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The equation r²cos²(θ) - 2rcos(θ) + r²sin²(θ) = 8 to rectangular form is (x - 1)² + y² = 9

From the question, we have the following parameters that can be used in our computation:

r²cos²(θ) - 2rcos(θ) + r²sin²(θ) = 8

The above equation is in its polar form, where

r² = x² + y²

cos(θ) = x/r

sin(θ) = y/r

Rearrange the terms in r²cos²(θ) - 2rcos(θ) + r²sin²(θ) = 8

So, we have

r²cos²(θ)  + r²sin²(θ) - 2rcos(θ) = 8

Factorize

r²(cos²(θ)  + sin²(θ)) - 2rcos(θ) = 8

In trigonometry

cos²(θ)  + sin²(θ) = 1

So, we have

r² - 2rcos(θ) = 8

Recall that

r² = x² + y²

cos(θ) = x/r

So, we have

x² + y² - 2r * x/r = 8

Evaluate

x² + y² - 2x = 8

Rearrange

x²  - 2x + y² = 8

Express as perfect squares

(x - 1)² + y² = 8 + 1

So, we have

(x - 1)² + y² = 9

Hence, the equation to rectangular form is (x - 1)² + y² = 9

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If your objective is to save $300,000 over 25 years, you would need to put away around $3077.41 every six months.

To solve this problem, we can use the formula for the future value of an annuity:

[tex]FV = P *\frac{((1 + r/n)^{(n*t)} - 1)}{(r/n)}[/tex]

where FV is the future value, P is the periodic payment, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

Since the interest is compounded annually, we have n = 1 and r = 0.05. To find the periodic payment P, we can rearrange the formula:

P = FV * (r/n) / ((1 + r/n)^(n*t) - 1)

Plugging in the given values, we have:

[tex]P = 300000 * (0.05/2) / ((1 + 0.05/2)^{(2*25)} - 1)[/tex]

P ≈ $3077.41

Therefore, you would need to invest approximately $3077.41 every six months to reach a savings goal of $300,000 at the end of 25 years.

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Answer:

6 × 3 = 18

18 ÷ 2 = 9

Therefore, the expression 6 × 3 ÷ 2 simplifies to 9

Step-by-step explanation:

To simplify the expression, we follow the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right.There

are no parentheses or exponents,so we multiply 6 by 3 to give us 18.

Then we divide 18 by 2 to give us 9.

There for the Expression simplifies to 9.

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The fraction in factored form is determined as  (x - 8)(x + 10) / (x - 10)(x + 8).

The quotient of the division is calculated as follows;

( x² - 3x - 40 )/ (x² + 8x - 20) ÷ (x² + 13x + 40)/(x² + 12x + 20)

Factorize each of the expressions;

x² - 3x - 40

= x² + 5x - 8x - 40

= x(x + 5) - 8(x + 5)

= (x + 5)(x - 8)

x² + 8x - 20

= x² + 2x - 10x - 20

= x(x + 2) - 10(x + 2)

= (x + 2)(x - 10)

x² + 13x + 40

=  x² + 5x + 8x + 40

= x(x + 5) + 8(x + 5)

= (x + 5)(x + 8)

x² + 12x + 20

=  x² + 2x + 10x + 20

= x(x + 2) + 10(x + 2)

= (x + 2)(x + 10)

The fraction in factored form is written as;

(x + 5)(x - 8)/(x + 2)(x - 10) ÷ (x + 5)(x + 8)/(x + 2)(x + 10)

(x + 5)(x - 8)/(x + 2)(x - 10)  x  (x + 2)(x + 10)/(x + 5)(x + 8)

= (x - 8)(x + 10) / (x - 10)(x + 8)

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When the y-value starts to decrease and the x-value remains constant over time, Y decreases and X increases to a consistent slope.

A consistent slope represents a constant rate of change between two variables, typically represented by the ratio of the change in y to the change in x.

When the y-value starts to decrease and the x-value remains the same over time, the consistent slope represents a negative correlation between y and x.

This means that as the value of y decreases, the value of x remains the same or may increase, depending on the context of the problem.

Therefore, the answer is option B: Y decreases and X increases.

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Answer:

70%

Step-by-step explanation:

63 / (63+27)

63 / 90 = 0.7

Turn 0.7 into a fraction, 70 under 100

Then rewrite the fraction 70 under 100 to 70%

The exponential function that models the relationship between the two points is [tex]y = 41.62 (1.0824)^x.[/tex]

We have,

To write an exponential model given two points (6,85) and (7,92), we can use the general exponential function form:

y = a[tex]b^x[/tex]

where y is the dependent variable, x is the independent variable, a is the initial value (when x = 0), and b is the base of the exponential function.

To find the values of a and b, we can use the two given points as follows:

When x = 6, y = 85:

85 = a b^6

When x = 7, y = 92:

92 = a b^7

We now have a system of two equations with two unknowns (a and b).

We can solve for a by dividing the second equation by the first equation:

92/85 = (ab^7) / (ab^6)

1.0824 = b

Substitute this value of b into one of the equations to solve for a:

85 = a (1.0824)^6

a = 41.62

Therefore,

The exponential function that models the relationship between the two points is [tex]y = 41.62 (1.0824)^x.[/tex]

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The machine's purchasing cost is $309,210.

The acquisition cost of the machine is the total cost incurred by the firm to purchase and put the machine into service.

To determine the acquisition cost of the machine, we need to add up all the costs listed in the question.

Invoice price = $299,000

Applicable sales tax = $2,100

Freight paid = $1,160

Cost of insurance coverage = $1,025

Installation costs = $1,900

Testing and adjusting costs = $1,375

Repair of damages caused by the firm's employees = $1,450

Prepaid maintenance contract = $1,200

Total acquisition cost = Invoice price + Sales tax + Freight paid + Cost of insurance coverage + Installation costs + Testing and adjusting costs + Repair of damages caused by the firm's employees + Prepaid maintenance contract

Total acquisition cost = $299,000 + $2,100 + $1,160 + $1,025 + $1,900 + $1,375 + $1,450 + $1,200

Total acquisition cost = $309,210

Therefore, the acquisition cost of the machine is $309,210.

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A type of association that is seen between the age and height of the children include the following: C. positive linear association.

In Mathematics and Statistics, a positive correlation is used to described a scenario in which two variables move in the same direction and are in tandem.

This ultimately implies that, a positive correlation exist when two variables have a linear relationship or are in direct proportion. Therefore, when one variable increases, the other variable generally increases, as well.

By critically observing the scatter plot shown in the image attached above, we can reasonably infer and logically deduce that there is a positive correlation between the x-values (age) and y-values (height) because they both increase simultaneously.

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The correct statements regarding the scatter plot are given as follows:

The scatter plot, as shown by the image given in the problem is a series of points of the data-set in the following format:

(SAT score, GPA).

There is a positive association between the data, as the line of best fit, which gives the best approximation between the SAT score and the GPA, and can be used to predict the GPA based on the SAT score, is increasing.

For a SAT score of 1700, an expected GPA is of about 3.1, considering the intersection of y = 3.1 with the line of fit when x = 1700.

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Step-by-step explanation:

Exponetial (compounding ) growth

  New Price = original price ( 1 + .10)^n      n is months  .10 is 10% per month

                     = $  110 ( 1.10)^3 = $ 146 . 41   three months later

The four math sentences that have an answer of 2.4 are 1.2 + 1.2, 1.2 * 2, 3.6 - 1.2 and  4.8/2

From the question, we have the following parameters that can be used in our computation:

An answer of 2.4

Using a different operation in each sentence., we have the following math sentences

1.2 + 1.2

1.2 * 2

3.6 - 1.2

4.8/2

The solution to each of the above sentences is 2.4

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The possible values of x, given the number of bets are $54,999, $14,999, and -$1.

The expected winnings would be -$0.97.

The possible values of the net winning would be the amount you win at net if you win first prize, second prize, or nothing.

The possible values would therefore be:
$54,999, $14,999, and -$1.

These show that if you win the first or second prize, you will deduct the $ 1 you bet. And if you win nothing, you would have lost $ 1.

The expected winnings would be:

= ∑ ( Probability of event x Potential net win / loss)
= ( 1 / 3, 000, 000 x 54, 999 ) + ( 2 / 3, 000, 000 x 14, 999 ) + ( 2,999, 997 / 3, 000, 000 x - 1 )

= - $ 0.97

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Since on average, 66% of the days in Seattle, Washington, are cloudy or rainy, a model that could be used to simulate this situation is an experimental probability model of about 2/3.

In this scenario and exercise, you are to required to describe a model that you could be used to simulate a situation in which 66% of the days in Seattle, Washington, are cloudy or rainy on an average.

In this context, we can reasonably infer and logically deduce that the most accurate and feasible model to use would be an experimental probability model of about 2/3;

66% = 66/100 = 33/50 ≈ 2/3.

In conclusion, you should select two (2) red marbles that each represent cloudy or rainy days and one (1) blue marble that represent non-cloudy or non-rainy days, and then pick one marble randomly. Repeat several times in order to determine experimental probability that a day would either be cloudy or rainy.

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The strategies that Alondra can use to complete the statement include:

When a decimal number is created by Alondra and possesses 4 or more in the tenths place, it will exceed 4.3, as stated to exemplify decimals including 4.31 or 4.32.

Similarly, if there are 3 units plus a greater number appended behind the second digit after the decimal point (hundredths place). Deriving an outcome extending further than 4.30 can occur for any value above that obtained from inputting the numbers 4 and 3 into their respective positions in a given sequence.

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Answer:

a)

[tex] log_{7}(3) - log_{7}(5) = log_{7}( \frac{3}{5} ) [/tex]

b)

[tex] log_{3}(11) + log_{3}(2) = log_{3}(22) [/tex]

c)

[tex] - 4 log_{8}(3) = log_{8}( {3}^{ - 4} ) = log_{8}( \frac{1}{ {3}^{4} } ) [/tex]

The amount of soil to fill the garden box is V = 1,134 inches³

Given data ,

Let the volume of the first rectangular box be V₁

Let the volume of the second rectangular box be V₂

where Volume of Rectangle = Length x Width x Height

So , V₁ = 21 x 7 x 2

V₁ = 294 inches³

And , V₂ = 12 x 10 x ( 7 )

V₂ = 120 x 7

V₂ = 840 inches³

And , the total soil in the garden box = V₁ + V₂

Amount of soil in garden box = 1,134 inches³

Hence , the volume is V = 1,134 inches³

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All the four ordered pairs that relate the side length of a pentagon to the perimeter of the pentagon are,

⇒ (1, 5) (2, 10) (3, 15) (4, 20)

Given that;

Fernando also learns in math class that the rule Add 1 to the side length of a pentagon with equal sides length results the rule Add 5 to the perimeter.

Now,

Here are four ordered pairs that relate the side length of a pentagon to the perimeter of the pentagon:

⇒ (1, 5) (2, 10) (3, 15) (4, 20)

Hence, In each pair, the first number represents the side length of the pentagon, and the second number represents the perimeter of the pentagon.

And,  If you add 1 to the side length in each pair, you'll see that the perimeter increases by 5, just as Fernando learned in math class.

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The angle in radians in simplest form is 7π/4 radians.

In Mathematics and Geometry, you will have to divide the central angle that is subtended by the arc of a circle by 360 degrees if you want to calculate the arc length formed by a circle, and then multiply this fraction by the circumference of the circle.

Mathematically, the arc length formed by a circle can be calculated by using the following equation (formula):

Arc length = rθ

Where:

By substituting the given parameters into the arc length formula, we have the following;

7π/2 = 2 × θ

θ = 7π/4 radians.

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The formula for the inverse function h⁻¹(P) is:

The population of insects in the lab experiment is 90, then the number of days it takes for the population to reach that size is approximately 17.5 days.

To find the inverse function h⁻¹(P), we need to solve the equation P = h(t) for t in terms of P.

Starting with P = h(t) = [tex]100/1+4e^-^0^.^0^5^t[/tex]

we can first multiply both sides by [tex]1+4e^-^0^.^0^5^t[/tex] to get:

[tex]P(1+4e^-^0^.^0^5^t) = 100[/tex]

t = -20ln(4/(P-100)+1)

Therefore, the formula for the inverse function h⁻¹(P) is:

h⁻¹(P) = -20ln(4/(P-100)+1)

(b) To find h⁻¹(90)

we substitute P = 90 into the formula for h^-1(P):

h⁻¹(90) = -20ln(4/(90-100)+1)

= 17.5

This means that if the population of insects in the lab experiment is 90, then the number of days it takes for the population to reach that size is  17.5 days.

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A) Maxine calculated the total mass of pH powder sold on day 1 by evaluating the algebraic expressions for small and large container sales.

B) If the variable s represents the number of containers sold, the value of the algebraic expression 0.75s represents the total mass of pH powder sold each day.

C) The algebraic expression to represent the mass (in kilograms) of powder sold in large containers daily is 1.25l.

D) The algebraic expression to show how Maxine calculated the total mass of pH powder sold on day 1 is 0.75s + 1.25l, which is evaluated to be 49.25 kg.

E). Using the algebraic expression, the total mass of powder sold on Day 2 and 3 is as follows:

An algebraic expression is a combination of of variables and constants, together with mathematical operands (addition, subtraction, division, and multiplication, etc.).

pH Plus Power Sales

Day       # of small containers    # of large containers    Total Mass

                    (0.75 kg)                            (1.25 kg)              (Kilograms)

Day 1               14                                       31                        49.25 kg

Day 2              19                                      42                        66.75 kg

Day 3             24                                      28                        53.00 kg

Let the number of small containers sold = s

Let the number of large containers sold = l

The mass of a small container = 0.75 kg

The mass of a large container = 1.25 kg

Total mass of small containers sold daily  = 0.75s

Total mass of large containers sold daily = 1.25l

Total mass of small and large containers sold daily = 0.75s + 1.25l

Day 1 = 0.75(14) + 1.25(31) = 49.25 kg

Day 2 = 0.75(19) + 1.25(42) = 66.75 kg

Day 3 = 0.75(24) + 1.25(28) = 53.00 kg

pH Plus Power Sales

Day       # of small containers    # of large containers    Total Mass

                    (0.75 kg)                            (1.25 kg)              (Kilograms)

Day 1               14                                       31                        49.25 kg

Day 2              19                                      42

Day 3             24                                      28

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Answer: f(x)=x-2

Step-by-step explanation:

Linear function means a straight line, so graph each one to find which is a straight line

Answer:

[tex] \frac{4}{5} [/tex]

Only afford a home with a market value of about $375,000, and she would require a down payment of $75,000 to do so.

First, we have to ascertain how much Kelly can comfortably afford to pay for her mortgage each month. Her income, expenses, and other financial commitments will determine this. Her monthly mortgage payment should not be more than 28% of her total monthly income, on average.

The loan is interest rate is the next thing we need to think about. Kelly will have a constant monthly payment for the duration of the loan thanks to a 30-year fixed rate loan at the current market rate of 4.25%.

With a 20% down payment and a loan with a 4.25% interest rate, we could be calculate the maximum purchase price to be roughly $375,000 using a mortgage calculator.

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The step that indicates the addition property of equality while solving the equation -1 - 6x = x - 15 is : (b) -1 - 6x + 15 = x - 15 + 15.

The "addition-property" of equality states that if we add the same quantity to both sides of an equation, the equation remains true.

In the equation -1 - 6x = x - 15, we need to isolate the variable term 'x'.

To isolate variable "x",  we are adding 15 to both sides of the equation, which is the addition property of equality. This step simplifies the equation and isolate the variable "x" on one side.

So, we get,  -1 - 6x + 15 = x - 15 + 15,

Therefore, the correct step indicating the "addition-property" is (b).

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The given question is incomplete, the complete, the complete question is

Which of the following steps indicates the addition property of equality while solving the equation -1 - 6x = x - 15?

(a) x = 14/2

(b) -1 - 6x + 15 = x - 15 + 15

(c) -1 - 6x = x - 15

(d) 23 - 6x - 24 = x - 15

Answer:

(x + 4)/x^2 - 8x)

Step-by-step explanation:

Using the sine ratio, the value of p in the right triangle, in simplest form is: p = 4.

To find the value of p, we have to determine what formula to use. We would apply the sine ratio, which is:

sin ∅ = opposite side / hypotenuse.

From the right triangle in the diagram, we have:

sin 30 = p / 8

p = sin 30 * 8

Note: sin 30 = 1/2

Therefore, we have:

p = 1/2 * 8

p = 4

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Answer:

poor photo quality.

Step-by-step explanation:

The picture for reduction and enlargement is attached

Scale factors are used to increase or decrease image. The situation of increment is usually called magnifying. while that of decrease is reduction.

The drawing of the the reduction is a square of side 8 units. The scale factor given is 1/2 hence we have the dimensions are as follows

preimage: 8 units x 8 units

image: 4 units x 4 units

The drawing of the the enlargement is a circle of radius 2 units. The scale factor given is 2, hence we have the dimensions are as follows

preimage: radius = 2 units

image: radius = 4 units

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The momentum change of the rock caused by its fall will be 35.49 x [tex]10{-24[/tex] m/sec and the magnitude of the earth velocity will be 212.94 kgm/s

We have,

Mass of the rock = 18 kg

Distance traveled = 7 m

Mass of earth = 6 x [tex]10^{24[/tex] Kg

Using second equation of motion

S= ut  + 1/2 at²

If the body freely falls initial velocity will be zero.

u = initial velocity = 0

Then, S = 1/2 at²

7 = 1/2 x 10 x t²

14/ 10= t²

t= 1.183 sec

Now, Change in the momentum of the rock will be,

ΔP= FΔT

ΔP = (mg ) ΔT

ΔP= (18) x  10 x 1.183

ΔP=212.94 kg-m/s²

Now,  Earth will also have same momentum change

ΔP(earth)= ΔP(rock)

6 x [tex]10^{24[/tex] Kg x Δv = 212.94

Δv = 35.49 x [tex]10{-24[/tex] m /sec

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