcomes). Now, we have specify that according to sydney smith written in on the conduct of the understanding about square peg that it is impossible to fit a square peg in a round hole. Let consider an event A of fit a square peg in a round hole. We have specify that it is impossible to fit a square peg in a round hole. So, the favourable possible outcomes for event A = 0. Therefore, the probability that to fit a square peg in a round hole, P(A) = 0

Hence, required probability value is zero.

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Finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer B. ) Least Common Multiple.

To determine whether finding the answer requires finding a greatest common factor (GCF) or a least common multiple (LCM), we need to analyze the given information.

In this scenario, the red lights flash every fifth second, the blue lights flash every third second, and the green lights flash every fourth second. We want to find the first time when all three lights flash simultaneously.

To find this time, we need to find the smallest number that is divisible by all three given numbers (5, 3, and 4). This means we are looking for the least common multiple (LCM) of these numbers.

To calculate the LCM, we can use the formula:

LCM(a, b) = (a * b) / GCF(a, b),

where GCF(a, b) represents the greatest common factor of numbers a and b.

Therefore, in this case, finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer:

B. ) Least Common Multiple.

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Given that 25% of students solved at least two tasks and there were 32 students who wrote the test, we can prove that there was at least one task that no more than 12 students solved.

There was at least one task that no more than 12 students solved, we can use a proof by contradiction.

Assume that all five tasks were solved by more than 12 students. This means that for each task, there were at least 13 students who solved it. Since there are five tasks in total, this implies that there were at least 5 * 13 = 65 students who solved the tasks.

However, we are given that only 25% of students solved at least two tasks. If we let the number of students who solved at least two tasks be S, then we can write the equation:

S = 0.25 * 32

Simplifying, we find that S = 8.

Now, let's consider the remaining students who did not solve at least two tasks. The maximum number of students who did not solve at least two tasks is 32 - S = 32 - 8 = 24.

If all five tasks were solved by more than 12 students, then the total number of students who solved the tasks would be at least 65. However, the maximum number of students who could have solved the tasks is 8 (those who solved at least two tasks) + 24 (those who did not solve at least two tasks) = 32.

This contradiction shows that our initial assumption is false. Therefore, there must be at least one task that no more than 12 students solved.

Hence, we have proven that there was at least one task that no more than 12 students solved if 32 students wrote the test.

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The measure of arc a must be 2 times measure of inscribed angle, which is 90 degrees.

What is arc?

In geometry, an arc is a segment of a circle's circumference. It is defined by two endpoints and all the points on the circle's circumference between them.

What is inscribed angle?

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint.

For any inscribed angle in a circle, the measure of the angle is always half the measure of the arc that it intercepts. This is known as the inscribed angle theorem.

So, if we have an inscribed angle with a measure of 45 degrees, then the measure of its corresponding arc would be 2 times that, which is 90 degrees.

Therefore, if the inscribed angle is associated with arc a, and the measure of the corresponding angle is 45 degrees, then we know that the measure of arc a must be 2 times that, which is 90 degrees.

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The volume of the composite figure is 96.3 cm³

First we need to find the volume of the cylinder, and then remove the volume of the rectangular prism.

The radius of the prism is 3cm and the height is 5cm, then the volume is:

V = 3.14*R²*H

V = 3.14*(3cm)²*5cm

V = 141.3 cm³

And the volume of the prism is:

V' = 3cm*3cm*5cm = 45 cm³

The difference gives:

volume = 141.3 cm³ -  45 cm³

volume = 96.3 cm³

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Answer: 5/6

Step-by-step explanation:

probabilities = possibilities/outcome

To get an even you can possible roll 2, 4, 6 with 6 total outcomes

P(even) = probability of getting an even = 3/6

To get a divisor of 3: only 3 and 6 can be divided by 3

P(divisor of 3)=2/6

Because of the or in the statement, you add the 2 probabilities

P(even or divisor of 3) =  3/6+2/6 = 5/6

The rate of change between August and October is

Rate is how a quantity changes over a period of time.

Therefore rate = change in quantity/change in time.

For example acceleration is defined as the rate of change of velocity with time. This means that acceleration = change in velocity/time

change in quantity = 85-81 = 4

change in time = 2 months

therefore rate of change = 4/2 = $2 per month

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The volume of the composite figure is 152 m³.

The volume of a rectangular prism can be found using the formula V = lwh, where l is the length, w is the width, and h is the height.

For the rectangular prism:

V_prism = lwh = 9m * 8m * 3m = 216 m³

For the cube:

V_cube = s^3 = 4m * 4m * 4m = 64 m³

Now, subtract the volume of the cube from the volume of the prism:

V_composite = V_prism - V_cube = 216 m³ - 64 m³ = 152 m³

The volume of the composite figure is 152 m³.

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The expanded form of the smallest bone in the human body located in the ear which is about 2. 8 millimeters in length is                                       2 × 1 millimeter + 8 × 0.1 millimeters.

Given that the smallest bone in the human body is the stapes bone. It is located in the ear and is about 2. 8 millimeters in length.

To write 2.8 millimeters in expanded form, we need to express each digit's place value in the number.

2.8 millimeters can be written as:

2 millimeters + 0.8 millimeters

or

2 millimeters + 8/10 millimeters

In expanded form, this is:

2 millimeters + 8 tenths of a millimeter

or

2 × 1 millimeter + 8 × 0.1 millimeters

Therefore, 2.8 millimeters in expanded form is:

2 × 1 millimeter + 8 × 0.1 millimeters = 2.0 + 0.8 = 2.8 millimeters

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The probability of selecting two country songs followed by an R&B song is 0.0202.

The probability of selecting two country songs followed by an R&B song is determined below as follows:

The total number of songs in the playlist = 40

The probability of the first song being country = 15/40.

The probability of the second song also being country = 14/39

The probability of the third song being R&B =s 10/38

Therefore, the probability of selecting two country songs followed by an R&B song is:

Probability(country, country, R&B) = (15/40) x (14/39) x (10/38)

Probability(country, country, R&B)  = 0.0202

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The solution to the composite function (f + g)(x) is: 8x + 7

Composite functions are said to occur when the output of one function is used as the input of another. If we have a function f and another function g, it means that the function fg(x), said as “ f of g of x”, is the composition of the two functions.

Now, we are given two functions as:

f(x) = x + 3

g(x) = 7x + 4

Thus, we can say that:

(f + g)(x) = f(x) + g(x)

= x + 3 + 7x + 4

= 8x + 7

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Curved surface area of cone: 255π or approx. 801.41 sq units with radius 15 and height 8.

The curved surface area of a right circular cone can be calculated using the formula:

A = πrℓ

where A is the area of the curved surface,

r is the radius of the base of the cone, and

ℓ is the slant height of the cone.

To find the slant height, we can use the Pythagorean theorem:

ℓ² = r² + h²

where h is the height of the cone.

Substituting the given values, we get:

ℓ² = 15² + 8²

ℓ² = 225 + 64

ℓ² = 289

ℓ = √289

ℓ = 17

Now, substituting the values of r and ℓ in the formula for curved surface area, we get:

A = πrℓ

A = π(15)(17)

A = 255π

Therefore, the area of the curved surface of the cone is 255π square units, or approximately 801.41 square units.

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The answer would be D

Since Lucy's number has a digit of 5 which is 10x greater than Ethan's number, 5 x 10 = 50; so 50 would be in the tenth place, and option d is the only one with 5 in the tenth place.

The value of a number moves one decimal place to the left for each position it moves to the right. So, the number that Lucy could have written where the number 5 has a value 10 times larger than Ethan's number is 4974.65.

In order for the value of the digit 5 to be 10 times larger in Lucy's number than it is in Ethan's, the '5' in Lucy's number should reside one place left compared to Ethan's. This means the 5 should be in the tens place, hundreds place, or beyond. So, we should look for a number having digit 5 at those places.

Let's examine the given options:

Therefore, the correct answer is C. 4974.65.

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The probability of the toy character wearing a polka-dotted shirt and a cowboy hat is P(polka-dot shirt and cowboy hat) = 2 / 12 = 1/6.

What is the probability of the toy character wearing a polka-dotted shirt and a cowboy hat in the number of outfits?

There are 3 choices for the shirt (striped, plain, or polka-dotted), 2 choices for the pants (denim or plaid), and 2 choices for the cap (cowboy or baseball). Therefore, there are a total of 3 x 2 x 2 = 12 different outfits that the toy character could be wearing.

The probability of the toy character wearing a polka-dotted shirt and a cowboy hat is the number of outfits with a polka-dotted shirt and a cowboy hat divided by the total number of outfits:

P(polka-dot shirt and cowboy hat) = number of outfits with a polka-dot shirt and cowboy hat / total number of outfits

To find the number of outfits with a polka-dotted shirt and a cowboy hat, we need to consider each clothing item separately. There is only 1 choice for the cowboy hat and only 1 choice for the polka-dotted shirt. There are 2 choices for the pants, but we don't care which pants the toy character is wearing. Therefore, the number of outfits with a polka-dotted shirt and a cowboy hat is:

1 (cowboy hat) x 1 (polka-dotted shirt) x 2 (pants) = 2

So the probability of the toy character wearing a polka-dotted shirt and a cowboy hat is:

P(polka-dot shirt and cowboy hat) = 2 / 12 = 1/6

Therefore, the answer is (B) 1/6.

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The net worth is about $106.

This can be calculated by the addition of liabilities and assets amount

Assets        -               Amount

Checking amount -  $14,532

Retirement savings- $43,675

Car                          - $12,225

Therefore total assets equal to: $70,432

Now, for liabilities

Liabilities            -       Amount

Student loan             $1,225

Credit card debt       $3,876

Home loan balance  $65,225

Therefore total liabilities equals to: $70,326

The net worth can be find out by subtracting the total liabilities from the total assets:

Hence Net worth = Total assets- Total liabilities

                             = $70,432- $70,326

                             = $106

Therefore the total net worth is about $106.

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