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Integer Operations

Answer: B

Explanation:

A + B = (2x - 7x² + 3) + (4x² - 12x) = 2x - 7x² + 3 + 4x² - 12x = -3x² -10x + 3

The fraction 15/365 has been simplified as 3/73.

In the fraction 15/365, the numerator is 15, and the denominator is 365. The numerator represents the number of parts we're interested in, and the denominator represents the total number of parts in the group. So, in this case, we're interested in 15 parts out of a group of 365.

To simplify this fraction, we need to find a common factor of the numerator and the denominator that we can divide both by. In this case, 5 is a common factor of both 15 and 365. Dividing both by 5 gives us:

15/365 = (15 ÷ 5) / (365 ÷ 5) = 3/73

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The area of Vanessa's patio is represented by the expression 3x⁴ - 4x³ - 13x² + 35x + 10, which is obtained by multiplying the expressions representing the length and width.

To find the area of Vanessa's patio, we need to multiply the expressions representing the length and width.

Length = 3x² + 5x + 10

Width = x² - 3x - 1

Area = Length x Width

Area = (3x² + 5x + 10)(x² - 3x - 1)

To multiply these expressions, we can use the distributive property and then simplify

Area = 3x⁴ - 4x³ - 13x² + 35x + 10

Therefore, the expression that represents the area of Vanessa's patio is 3x⁴ - 4x³ - 13x² + 35x + 10.

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Answer:

[tex]{ \boxed{ \mathfrak{amswer}}} :{ \underline{ \tt{ \: { \alpha }^{2} + { \beta }^{2} = - 1} \: }}[/tex]

Step-by-step explanation:

From General equation of a polynomial:

✍️ x² - (Sum of roots)x + (Product of roots) = 0

Therefore;

[tex]{ \tt{f(x) = {x}^{2} - x + 1 = 0 }}[/tex]

From the equation,

[tex]{ \tt{ \alpha + \beta = 1}} - - - { \rm{sum}} \\ { \tt{ \alpha \beta = 1 }} - - - { \rm{product}}[/tex]

From the question:

[tex]{ \tt{ {( \alpha + \beta )}^{2} = ( \alpha + \beta )( \alpha + \beta ) }} \\ \\ { \tt{ {( \alpha + \beta )}^{2} = { \alpha }^{2} + 2 \alpha \beta + { \beta }^{2} }} \\ \\ { \tt{ {( \alpha + \beta )}^{2} = ( { \alpha }^{2} + { \beta }^{2} ) + 2 \alpha \beta }} \\ \\ { \boxed{ \tt{( { \alpha }^{2} + { \beta }^{2} ) = {{( \alpha + \beta )}^{2} - 2 \alpha \beta }}}}[/tex]

Therefore, from sum and product

[tex]{ \tt{ { \alpha }^{2} + { \beta }^{2} = (1) {}^{2} - 2(1) }} \\ = { \tt{1 - 2}} \\ = - 1[/tex]

Answer:

Slope of green line:

[tex]m = \frac{ - 3 - 3}{4 - ( - 4))} = - \frac{6}{8} = - \frac{3}{4} [/tex]

Slope of red line:

[tex]m = \frac{3 - ( - 5)}{3 - ( - 3)} = \frac{8}{6} = \frac{4}{3} [/tex]

The slopes of these two lines are negative reciprocals of each other.

Therefore, the correct answer is True (A).

The equation of the ellipse is given by  ( x - 2 )² / 25 + ( y - 2 )² / 16 = 1

Given data ,

Let the center of the ellipse be ( h , k ) = ( 2 , 2 )

Now , the length of the major axis is 2a = 10 units

The length of the minor axis 2b = 8 units

So , a = 5 and b = 4

Now , the standard for of equation of ellipse is

( x²/a² ) + ( y²/b² ) = 1

On simplifying , we get

( x - 2 )² / ( 5 )² + ( y - 2 )² / ( 4 )² = 1

( x - 2 )² / 25 + ( y - 2 )² / 16 = 1

Hence , the equation of ellipse is ( x - 2 )² / 25 + ( y - 2 )² / 16 = 1

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The quadrilateral is plotted on the graph

Given data ,

Let the quadrilateral be represented as UVWX

Now , the coordinates of the quadrilateral are

U (2,-5), V (6, 2), W (-1,6) and X (-5, -1)

Now , the distance between the points is given by the distance formula ,

Distance D = √ ( x₂ - x₁ )² + ( y₂ - y₁ )²

D = √ ( 6 - 2 )² + ( -5 -2 )²

D = √ ( 4 )² + ( 7 )²

D = √ ( 16 + 49 )

D = √65 units

And , the distance between VW is given by

D = √ ( 6 + 1 )² + ( 2 - 6 )²

D = √ ( 7 )² + ( 4 )²

D = √ ( 16 + 49 )

D = √65 units

Hence , the quadrilateral is a square

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The class interval containing the median speed is 30 < s < 40.

We must first establish the median position before we can locate the class interval holding the median speed.

By dividing the overall frequency by two, the median position can be calculated.

Total frequency = 12 + 16 + 18 + 2 + 2 = 50

Median position = 50 / 2 = 25

Now, we have to solve for the class interval that has the median position which is the 25th value.

20 < s < 30: Frequency is 12, cumulative frequency is 12 (Not containing the 25th value)

30 < s < 40: Frequency is 16, cumulative frequency is 12 + 16 = 28 (Contains the 25th value)

So, the class interval containing the median speed is 30 < s < 40.

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The formula to calculate the future value of an investment with compound interest is:

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

where:

A = future value

P = principal (initial investment)

r = annual interest rate (as a decimal)

n = number of times interest is compounded per year

t = time (in years)

In this case, we have:

P = $200 (initial investment)

r = 20% = 0.20

n = 1 (compounded annually)

t = 6 (6 years)

So, we can plug these values into the formula:

A = 200(1 + 0.20/1)^(1*6)

A = 200(1.20)^6

A = $766.97 (rounded to the nearest cent)

Therefore, Robert will have approximately $766.97 in his retirement account after 6 years.

Answer: x = 10[tex]\sqrt{2}[/tex]

Step-by-step explanation:

    We are given that this is an isosceles triangle and the little box represents 90 degrees. This means that both the "legs" of the triangle are equal to 10 and the hypotenuse is equal to x. Since this is a right triangle, we can use the Pythagorean theorem to solve.

a² + b² = x²

10² + 10² = x²

100 + 100 = x²

200 = x²

[tex]\sqrt{200}[/tex] = x

x = [tex]\sqrt{200}[/tex]

x = [tex]\sqrt{2*100}[/tex]

x = 10[tex]\sqrt{2}[/tex]

The key features of this exponential function include the following:

y-intercept of f: (0, 10).

y-intercept of g: (0, 5).

Domain of both f and g: All real numbers or (-∞, ∞).

Range of both f and g = (0, ∞) or {y|y ≥ 0}.

End behavior of both f and g: as x → -∞, y → 0 and x → ∞, y → ∞

In Mathematics and Geometry, an exponential function is typically represented by the following mathematical equation:

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

Where:

Based on the given exponential function f, we can logically deduce that the y-intercept is represented by a, which is equal to 10 while the y-intercept of g is equal to 5.

In conclusion, there is a horizontal asymptote at y = 0 because it is a line which the graph of a given function approaches, but never touches or crosses.

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the answer to 40 divided by 180 is 4.5

The choice correctly compares the volumes, in milliliters of two of these liquids is,

⇒ 1.078 < 1.23

We have to given that;

A scientist listed the volumes, in milliliters, of some liquids used in an experiment.

1.23

1.078

1.203

1.17

Now, We get;

By all the options;

⇒ 1.078 < 1.23

Thus, The choice correctly compares the volumes, in milliliters of two of these liquids is,

⇒ 1.078 < 1.23

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The constant of proportionality relating the number of miles y to the number of minutes x is 1/6.

To find the constant of proportionality, we can use the formula: y = kx. Where y represents the number of miles run, x represents the time taken in minutes, and k is the constant of proportionality. Coach Liba ran 9 miles in 54 minutes. We can use these values to solve for k: 9 = k(54)

Dividing both sides by 54, we get: k = 9/54 = 1/6. This means that for every 1 minute Coach Liba runs, he covers 1/6th of a mile. Or for every 6 minutes, he covers 1 mile.

The ratio of the number of miles run to the time taken in minutes is constant and equal to 1/6. This can help us predict how far Coach Liba will run in a given amount of time or how long it will take him to run a certain distance.

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The probability that a randomly selected student is interested in a "spinning-room" and also that they are a "graduate-student" is 0.428.

The total number of students that are surveyed is = 500 students,

Out of the 500 students surveyed, 41 are interested in a spinning room and are in year 1-2,

⇒ 75 are interested and are in year 3-4, and

⇒ 87 are interested and are graduate students.

So, the total number of students interested in a spinning room is

⇒ 41 + 75 + 87 = 203,

Out of these 203 students interested in a spinning room, 87 are graduate students.

So, probability that randomly selected student is interested in "spinning-room" and is a "graduate-student" is :

⇒ P(spinning room and graduate student) = 87/203,

⇒ P(spinning room and graduate student) = 0.428,

Therefore, the required probability is 0.428.

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The given question is incomplete, the complete question is

A university recently surveyed 500 students to determine which new fitness area to offer in its recreation facility. the results of the survey are summarized in the following table.

                              Spinning Room   Climbing Wall     Ellipticals

Year 1-2                              41                     81                         25

Year 3-4                             75                    78                         44

Graduate Student             87                     49                        20

What is the probability that a randomly selected student is interested in a spinning room and that they are a graduate student?

Answer:

Step-by-step explanation:

D. possible angle is 60 degrees out out 360

60/360 = .1666 = 16.7%

P(D)=16.7%

P(B or C) = (30+90)/360 = .333 =33.3%   Or means add your probabilities

P(A)=180/360=.5 = 50.0%

P(not A)= 180/360=.5= 50.0%

Answer:

Step-by-step explanation:630

Enrique has 16 cups of milk and 2 cups of orange juice, making a total of 18 cups of liquid in all.

The number of cups of milk and orange juice Enrique have in all is calculated by converting the gallon of milk and the pint of orange juice in unit of cup as shown below;

For the gallon of milk:

1 gallon of milk = 1 gallon x 16 cups/gallon

= 16 cups

For the pint of orange juice:

1 pint of orange juice = 1 pint x 2 cups/pint

= 2 cups

Total number of cups = 16 cups + 2 cups = 18 cups

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Having simplified the above equation, the solution to the expression x = 10³⁰

We can simplify the equation using logarithmic rules

1/5 log x - log 1000 = 3

log[tex]x^{1/5}[/tex] - log 1000 = 3

log ([tex]x^{1/5}[/tex] /1000) = 3

[tex]x^{1/5}[/tex] /1000 = 10³

[tex]x^{1/5}[/tex]= 1000  * 10³

[tex]x^{1/5}[/tex] = 10⁶

x = (10⁶)⁵

x =  10³⁰

Therefore, the solution to the equation is x = 10³⁰

Note that the logarithm is the inverse function of exponentiation in mathematics. That is, the exponent to which b must be increased to obtain x is the logarithm of a number x to the base b. For example, since 1000 = 10³, 1000's logarithm base 10 is 3, or log₁₀ = 3.

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The required circumference of the circle with a diameter of 97 cm is 304 cm.

The circumference of a circle can be calculated using the formula:

C = πd

Where 'd' is the diameter of the circle, and 'π' is a mathematical constant approximately equal to 3.14159.

In this case, the diameter of the circle is given as 97 cm. So we can substitute this value into the formula to get:

C = πd

C = π(97 cm)

C ≈ 304.35 cm

Rounding this answer to 3 significant figures gives:

C ≈ 304 cm

Therefore, the circumference of the circle with a diameter of 97 cm is 304 cm.

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Answer:

V =base area × height or [(3√3)/2]a2h

where

a = base length

h = height of the prism

Step-by-step explanation:

a.  A statistical test used to help answer the question of whether an observed difference is real or just due to chance error- true

The statistical test referred to in the question is known as a hypothesis test. It is used to determine the probability that the observed difference between two groups or variables is not due to chance but rather represents a true difference. The test involves setting up a null hypothesis, which assumes that there is no difference between the groups or variables, and an alternative hypothesis, which assumes that there is a real difference.

The test calculates a p-value, which represents the probability of obtaining the observed difference by chance if the null hypothesis is true.

If the p-value is below a predetermined level of significance (usually 0.05), then the null hypothesis is rejected and the alternative hypothesis is accepted.

This means that the observed difference is considered real and not just due to chance error.

Hence, If the p-value is below a certain level of significance, the null hypothesis is rejected and the alternative hypothesis is accepted, indicating that the observed difference is real.

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The negative change in the y value will shift the graph downwards.

When a graph of a figure or a point get a negative change in its y value it gets shifted downwards.

When, f(x) - a is f(x) shifted downward a units.

Ex. Shift 3 units down,

Before

y = x² +x, Point (x, y)

After

y = x² +x -3, Point (x, y-3)

Hence, the negative change in the y value will shift the graph downwards.

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The equation for the scaled version of the function f(x) = x² is g(x) = a × x²

The function g(x) can be considered a scaled version of the function f(x) = x². To create a scaled version of a function, we can multiply the original function by a scaling factor. Let's call this scaling factor "a." Now, the equation for g(x) can be written as:

g(x) = a ₓ f(x)

Since f(x) = x², we can substitute it into the equation for g(x):

g(x) = a ₓ x^2

In this equation, "a" represents the scaling factor. If "a" is greater than 1, the function g(x) will stretch vertically, meaning its parabola will be more narrow compared to f(x). If "a" is between 0 and 1, the function g(x) will be compressed vertically, resulting in a wider parabola. If "a" is negative, the parabola will be reflected over the x-axis.

In summary, the equation for the scaled version of the function f(x) = x² is g(x) = a × x², where "a" is the scaling factor. Depending on the value of "a," the resulting parabola will be either stretched or compressed vertically, and may be reflected over the x-axis if "a" is negative.

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To find the values of a and b, we need to use the given information about the population and the growth rate.

At the start (x=0), the population is 240 birds. This means that f(0) = 240, which gives us the value of a in the function:

f(0) = ab^0 = a = 240

Next, we need to find the value of b. We know that the population increases at a rate of 16% annually, which means that it grows by a factor of 1.16 each year. In other words, b = 1.16.

Therefore, the exponential function that represents the number of birds after x years is:

f(x) = 240(1.16)^x

Note that the value of b is greater than 1, which means that the function represents exponential growth.

Answer:

Step-by-step explanation:

To find the values of a and b in the exponential function, we need to use the given information about the population growth rate.

The formula for exponential growth is:

f(x) = a * (1 + r)^x

where:

a = initial value

r = growth rate

x = time

In this case, we know that the initial population is 240 birds and the growth rate is 16% per year. We can convert the percentage to a decimal by dividing by 100:

r = 16% / 100 = 0.16

Now we can plug in the values we have:

f(x) = a * (1 + r)^x

f(x) = 240 * (1 + 0.16)^x

Simplifying:

f(x) = 240 * 1.16^x

So the values for a and b are:

a = 240

b = 1.16

The addition of the given expression will gives 6a - 5b

Since we know that numerical expression is an algebraic information stated in the form of numbers and variables that are unknown.

WE are given the expression as;

2a + 3b; 4a - 8b​

We need to add the expression so,

2a + 3b + 4a - 8b​

Combine the like terms then we get;

2a + 4a + 3b - 8b

6a - 5b

The simplification of the given expression will be 6a - 5b.

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Each child, Miriam, Joe, and Paul, received one-sixth (1/6) of the entire cake.

To determine what part of the entire cake each child got, we need to divide the one-half of a cake that Stanley has equally among his three children, Miriam, Joe, and Paul.

The result is (1/6), which means that each child, Miriam, Joe, and Paul, received one-sixth (1/6) of the entire cake.

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In case of a helicopter and is flying at 160 miles per hour at a constant altitude above a straight road, the speed of the car is equals to the -197.57 mph.

There is a helicopter which is flying at 160 miles per hour at a constant altitude of 0.75 miles above a straight road. During this radar is used to determine the distance of oncoming car is exactly 1 mile from helicopter. Distance decreasing rate = 130 miles per hour

Here, we want to calculate the speed of the car. Using geometry, there is formed a right angled triangle, see above figure. Let the length of base, height and hypothenuse be x,y and z respectively. The distance from the helicopter to the car, represents hypotenuse, say z. The hypotenuse of the triangle is decreasing at 66 mph, i.e. dz/dt = -130mph

From Pythagoras theorem, z² = x² + y²

differentiating wrt time, [tex]2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}[/tex]

we know, dy/dt is zero since copter never changes height. Also since y = 0.75 miles and z = 1 mile, [tex] x = \sqrt{1²-0.75²} [/tex]

= 0.658 miles

As we know speed is rate of change of position, so speed of car is dx/dt. Now, Substitute all known values in above formula, 2 mi × 1 × -130 mph = 2× 0.658 m dx/dt + 2×0

=> dx/dt = - 130/0.658

= -197.57 mph

Hence, required value is - 197.57 mph.

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