Suppose that you start working for a company at age 25. You are offered two rather unlikely, but quite enticing, retirement plans from which you are allowed to choose one. [Round all answers to the nearest dollar.]Retirement plan 1: When you retire, you will receive $20,000 for each year of service.Retirement plan 2: When you start work, the company deposits $2500 into a savings account that is guaranteed to pay a yearly rate of 18%. When you retire, the account will be closed and the balance given to you

a) The system has infinitely many solutions, which are:

(1 - y, y, -3).

b) The system has no solution.

To obtain the number of solutions of each system, the equations are build from the augmented matrix presented.

For item a, the system is given as follows:

Both the left side of the matrix (3 x 3) and the right side (3 x 1), have the same rank, that is, the same number of non-zero rows, hence the system has infinitely many solutions, which are:

For item b, the system is given as follows:

The last statement is false, hence the system has no solutions. It can also be seem by the different ranks of each matrix, with the 3 x 3 matrix having a rank of 2 and the 3 x 1 matrix having a rank of 3.

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Isosceles triangle problem can be solved by given steps

what is isosceles triangle ?

An isosceles triangle in geometry is a triangle with two sides that are the same length. Sometimes it is stated as having exactly two sides that are the same length, and other times it is specified as having at least two sides that are the same length, the latter form containing the equilateral triangle as a particular case.

explanation:

In isosceles triangle

1. Two sides of the triangle are congruent

2. The angle opposite those sides are congruent

x + 72 +72 = 180

             x  = 180 -144

             x = 36

Hence the isosceles triangle problem can be solved by above steps and explanation

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The domain of the function seems to be the set of values that can be plugged into it.

A function is the expression that establishes the relationship between the dependent variable and the independent variable. As the value of the independent variable varies in the function, so does the value of the dependent variable.

Here given,

f(x) = x² − 9x and g(x) = x + 1.

To find,

f + g =  x² − 9x +  x + 1

f + g = x² − 8x  + 1

f - g = x² − 9x - x - 1

f - g =  x² − 10x - 1

f x g = (x² − 9x) x (x + 1)

f x g = (x³ +x ² - 9x² -9x)

f x g = x³ - 8x² - 9x

Therefore, values of function are:

f + g = x² - 8x + 1.

f - g = x² - 10x - 1.

f x g = x³-8x²-9x.

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Answer: -1/2

Step-by-step explanation:

Because the degrees given is greater than 360, we must find out what its simplest degree is. The easiest way to do this is to find the remainder of the argument divided by 360.

So let's start by doing 930 / 360. This will give you a remainder of 210 degrees. So now we are finding sin 210 degrees.

Remember that sin is the y-values of the unit circle. From 0 < θ  < 180, sin θ  is positive, so we automatically know it must be a negative answer. Now, 210 is 30 degree below the x-axis, meaning that coordinate pair of (cos 210 degrees, sin 210 degrees) is (-sqrt(3)/2 , -1/2). Tje y-value, -1/2, is your answer.

Answer:

c

Step-by-step explanation:

The lowest form of 6/42 is 1/7 and,

the lowest form of 14/28 is 1/2.

What is the lowest form of fraction?

When there is no common factor between the fraction's numerator (top) and denominator (bottom), the fraction is said to be in its simplest form. This is also known as its Lowest form.

In fraction 6/42, both denominator and numerator have common factor as 6, So we divide the denominator and numerator by 6.

Hence, we get 1/7.

Similarly, In fraction 14/28, both denominator and numerator have common factor as 14, So we divide the denominator and numerator by 14.

Hence, we get 1/2.

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a. The number of ways that this is done, if the order of the choices is relevant is 12 ways.

b. The number of ways that this be done, if the order of the choices is irrelevant is 6 ways

Combinations are also referred to as selections. Combinations implies the selection of things from a given set of things. In this case, we intend to select the objects.

Combination formula

= n! / ((n – r)! r!)

n = the number of items.

r = how many items are taken at a time.

When the orders are relevant, this will be:

= 4! / (4 - 2!)

= 4! / 2!

= 12

When orders are irrelevant, this will be:

= 4! / (4 - 2!)2!

= 6 ways

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The value of x is in the parallelogram is 10

In a parallelogram, the opposite sides are parallel to each other.

The opposite sides of a parallelogram are equal in length.

The measurement of opposite angles of a parallelogram is equal.

Like all other quadrilaterals, the sum of all the angles of a parallelogram is 360°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

9x +8 and 8x + 2 are adjacent angles of the parallelogram

Mathematically, 9x + 8 + 8x + 2 = 180

17x + 10 = 180

17x + = 180 - 10

17x = 170

x = 170/17

x = 10

In conclusion the value of x is 10

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There is no fixed range for a square root function.

Range is the set of all the output values that a function can take. In other words, it is simply the set of all values that the dependent variable can take for the function to exist. It is also  known as the co-domain.

Let us take a simple square root function given as-

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

Here, y is the dependent variable. We know that the square root of any number cannot be negative. Thus, y can take only positive values. Therefore, the range of a square root function is [0, ∞)

However, this is not always the case, there are various translations of a square root function for which the range might differ. For example, if we have a function as follows-

y = [tex]-\sqrt{x}[/tex]

Here, because of the minus sign, y will always be negative and thus the range will be (-∞, 0]

Hence, we conclude that there is no fixed range for a square root function.

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The exponential decay formula can take one of three forms:

f (x) = abˣ , f (x) = a (1 – r)ˣ , P = P₀[tex]e^{-kt}[/tex]

Where,

a (or) P₀ = Initial amount ,b = decay factor, e = Euler’s constant, r = Rate of decay (for exponential decay), k = constant of proportionality

What is Exponential Decay ?

Exponential decay occurs in mathematical functions when the pace by which changes are occurring are decreasing and must thus reach a limitation, which is the horizontal asymptote of an exponential function.

The amount drops gradually, followed by a quick reduction in the speed of change and increases over time. The exponential decay formula is used to determine the decrease in growth. The exponential decay formula can take one of three forms:

f (x) = abx

f (x) = a (1 – r)x

P = P0 e-k t

Where,

a (or) P0 = Initial amount

b = decay factor

e = Euler’s constant

r = Rate of decay (for exponential decay)

k = constant of proportionality

x (or) t = time intervals (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

In exponential decay, the quantity decreases very rapidly at first, and then more slowly. The rate of change decreases over time. The rate of decay becomes slower as time passes. Hence, the exponential decay graph is denoted as

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The inverse of the function f(x) = [tex]2x^2 - 4[/tex] is

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here, f(x) = [tex]2x^2 - 4[/tex]

Let f(x) = y

[tex]2x^2 - 4[/tex] = y

[tex]2x^2 = y + 4\\x^2 = \frac{y + 4}{2}\\x = \pm \sqrt{\frac{y + 4}{2}}[/tex]

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

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The population increases by 4.3% each year

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

Population of a city next year = Multiply its current population by 1.043

This means that

Growth factor =1.043

The percentage population increment is then calculated as

Percentage population increment = Growth factor - 1

Substitute the known values in the above equation, so, we have the following representation

Percentage population increment = 1.043 - 1

Evaluate

Percentage population increment = 0.043

Express as percentage

Percentage population increment = 4.3%

Hence, the percentage is 4.3%

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First, 108/9=12

This means that are 12 groups of people.

If there are 8 students in each group, 12 x 8 = 96.

96 should be your answer.

The claim that all isosceles triangles are equivalent to one another is untrue.

There are two equal sides and two equal angles in an isosceles triangle.

The term is derived from the Greek words iso (same) and Skelos (leg).

A scalene triangle is one in which none of its sides are equal, as opposed to an equilateral triangle, which has all of its sides equal.

A right triangle with two equal-length legs is called an isosceles right triangle.

The corresponding angles would also be congruent because the lengths of the two legs of the right triangle are equal.

As a result, the two acute angles and two legs of an isosceles right triangle are congruent.

Therefore, the claim that all isosceles triangles are equivalent to one another is untrue.

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The ratio of the length of each transversal will be  [tex]\frac{\overline {CB}}{ \overline {AB}} = \frac{\overline {EF}}{\overline{DE}} =1[/tex]

The ratio of the lines can be compared using Three Parallel Lines Theorem.

What is Three Parallel Lines Theorem?

When two transversals are intersected by three parallel lines, the transversals are proportionally divided.

According to the three parallel lines theorem, the segments created by each transversal and the three parallel lines are proportionate.

The findings are:

Reasons:

The question is a four part question

Let the equations of the parallel lines be as follows;

Line, x; y = x

Line, y; y = x + 1

Line z; y = x + 2

The points at which transversal 1 intersect the lines x, y, and z, are;

A(0.4, 0.4), B(0.6, 1.6), and C(0.8, 2.8)

The length of segment [tex]{\overline {AB}}[/tex]  = √((0.6 - 0.4)² + (1.6 - 0.4)²) = 0.2·√(37)

The length of segment [tex]{\overline {CB}}[/tex] = √((0.8 - 0.6)² + (2.8 - 1.6)²) = 0.2·√(37)

The ratio of the lengths of the segment formed by transversal 1 is therefore;

         [tex]\sqrt{x} \ ratio\ of\ the\ length \ of segments =\frac{\overline {CB}}{ \overline {AB}} = \frac{2.\sqrt{37} }{2.\sqrt{37} } =1[/tex]

The points at which transversal 2 intersect the lines x, y, and z, are;

D(1.1, 3.1), E(1.3, 2.3), and F(1.5, 1.5)

The length of segment  [tex]{\overline {DE}}[/tex]= √((1.3 - 1.1)² + (2.3 - 3.1)²) = 0.2·√(17)

The length of segment [tex]{\overline {EF}}[/tex] = √((1.5 - 1.3)² + (1.5 - 2.3)²) = 0.2·√(17)

The ratio of the lengths of the segment formed by transversal 2 is therefore;

[tex]\sqrt{x} \ ratio\ of\ the\ length \ of segments =\frac{\overline {EF}}{ \overline {DE}} = \frac{0.2.\sqrt{17} }{0.2.\sqrt{17} } =1[/tex]

Therefore;

            [tex]\frac{\overline {CB}}{ \overline {AB}} = \frac{\overline {EF}}{\overline{DE}} =1[/tex]

Which gives;

The the comparison can also be made with the triangle proportionality theorem.

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Answer:

2

Step-by-step explanation:

4 divided by 2 equals 2.

By using the law of exponents, it has been proved that

[tex]a^{-n} = \frac{1}{a^n}[/tex]

What are exponent?

Exponent tells us how many times a number is multiplied by itself.

For example : In [tex]2^4 = 2 \times 2 \times .... \times 2[/tex]

Here, 2 is multiplied by itself 4 times.

If [tex]a^m = a \times a \times a \times ..... \times a[/tex] (m times),

a is the base and m is the index.

The laws of exponents are

[tex]a^m \times a^n = a^{m+n}\\\\a^ m \div a^n = a^{m - n}\\\\a^ 0 =1\\\\a^{-m} = \frac{1}{a^m}\\\\(ab)^m = a^m \times b^m\\\\(\frac{a}{b})^m = \frac{a^m}{b^m}\\\\(a^m)^n = a^{mn}[/tex]

We know this law

[tex]a^p \times a^q = a^{p + q}[/tex]

Putting  p = n and q = -n

[tex]a^n \times a^{-n} = a^{n + (-n)} = a^{n - n} = a^0 = 1[/tex]

So,

[tex]a^n \times a^{-n} = 1\\a^{-n} = \frac{1}{a^n}\\a^{-n} = \frac{1^n}{a^n}\\a^{-n} = (\frac{1}{a})^n[/tex]

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B: 1000 + 25 + Water deposit (50) + Phone connection (40) + Elect. connection (60)

1000 + 25 + 200 = 1225

The rate of change in the given interval would be 1.25 ft/s.

What is the rate of change?

The slope of a graphed function is determined using the average rate of change formula. Divide the change in y-values by the change in x-values to find the average rate of change.

By observation, we can take the points from the graph in between the given interval.

The points would be (3, 2.5) and (5, 4)

Then the rate of change = y2 - y1/ x2 - x1

                                         = 4 - 2.5/5 - 3

                                         = 2.5/2

                                         = 1.25 ft/s.

Hence, the rate of change in the given interval would be 1.25 ft/s.

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Answer:

  1,362,247

Step-by-step explanation:

You want the bacteria population after 20 days if the initial population of 3,800,000 is declining at the rate of 5% per day.

The population of bacteria can be expressed by the exponential function ...

  population = (initial population) × (decay factor)^(time)

where ...

  decay factor = 1 - decay rate . . . . . . . in the period equal to 1 time unit

For a decay rate of 5% per day, this means our bacteria population in 20 days will be ...

  p = 3800000·(1 -0.05)^20 ≈ 1,362,247

__

Additional comment

The initial population is given to 2 significant figures, so it makes a certain amount of sense to give the result to approximately that precision:

  1.36 million or 1.4 million bacteria after 20 days

(Sometimes when the leading digit is 1, it is useful to have one more significant figure.)

The quadratic equations and their factors are simplified below

A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term. The first condition for an equation to be a quadratic equation is the coefficient of x² is a non-zero term(a ≠ 0). For writing a quadratic equation in standard form, the x² term is written first, followed by the x term, and finally, the constant term is written. The numeric values of a, b, c are generally not written as fractions or decimals but are written as integral values.

1) (x - 5)(x - 5) = x² - 5x - 5x + 25 = x² -10x + 25

2) (x + 2)(x + 9) = x² + 2x + 9x + 18 = x² + 11x + 18

3) (x + 2)(x + 3) = x² + 2x + 3x + 6 = x² + 5x + 6

4) (x + 1)(x + 2) = x² + x + 2x  + 2 = x² + 3x + 2

5) (x - 3)(x - 3) = x² - 3x - 3x + 9 = x² - 6x + 9

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Answer:

Step-by-step explanation:

1) 4 1/4 + 5 1/2

= 9 3/4

2) 10 4/5 - 6 1/5

= 4 3/5

3) 3 1/4 - 1 3/4

= 1 1/2

4) 3 2/5 + 7 7/10

= 11 1/10

5 ) 10 3/13 + 2 10/13

= 13

Answer:

1

Step-by-step explanation:

Its for sure 1

The 10 percent of a number can be found using multiplying the given number by 0.1.

According to the question,

We have the following information:

10% of a number

Now, we have to know the process by which we can find the 10% of a number.

Now, we know that when we remove the sign of the percentage then the number is divided by 100.

So, we have the following expression:

10/100

0.1

Now, we can find 10% of a number by multiplying this number with the given number.

So, we will have:

0.1*number

Hence, the 10% of a number can be found using multiplying the given number by 0.1.

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Naomi's meal cost $18.

Cost is the amount or equivalent paid or charged for something.

If a car is bought for $20,000 then the cost of the car is the money exchanged for it.

From the word problem;

let the first letters of their names represent each variable

n + r + s + v = 108 ----- (1)

but r = 2n ----- (2)

s = 2v ----- (3)

s = r ----- (4)

v = s ----- (5)

n = ?

Substituting the values in equation 1

n + 2n + 2n + n = 108

6n = 108

n = 18

Naomi's meal costs $18

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Answer:

Step-by-step explanation:

SLOPE OF THE GIVEN LINE

6x + 18y = 36

x + 3y = 6

3y = -x + 6

3/3 y = -1/3 x + 6/3

y = -1/3 x + 2

Slope = -1/3

slope of a perpendicular line  = 3

POINT SLOPE FORM

y-4 = 3(x-(-5)

y - 4 = 3(x+5)

SLOPE INTERCEPT FORM

y-4 = 3x + 15

y = 3x + 15 + 4

y = 3x + 19

STANDARD FORM

3x - y + 15 + 4 = 0

3x - y + 19 = 0

Equation of line in point slope form is y-4= 3 (x+5), y=3x+19 is equation in slope intercept form and equation in standard form is 3x-y+19=0.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

6x+18y=36

18y=-6x+36

Divide both sides by 18

y=-6/18x+36/18

y=-1/3x+2

Slope of given line is -1/3, so slope of perpendicular line is 3.

The equation of line in point slope form with slope 3 and goes through the point (-5,4) is given below

y-4= 3 (x+5)

To find the equation in slope intercept form we need to find y intercept

4=3(-5)+b

4=-15+b

19=b

y=3x+19 is equation in slope intercept form.

The equation in standard form is 3x-y+19=0

Hence,  equation of line in point slope form is y-4= 3 (x+5), y=3x+19 is equation in slope intercept form and equation in standard form is 3x-y+19=0.

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We make a ten in first grade by taking 1 from the next left digit of the number in which we are doing the indicated operation.

According to the question,

We have the following information:

Making a ten in first grade

We consider that in making a ten we take 1 from the next left digit of the number from which we are subtracting any other number. Or it is often said to be taking a carry.

Now, this number 1 is treated as 10 instead of 1 and then 10 is added to the digit to ease subtraction in lower grades.

For example, 24-16 can be performed when we add a ten to 4 by taking 1 from 2. Now, 6 can be easily subtracted from 14.

Hence, we make a ten in first grade by taking 1 from the next left digit of the number in which we are doing the indicated operation.

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There are 125 of the smallest cubes that will fill the largest cube using the size of their respective volume

The volume of cube is calculated by using a × a × a = a³ where "a" is the edge.

The volume of the smallest cube = 1.5inches × 1.5inches × 1.5inches

The volume of the smallest cube = 3.375inches³

The volume of the largest cube = 7.5inches × 7.5inches × 7.5inches

The volume of the largest cube = 421.875inches³

Thus, the number of the smallest cubes it would take to fill the largest cube is derived by

421.875inches³/3.375inches³ = 125.

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A mathematical proof is an inferential justification for a mathematical claim that demonstrates how the premises logically support the conclusion.

Result for an image A proof formula:

There are numerous techniques to demonstrate something; we'll talk about three of them: direct proof, proof by contradiction, and proof by induction. We'll discuss each of these proofs, including what they are and when and how to use them.

Proof: If n is an integer, then. We shall demonstrate the converse statement, "if n is divisible by 4, then n is divisible by 2," in order to demonstrate that "if n is not divisible by 2, then n is not divisible by 4." Let's say n can be divided by 4.

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Answer:

B) x + 1

Step-by-step explanation:

Consecutive integers mean continuous integers ( 1, 2 , 3 ,....). We will get the next number by adding 1 to the previous number.

Eg: 5 , 6 , 7 ,.....

6 = 5 + 1

7 = 6 + 1

So, if first number is x, then the next number will be x + 1