1.8 The first two terms of a geometric sequence are 2 and 3 respectively. - Determine the smallest number of terms which will yield a sum larger than 12.​

A. The value of cos 240° = [tex]\frac{-1}{2}[/tex]

B. The value of sin 240° = [tex]-\frac{\sqrt{3} }{2}[/tex]

What are basic Trigonometric functions?

The angles of sine,  cosine, and tangent are the primary classification of functions of trigonometry. And the three functions which are cotangent, secant, and cosecant can be derived from the primary functions. Basically, the other three functions are often used as compared to the primary trigonometric functions.

Part A:

we know that, cos (a +b)=cos a cos b - sin a sin b

cos 240° = cos (60° +180°)

=cos 60° cos 180° - sin 60° sin 180°

=[tex]\frac{1}{2} *(-1)-\frac{\sqrt{3} }{2} *0[/tex]

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

Part B:

we know that, sin(a- b)=sin a cos b - cos a sin b

sin 240° = sin(270° - 30°)

=sin 270° cos 30° - cos 270° sin 30°

=[tex]-1 *\frac{\sqrt{3} }{2}-0*\frac{1}{2}[/tex]

=[tex]-\frac{\sqrt{3} }{2}[/tex]

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Step-by-step explanation:

We can start by using an equation to represent the given information. Let's call the number p, and the number k, then:

k = (1/5)p + 6

So, "six more than one-fifth of a number p is equal to a number k" can be represented mathematically as k = (1/5)p + 6. To find the value of p, we can solve for p:

p = 5k - 30

So, p is equal to 5 times the value of k minus 30.

Answer:

choice B

Step-by-step explanation:

plug in a few test values of x to prove it.  The other choice would be a cube root function of x.

Answer:

To approximate the area between the x-axis and the graph of f(x) = x^2 + 4 over the interval [0, 2], we can use the right rectangle method, where the heights of the rectangles are given by the value of the function at the right endpoint of each subinterval. If we divide the interval [0, 2] into 4 equal subintervals of width 0.5, the right endpoint of each subinterval would be 0.5, 1, 1.5, 2.

The height of the first rectangle would be f(0.5) = 0.5^2 + 4 = 4.25, the height of the second rectangle would be f(1) = 1^2 + 4 = 5, the height of the third rectangle would be f(1.5) = 1.5^2 + 4 = 6.25, and the height of the fourth rectangle would be f(2) = 2^2 + 4 = 8.

The sum of the areas of the rectangles is equal to (0.5) × (4.25 + 5 + 6.25 + 8) = (0.5) × 24 = 12.

So, the approximate area between the x-axis and the graph of f(x) = x^2 + 4 over the interval [0, 2] is 12.

Answer:

a= 12, b=4, Choices B and F

Step-by-step explanation:

2/2=1

8/2=4

12/2=6

Answer:

20

Step-by-step explanation:

[tex]\sqrt[3]{64xy^{3}}[/tex] = [tex]\sqrt[3]{64}[/tex] × [tex]\sqrt[3]{x}[/tex] × [tex]\sqrt[3]{y^{3}}[/tex]

a = coefficient

  = [tex]\sqrt[3]{64}[/tex]

∴a = 4

[tex]x^{b}[/tex] = [tex]x^{-2}[/tex]×[tex]\sqrt[3]{x}[/tex]

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

   = [tex]x^{-2+\frac{1}{3}}[/tex]

   = [tex]x^{\frac{-6}{3} + \frac{1}{3}}[/tex]

   = [tex]x^{\frac{-6+1}{3}}[/tex]

[tex]x^{b}[/tex] = [tex]x^{-\frac{5}{3}}[/tex]

∴b = [tex]-\frac{5}{3}[/tex]

[tex]y^{c}[/tex] = [tex]y^{-4}[/tex]×[tex]\sqrt[3]{y^{3}}[/tex]

   = [tex]y^{-4}[/tex]× [tex](y^{3})^{\frac{1}{3}}[/tex]

   = [tex]y^{-4}[/tex] × [tex]y[/tex]

   = [tex]y^{-4 + 1}[/tex]

[tex]y^{c}[/tex] = [tex]y^{-3}[/tex]

∴c = [tex]-3[/tex]

∴Product of a, b and c = [tex](4)[/tex]×[tex](-\frac{5}{3})[/tex]×[tex](-3)[/tex]

                                      = 20

The x-components  and y-components of the vector [tex]\mathbf{\hat v}[/tex] are -3.0 cm/s and 0 cm/s, respectively.

In mathematics, an axis is a reference line used to locate points in space. It is often used to define a coordinate system, which is a system for representing points and geometric shapes in space using numbers or coordinates.

To find the x- and y-components of the vector [tex]\mathbf{\hat v}[/tex] = (3.0 cm/s, −x-direction), we need to determine the direction of the vector. The notation "-x-direction" means that the vector points in the opposite direction of the positive x-axis. Therefore, the angle between the vector and the negative x-axis is 180 degrees.

The magnitude (or length) of the vector is given by the first component, which is 3.0 cm/s.

The x-component of the vector can be found by multiplying the magnitude by the cosine of the angle:

x-component = magnitude x cos(angle) = 3.0 cm/s x cos(180°) = -3.0 cm/s

The negative sign indicates that the x-component is in the opposite direction of the positive x-axis.

The y-component of the vector can be found by multiplying the magnitude by the sine of the angle:

y-component = magnitude x sin(angle) = 3.0 cm/s x sin(180°) = 0 cm/s

Therefore, the x-components  and y-components of the vector [tex]\mathbf{\hat v}[/tex] are -3.0 cm/s and 0 cm/s, respectively.

Expressed as an ordered pair, the components are (-3.0 cm/s, 0 cm/s).

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A. The indifference curve of a consumer is negatively sloped to the right because it represents the consumer's preference for one good over another. As the consumer's income increases, their willingness to substitute one good for the other increases.

B. The indifference curve is convex to the origin because it shows the diminishing marginal rate of substitution of the two goods. As the consumer reaches the optimal combination of goods, the marginal rate of substitution between the two goods decreases.

C. Using the Lagrange method, the optimum combination of goods can be determined as follows:

Let L(X,Y) be the Lagrangian of this problem:

L(X,Y) = √X² + Y² - λ(200 - 3X - 4Y)

Where λ is the Lagrange multiplier.

The optimal solution is obtained by setting the derivatives of L with respect to X and Y to 0 and solving the equations simultaneously.

∂L/∂X = 0 -> X = 11.85

∂L/∂Y = 0 -> Y = 8.57

Therefore, the optimum combination of goods is X = 11.85 and Y = 8.57.

D. The slope of the indifference curve is equal to the slope of the budget line. This can be shown by comparing their respective equations and noting that their slopes are the same. The equation for the indifference curve is Y/X = (-2/3) and the equation for the budget line is Y/X = (-3/4). Since these equations are the same, it follows that the slopes of the two lines are equal.

The new triangle with vertices A'(0, 6), B'(12, -6), and C'(6, -6) would be larger than the original triangle by a factor of 120%.

A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.

To graph a polygon with vertices A(0,5), B(10,-5), and C(5,-5), we can plot these points on a coordinate plane and connect them to form a triangle.

Then, to find the image of this triangle after a dilation with scale factor k = 120%, we need to multiply each coordinate of the vertices by k = 1.2.

For vertex A, the new coordinates would be (0 × 1.2, 5 × 1.2) = (0, 6).

For vertex B, the new coordinates would be (10 × 1.2, -5 × 1.2) = (12, -6).

For vertex C, the new coordinates would be (5 × 1.2, -5 × 1.2) = (6, -6).

So, the new triangle with vertices A'(0, 6), B'(12, -6), and C'(6, -6) would be larger than the original triangle by a factor of 120%.

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Answer:

$3.25 per hotdog

Step-by-step explanation:

This question can be written as a system of equations:

x = price of hotdogs

y = price of drinks

First family:

4x + 4y = 18

Second family

5x + 3y = 20

Multiply each equation so that one of the variables can cancel out:

-3(4x + 4y = 18)

4(5x + 3y = 20)

First equation:

-12x - 12y = -54

Second equation

20x + 12y = 80

Add the equations together

-12x - 12y = -54

20x + 12y = 80

8x = 26

Divide

x = $3.25

Insert the x into one of the original equations to find y:

4(3.25) + 4y = 18

13 + 4y = 18

4y = 5

y = $1.25

Answer:

Answer is in the attached photo.

Step-by-step explanation:

The solution is in the attached photo, do take note a variable is a alphabet can be assigned to any value, while a constant is a fixed number or value.

Answers:

variable = t

constant = -7

Step-by-step explanation:

So we know that a variable is a letter while a constant is a number

in here -7 is the constant (the number) and t is the variable (the letter)

Therefore, the variable is t and the constant is -7.

The height of the tree to the nearest hundredth of a meter is 10.77m.

Detailed explanation:

An illustration of identical triangles is shown in the diagram. View the illustration I included with my response to see how triangles can be used to solve this issue.

Because they both share an internal angle, the triangles made by Alyssa and her shadow and the triangle created by the tree and its shadow are examples of comparable triangles.

The ratios of the comparable sides on both sides are proportionate according to the similar triangles theorem.

Therefore, in order to determine the height of the tree, we must make a percentage of the corresponding sides of both triangles. Let x equal the tree's height.

[tex]\frac{BE}{BC} =\frac{DE}{AC}[/tex]

BE = 4.25 meters

DE = 1.65 meters

BC = 27.75 meters

AC = x meters

[tex]\frac{4.25}{27.75} =\frac{1.65}{x}[/tex]

Cross multiply is now used to find x.

4.25x= 1.65(27.75)

x = 10.77

Therefore, the height of the tree is 10.77 meters.

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Complete question:

Alyssa is 1. 65 meters tall. At 11 a. M. , she measures the length of a tree's shadow to be 27. 75 meters. She stands 23. 5 meters away from the tree, so that the tip of her shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

With percentage=76% of foreign coins of total 150 coins, Other coins collected were 36.

What is the percentage?

In mathematics, a percentage is a statistic or ratio that may be expressed as a fraction of 100. To find the percentage of a number, divide it by its total and multiply by 100. Hence, the percentage represents one part of a hundred. The phrase % stands for one hundred percent. It is represented by the symbol "%".

Here are some instances of percentages:

Now,

Given that total coins=150

Foreign coins=76% of total

then other coins are 24% of total

so other coins=150*24/100

=72/2

=36

Hence,

          Other coins collected were 36.

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Answer:

C.Only Maryanne's equation has the same solution as the original equation.

The solution is : 39 is the solution to Mr. Smith's equation.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

here, we have,

given that,

the equation is:

(16-2w)^2 + (3y÷3z)

Let w=5 y=9 and z=3

(16-2*5)^2 + (3*9÷3*3)

PEMDAS says parentheses first

Multiply and divide in the parentheses

(16-10)^2 + (27÷9)

Then add and subtract in the parentheses

(6)^2 + (3)

Now the exponent

=36 +3

=39

Hence, The solution is : 39 is the solution to Mr. Smith's equation.

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complete question:

What is the solution to Mr. Smith's equation?

There will be about 95 counselors there with the help of expression 2375/50.

What are expressions exactly?

In mathematics, expressions are mathematical claims that consist of at least two sentences that contain numbers, variables, or both, and are linked by an operator in between. Mathematical operations include addition, subtraction, multiplication, and division. For instance, x + y is an equation in which the words x and y are separated by an addition operator. In mathematics, there are two types of expressions: numerical expressions, which include only numbers, and algebraic expressions, which contain both numbers and variables.

e.g. A number is 6 more than half of another number, x. This proposition can be stated mathematically as x/2 + 6. Mathematical expressions are used to answer complex problems.

Now,

Given that Total campers = 2375

and for each cabin there are 2 counselors and 50 campers

then no. of cabins = 2375/50

No. of counselors= 2*no. of cabins

=2*2375/50

=95

Hence,

            There will be about 95 counselors there with the help of expression 2375/50.

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The perimeter and area of the original square are 4s and s², respectively, and the perimeter and area of the dilated square are 192 and 2304, respectively.

What is area of square ?

Area of square can be defined as the square of given side.

Given ,

Let's assume that the side length of the original square is 's'. The perimeter of the original square is then 4s and its area is s².

The perimeter of the dilated square will be 4 times longer than the original square, so it will be 4 * 4s = 16s.

The area of the dilated square will be 4 times greater than the original square, so it will be 4^2 * s² = 16s².

Here s is 12

so ,

16s = 16 * 12 = 192

16s*s = 16 * 12 * 12 = 2304

Therefore, The perimeter and area of the original square are 4s and s², respectively, and the perimeter and area of the dilated square are 192 and 2304, respectively.

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Answer:

x = 80

Step-by-step explanation:

supplementary angles sum to 180°

sum the 2 angles and equate to 180

x + 40 + x - 20 = 180

2x + 20 = 180 ( subtract 20 from both sides )

2x = 160 ( divide both sides by 2 )

x = 80

The intervals of each behavior of the function are given as follows:

a) Increasing: (-∞, -1).

b) Decreasing: (-1, 1).

c) Constant: (1, 3).

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable .

here, we have,

to classify the function as increasing, decreasing or constant:

The function is increasing when the graph moves right and up.

The function is decreasing when the graph moves right and down.

The function is constant when the graph of the function is an horizontal line.

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In the cyclic quadrilateral the value of R and S are

< S = 87 degrees

< R = 216 degrees

In a cyclic quadrilateral, the opposite angles are supplementary hence

< S + < Q = 180

in the problem we have that < Q = 93

< S + 93 = 180

< S = 180 - 93

< S = 87 degrees

Angle R is adjacent angle to angles 126 and 90 hence we have that

< R = 126 + 90

< R = 216 degrees

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The values of x for which z is greater than y are 7.

In mathematics, an equation is a statement that two mathematical expressions are equal. The expressions can contain variables, constants, mathematical operations, and functions. An equation can be used to represent a relationship between two or more variables, and it is often used to solve problems by finding the values of the variables that satisfy the equation.

Equations can be represented in different forms, such as algebraic equations, differential equations, integral equations, and partial differential equations. Algebraic equations are the most common type of equations, and they involve algebraic operations such as addition, subtraction, multiplication, and division.

Equations can be classified according to their degree, which is the highest power of the variable in the equation. Linear equations are those with a degree of one, and they can be represented by a straight line on a graph. Quadratic equations have a degree of two, and they can be represented by a parabolic curve. Higher degree equations are generally more complex, and they may not have a simple geometric representation.

For x = -4, y = 4(-4) - 3 = -19 and z = -18, so y is greater.

For x = 2, y = 4(2) - 3 = 5 and z = 6, so z is greater.

For values of x where z is greater than y, we need to compare the values of z and y for each x value in the table.

At x = 3, y = 4(3) - 3 = 9 and z = 9, so they are equal.

At x = 7, y = 4(7) - 3 = 25 and z = 27, so z is greater.

At x = 10, y = 4(10) - 3 = 37 and z = 35, so z is not greater than y.

Therefore, the values of x for which z is greater than y are 7.

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Answer:

So, the answer is (b) 320 feet.

Step-by-step explanation:

Let's call the length and width of the first playground "x".

The perimeter of the first playground is 160 feet, so 2 times the length plus 2 times the width is equal to 160:

2x + 2x = 160

Simplifying:

4x = 160

Dividing both sides by 4:

x = 40

So the length and width of the first playground are both 40 feet.

The second playground is twice as long and twice as wide as the first playground, so the length and width of the second playground would be 2 * 40 = 80 feet.

The perimeter of the second playground would then be 2 * 80 + 2 * 80 = 320 feet.

So, the answer is (b) 320 feet.

Answer: 266

Step-by-step explanation:

I = prt

p= 1900

r=.07

t=2

1900x.07x2=266

Answer:

----------------------

Given inequality:

Consider x ≠ 0 and multiply all terms by x, 4 and 3:

The x-intercepts are:

This quadratic function has a positive leading coefficient and two zeros, and hence is positive when:

The matching choice is C.

Answer:

Step-by-step explanation:

The formula for the circumference of a circle is C = 2 * π * r, where r is the radius of the circle. We can find the radius of the circle by rearranging the formula to r = C / (2 * π).

Substituting the given values into the formula, we get:

r = 21.98 inches / (2 * 3.14) = 21.98 inches / 6.28 = 3.50 inches

The diameter of the circle is equal to twice the radius, so we can find the diameter by multiplying the radius by 2:

d = 2 * r = 2 * 3.50 inches = 7.00 inches.

To find the diameter of a circle with a given circumference, you can use the following steps:

Define the formula for the circumference of a circle: The circumference of a circle is given by the formula C = 2πr, where π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Substitute the given circumference into the formula: If the circumference of the circle is 21.98 inches, we can substitute that into the formula to get 21.98 = 2πr.

Solve for the radius: To solve for the radius, we can divide both sides of the equation by 2π: 21.98 / 2π = r. Using 3.14 for π, this gives us 21.98 / (2 * 3.14) = r, which can be simplified to r = 2.01 inches.

Calculate the diameter: The diameter of a circle is twice the radius, so the diameter can be calculated as 2 * r = 2 * 2.01 = 4.02 inches.

So, the diameter of the circle with a circumference of 21.98 inches is 4.02 inches.

Lowell's verbal representation of the equation y = x - 25 is "y is equal to x minus 25".

This verbal representation accurately describes the equation. In the equation, y is the dependent variable and x is the independent variable. The equation says that y is equal to x minus 25, which means that for every value of x, there is a corresponding value of y that is x minus 25.

So, Lowell has completed the task correctly.

Answer:

Alex spends more time 2/5 of 100 is 40%

The requried hours spent babysitting and tutoring are 5 and 8 hours respectively.

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
Let the hours spent babysitting and tutoring be x and y respectively,
According to the quesiton,

Mark gets paid $6 per hour for babysitting and $15 an hour for tutoring. He makes $150 in one week.
6x + 15y = 150 - - - -(1)

John Makes $10 an hour babysitting and $8 an hour tutoring and gets paid a total of $114.
10x + 8y = 114 - - - - -(2)

Solving above equations 1 and 2 by elimination method we get x = 5 and y = 8

Thus, the requried hours spent for babysitting and tutoring are 5 and 8 hours respectively.

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