Simplify negative 2 and 1 over 9 – negative 4 and 1 over 3.

Answer:

The statement is false

Step-by-step explanation:

we have

[tex]4y=x[/tex]

Solve for y

That means ----> Isolate the variable y

Divide by 4 both sides

[tex]y=\frac{1}{4}x[/tex]

This is the equation of a proportional relationship between the variable x and the variable y

where the constant of proportionality k or slope m is equal to 1/4

therefore

The statement is false

Abdul drove the truck for 177 miles.

To find the number of miles driven by Abdul, we need to subtract the base fee from the total amount he paid and then divide the result by the additional charge per mile. Let's represent the number of miles driven by x.

Total amount paid - Base fee = Additional charge per mile * Number of miles driven

$146.16 - $16.95 = $0.73 * x

Simplifying the equation: $129.21 = $0.73x

Dividing both sides by $0.73, we get: x = 177

Therefore, Abdul drove the truck for 177 miles.

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Answer:

D. None of the above.

Step-by-step explanation:

OPTIONS A and C. The Olympic final had faster average times and more noticeable variation compared to the top 888 U.S. qualifier times.

Looking at the dot plots, we can draw the following conclusions:

A. The times in the Olympic final were faster on average than the top 888 U.S. qualifier times. - True, since the lower dot plot represents Olympic final times and the lower values (faster times) are more concentrated.

B. All of the times in the Olympic final were faster than all of the top 888 U.S. qualifier times. - False, as there are some times in the U.S. qualifier dot plot that are faster than some times in the Olympic final dot plot.

C. The Olympic final times varied noticeably more than the times of the top 888 U.S. qualifiers. - True, since the spread of dots in the lower plot seems to be wider compared to the upper plot, indicating greater variation in Olympic final times.

So, the correct choices are A and C.

Answer:

25.6

Step-by-step explanation:

Convert the percentage to decimal form by dividing by 100.

4% of 640

= 0.04(640)

= 25.6

Answer:

Step-by-step explanation:

1)w=−4/3x+−1/3y+2/3z+26/3

2)w=−3/2x+1/2y+−1/2z+7

3)w=1/2x+y+3/2z−9

4)w=2x+2y−z−10

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Answer:

The aircraft has a height of 1000 m at t=2 sec, and at t=8 sec

Step-by-step explanation:

Finding Exact Roots Of Polynomials

A polynomial can be expressed in the general form    

[tex]\displaystyle p(x)=a_nx^n+a_{n-1}\ x^{n-1}+...+a_1\ x+a_0}[/tex]  

The roots of the polynomial are the values of x for which    

[tex]P(x)=0[/tex]

Finding the roots is not an easy task and trying to find a general solution has been discussed for centuries. One of the best possible approaches is trying to factor the polynomial. It requires a good eye and experience, but it gives excellent results.    

The function for the trajectory of an aircraft is given by    

[tex]\displaystyle h(x)=0.5(-t^4+10t^3-216t^2+2000t-1200)[/tex]

We need to find the values of t that make H=1000, that is

[tex]\displaystyle 0.5(-t^4+10t^3-216t^2+2000t-1200)=1000[/tex]

Dividing by -0.5

[tex]\displaystyle t^4-10t^3+216t^2-2000t+1200=-2000[/tex]

Rearranging, we set up the equation to solve

[tex]\displaystyle t^4-10t^3+216t^2-2000t+3200=0[/tex]

Expanding some terms

[tex]\displaystyle t^4-8t^3-2t^3+200t^2+16t^2-1600t-400t+3200=0[/tex]

Rearranging

[tex]\displaystyle t^4-8t^3+200t^2-1600t-2t^3+16t^2-400t+3200=0[/tex]

Factoring

[tex]\displaystyle t(t^3-8t^2+200t-1600)-2(t^3-8t^2+200t-1600)=0[/tex]

[tex]\displaystyle (t-2)(t^3-8t^2+200t-1600)=0[/tex]

This produces our first root t=2. Now let's factor the remaining polynomial

[tex]\displaystyle t^2(t-8)+200(t-8)=0[/tex]

[tex]\displaystyle (t^2+200)(t-8)=0[/tex]

This gives us the second real root t=8. The other two roots are not real numbers, so we only keep two solutions

[tex]\displaystyle t=2,\ t=8[/tex]

Answer:

The weight of the larger box   = 15.75 kg.

The weight of the smaller box = 13.75 kg.

Step-by-step explanation:

We have the two equations based on the scenario.

The equations are:

[tex]$ 7x + 9y = 234 \hspace{20mm} \hdots {(1)} $[/tex]

[tex]$ 5x + 3y = 360 \hspace{20mm} \hdots {(2)} $[/tex]

'x' represents the weight of the larger box.

and 'y' the weight of the smaller box.

Multiplying Equation (2) by 3 throughout, we get:

[tex]$ 15x + 9y = 360 $[/tex]

Subtracting this and Equation (1), we get:

[tex]$ -8x = -126 $[/tex]

[tex]$ \implies x = 15.75 $[/tex] kg.

∴ The weight of the Larger box is 15.75 kg.

Substituting the value of 'x' in either of the equations will give us the value of 'y', the weight of the smaller box.

We substitute in Equation (1).

We get: 9y = 234 - 7(15.75)

⇒ 9y = 123.75

⇒ y = 13.75

∴ The weight of the smaller box is 13.75 kg.

Answer:

y-4=3/4(x+8)

Step-by-step explanation:

y-y1=m(x-x1)

y-4=3/4(x-(-8))

y-4=3/4(x+8)

The half of 14 is 7 and the half of 77 is 38.5

Hope this helped! (Plz mark me brainliest)

Answer:

Step-by-step explanation:

So the first step is to simply set up the problem based on what we are given. So here have two numbers, we are going to call the first number x and the second one y. With that now addressed, we can now proceed with the setup.

So the difference between the squares of the numbers is 24. So we have:

[tex]x^{2} -y^{2} = 24[/tex]

Then it says that three times the square of the first number (which we said was x) increased by the square of the second number is 76. So:

[tex]3x^{2} + y^{2} = 76[/tex]

Now we can see that this is simply a system of equations and we can use elimination to solve this! We even have the setup already as the coefficients in front of our y are opposite in sign and are equal. So:

[tex]x^{2} -y^{2} = 24\\3x^{2} +y^{2} = 76[/tex]

We can cancel our y squared terms out and that leaves, when we add the equations together:

[tex]4x^{2} = 100[/tex]

We can then solve for x by diving by four and taking the square root of the result.

[tex]x^{2} = 25\\[/tex]

Therefore, x = ±5

We have both negative and positive answers because if we squared -5 or +5 they would both give us 25. So we cant rule a negative answer out yet.

So now we can plug in x = -5 or +5 to either equation to solve for y as so:

[tex]5^{2} - y^{2} = 24\\25 - y^{2} = 24\\-y^{2} = -1 \\y^{2} = 1[/tex]

So y = ±1

In this case both negative and positive versions of our answer work (you can also double check), so we are left with:

x = ±5 and y = ±1

We can solve the problem by expressing one variable through another from one equation and substituting it into the second equation. Then solve for the first variable and substitute the found value in one of the equations to find the corresponding other variable.

The subject of this question is algebra, specifically equations involving squares of numbers and systems of equations. Let's denote the unknown numbers as 'x' and 'y'. From the problem, we know two equations:

One possible approach to finding the values of x and y would be to use the substitution method. From equation (1), we can express y² as x² - 24 and substitute this into equation (2):

3x² + (x² - 24) = 76

If we simplify and solve for x, we find that x = 4, -4.

Then, substituting 'x' into the first equation will yield corresponding 'y' values. That's how you can solve this kind of problems by using squares of numbers and method of substitution.

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Answer:

the number is X=2

Step-by-step explanation:

Suppose number is X

8X-1 =X+13

8X-X= 13+1

7X=14

X=2

Answer:

8n-1= 13+n is the equation, n=2 is the answer for the variable.

Step-by-step explanation:

8n-1= 13+n

Move the variable to one side.

8n-1= 13+n

-1n         -1n

7n-1= 13

Now, add to both sides, since that is the opposite of subtracting.

7n-1= 13

   +1  +1

Then, divide to both sides by 7 to leave the variable by itself.

7n= 14

7     7

n= 2

Answer: the answer is correct

Step-by-step explanation: edge 2020 trust me

The slope of given coordinates (0,-2) and (2,-1) is [tex]\frac{1}{2}[/tex]

The slope intercept form is [tex]y = \frac{1}{2}x -2[/tex]

Given that the coordinates are (0,-2) and (2,-1)

To find: slope and slope intercept form

The slope of line is given as:

For a line containing two points [tex](x_1 , y_1)[/tex] and [tex](x_2, y_2)[/tex] , slope of line is given as:

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

Here in this problem,

coordinates are (0,-2) and (2,-1)

[tex]x_{1}=0 ; y_{1}=-2 ; x_{2}=2 ; y_{2}=-1[/tex]

Substituting the values in above formula,

[tex]m=\frac{-1-(-2)}{2-0}=\frac{-1+2}{2}=\frac{1}{2}[/tex]

Thus slope of line is [tex]\frac{1}{2}[/tex]

To find slope intercept form:

The slope intercept form is given as:

y = mx + b

Where "m" is the slope of line and "b" is the y-intercept

Substitute [tex]m = \frac{1}{2}[/tex] and (x, y) = (0, -2) in above slope intercept we get,

[tex]-2 = \frac{1}{2} \times 0 + b[/tex]

b = -2

Thus the required slope intercept is given as:

Substitute [tex]m = \frac{1}{2}[/tex] and b = -2 in slope intercept form,

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

Thus the slope intercept form is found

Answer:

6 bicycles and 3 tricycles

Step-by-step explanation:

bicycles have two wheels therefore 6*2=12 total wheels

tricycles have three wheels therefore 3*3=9 total wheels

add the totals up and you have 21 total wheels

Final answer:

The shop can build 6 bicycles and 3 tricycles with the 21 wheels available.

Explanation:

To solve the problem of figuring out how many tricycles and bicycles can be built with 21 wheels, under the condition that there will be half as many tricycles as bicycles, we use algebraic methods.

Let's denote the number of bicycles as B and the number of tricycles as T.

Since each bicycle needs 2 wheels and each tricycle needs 3, we can establish the following equations based on these facts and the condition given:

Substituting T from the second equation into the first gives:

2B + 3(0.5B) = 21

2B + 1.5B = 21

3.5B = 21

B = 21 / 3.5 = 6

So, there are 6 bicycles. To find T, plug B back into the equation T = 0.5B:

T = 0.5×6 = 3

Therefore, the shop can build 6 bicycles and 3 tricycles with the 21 wheels available.

Answer:

[tex]\angle{A}=53.1^{\circ}[/tex]

[tex]\angle{A}=80.9^{\circ}[/tex]

[tex]b=12.4[/tex]

Step-by-step explanation:

Please find that attachment.

We have been given that in ΔABC, ∠C measures 46° and the values of a and c are 10 and 9, respectively.

First of all, we will find measure of angle A using Law Of Sines:

[tex]\frac{\text{sin(A)}}{a}=\frac{\text{sin(B)}}{b}=\frac{\text{sin(C)}}{c}[/tex], where, A, B and C are angles corresponding to sides a, b and c respectively.

[tex]\frac{\text{sin(A)}}{10}=\frac{\text{sin(46)}}{9}[/tex]

[tex]\frac{\text{sin(A)}}{10}=\frac{0.719339800339}{9}[/tex]

[tex]\frac{\text{sin(A)}}{10}=0.0799266444821111[/tex]

[tex]\frac{\text{sin(A)}}{10}*10=0.0799266444821111*10[/tex]

[tex]\text{sin(A)}=0.799266444821111[/tex]

Upon taking inverse sine:

[tex]A=\text{sin}^{-1}(0.799266444821111)[/tex]

[tex]A=53.060109978759^{\circ}[/tex]

[tex]A\approx 53.1^{\circ}[/tex]

Therefore, the measure of angle A is 53.1 degrees.

Now, we will use angle sum property to find measure of angle B as:

[tex]m\angle{A}+m\angle{B}+m\angle{C}=180^{\circ}[/tex]

[tex]53.1^{\circ}+m\angle{B}+46^{\circ}=180^{\circ}[/tex]

[tex]m\angle{B}+99.1^{\circ}=180^{\circ}[/tex]

[tex]m\angle{B}+99.1^{\circ}-99.1^{\circ}=180^{\circ}-99.1^{\circ}[/tex]

[tex]m\angle{B}=80.9^{\circ}[/tex]

Therefore, the measure of angle B is 80.9 degrees.

Now, we will use Law Of Cosines to find the length of side b.

[tex]b^2=a^2+c^2-2ac\cdot\text{cos}(B)[/tex]

Upon substituting our given values, we will get:

[tex]b^2=10^2+9^2-2(10)(9)\cdot\text{cos}(80.9^{\circ})[/tex]

[tex]b^2=100+81-180\cdot 0.158158067254[/tex]

[tex]b^2=181-28.46845210572[/tex]

[tex]b^2=152.53154789428[/tex]

Upon take square root of both sides, we get:

[tex]b=\sqrt{152.53154789428}[/tex]

[tex]b=12.3503663060769173[/tex]

[tex]b\approx 12.4[/tex]

Therefore, the length of side b is approximately 2.4 units.

Answer:

∠A = 53.1°, ∠B = 80.9°, b = 12.4

Step-by-step explanation:

i got it right on my test

Answer:

the answer is (6, 1)

Step-by-step explanation:

x² + y² - 12 x - 2 y + 12 = 0

(x²-12x) +(y² -2y) +12 = 0

(x²-2(6)(x)+6²)-6² +(y² -2y+1) -1+12 = 0

(x-6)² +(y-1)² = 5²

the center of a circle is (6, 1)

Answer:

[tex]\frac{n^{2} }{n+5}[/tex]

Step-by-step explanation:

We can see that for nth term of this sequence,numerator is square of the term number([tex]n^{2}[/tex]) and denominator is 5 added to the term number(n+5).

Therefore , the nth term of this sequence is

[tex]\frac{n^{2} }{n+5}[/tex]

The factorization of [tex]\(3x^3+7x^2-18x+8\)[/tex] is [tex]\((x-1)(3x^2 + 10x - 8)\)[/tex]. So, the factors are [tex](x-1), (3x^2 + 10x - 8)[/tex], and any additional factors that [tex]\(3x^2 + 10x - 8\)[/tex] may have, which can be further factored if possible.

Given that  x-1 is a factor of the expression [tex]\( 3x^3+7x^2-18x+8 \)[/tex], we can use the long division method to find the other factors.

Set up the division as follows:

            [tex]3x^2 + 10x - 8[/tex]

       ______________________

[tex]x - 1 | 3x^3 + 7x^2 - 18x + 8 \\ - (3x^3 - 3x^2)[/tex]

        ___________________

                [tex]10x^2 - 18x + 8 \\ - (10x^2 - 10x)\\[/tex]

                 ______________

                          -8x + 8

                          - (-8x + 8)

                          __________

                                 0

The quotient is [tex]\(3x^2 + 10x - 8\)[/tex], and since there is no remainder, \(x-1\) is indeed a factor.

Therefore, The factorization of [tex]\(3x^3+7x^2-18x+8\)[/tex] is [tex]\((x-1)(3x^2 + 10x - 8)\)[/tex]

So, the factors are [tex](x-1), (3x^2 + 10x - 8)[/tex], and any additional factors that [tex]\(3x^2 + 10x - 8\)[/tex] may have, which can be further factored if possible.

The complete factorization of the original expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] is:[tex]\[ (x - 1)(x + 4)(3x - 2) \][/tex]

To factorize the expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] given that one of the factors is [tex]\(x - 1\)[/tex], we can use polynomial division to divide the expression by [tex]\(x - 1\)[/tex] and find the quotient. The factors of the expression will then be [tex]\(x - 1\)[/tex]  and the factors of the quotient.

The expression to be factorized is: [tex]\(3x^3 + 7x^2 - 18x + 8\).[/tex]

Let's perform the division step-by-step.

1.Divide the first term of the dividend (3x³) by the first term of the divisor (x):

  [tex]\(3x^3 ÷ x = 3x^2\)[/tex].

Write this as the first term of the quotient.

2. Multiply the divisor by this term and subtract the result from the dividend:

Multiply [tex]\(x - 1\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(3x^3 - 3x^2\)[/tex].

Subtract this from the original polynomial: [tex]\(3x^3 + 7x^2\)[/tex]becomes [tex]\(10x^2\)[/tex].

3.Bring down the next term of the original polynomial to form a new dividend:

The new dividend is [tex]\(10x^2 - 18x\)[/tex].

4.Repeat this process for the new dividend**:

  Divide [tex]\(10x^2\) by \(x\) to get \(10x\)[/tex].

  Multiply [tex]\(x - 1\) by \(10x\)[/tex] to get [tex]\(10x^2 - 10x\)[/tex].

  Subtract this from the new dividend:[tex]\(10x^2 - 18x\)[/tex]  becomes [tex]\(-8x\)[/tex].

5.Bring down the next term of the original polynomial to form a new dividend:

  The new dividend is [tex]\(-8x + 8\)[/tex].

6.Repeat this process for the new dividend**:

  Divide [tex]\(-8x\) by \(x\)[/tex] to get [tex]\(-8\)[/tex].

  Multiply [tex]\(x - 1\) by \(-8\)[/tex] to get [tex]\(-8x + 8\)[/tex].

  Subtract this from the new dividend: [tex]\(-8x + 8\)[/tex] becomes 0.

The quotient we obtain from this division is [tex]\(3x^2 + 10x - 8\)[/tex]. Now, we need to factorize this quadratic expression. Let's proceed with the factorization.

The roots of the quadratic expression [tex]\(3x^2 + 10x - 8\)[/tex] are [tex]\(-4\)[/tex] and[tex]\(\frac{2}{3}\)[/tex]. This means that the quadratic expression can be factored as [tex]\((x + 4)(x - \frac{2}{3})\)[/tex].

However, to express the factors in a more standard form, we'll rewrite the factor [tex]\(x - \frac{2}{3}\) as \(3x - 2\)[/tex], which is obtained by multiplying the numerator and denominator of [tex]\(\frac{2}{3}\)[/tex] by 3.

Therefore, the factorized form of [tex]\(3x^2 + 10x - 8\)[/tex] is [tex]\((x + 4)(3x - 2)\)[/tex].

Combining this with the given factor [tex]\(x - 1\)[/tex], the complete factorization of the original expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] is:[tex]\[ (x - 1)(x + 4)(3x - 2) \][/tex]

Answer:

Step-by-step explanation:

4 + x = 133......to solve this, u would subtract 4 from both sides

so that would be the subtraction property of equality

Answer:

Step-by-step explanation:

4 + x = 13              subtract 4 from both sides

4 - 4 + x = 13 - 4

[tex]\pi[/tex] is a Irrational Number .

[tex]\sqrt{12}[/tex] is a Irrational Number .

3.14 is a Rational Number .

[tex]4 . \overline{123}[/tex] is a Rational Number .

[tex]\sqrt{49}[/tex]  is Natural, Whole, Rational Number and Integer .

[tex]-\frac{240}{6}[/tex] is Integer and Rational number.

In order to answer this question, we need to know what are Natural Numbers, Whole Numbers, Integer, Rational and Irrational Numbers.  

We cannot represent [tex]\pi[/tex] as a fraction of two integers and hence is Irrational.

[tex]\sqrt{12}[/tex] can be simplified as [tex]\sqrt{(4 \times 3)}=2 \times \sqrt{3} \times \sqrt{3}[/tex]  again cannot represent the same as a fraction of two integers, and [tex]\sqrt{12}[/tex]  is irrational.

3.14 can be represented in terms of [tex]\frac{314}{100}=\frac{157}{50}[/tex], and hence it is a Rational Number.

[tex]4 . \overline{123}[/tex] is the case of Repeating decimals, and repeating decimals are always Rational Numbers.

[tex]\sqrt{49}[/tex] is equal to 7 and hence is Natural Number. 7 can also be included in the whole number, rational number and Integer.

[tex]-\frac{240}{6}[/tex] is equal to -40, so it is Integer and Rational Number.

Answer:

We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis. The most common graphs name the input value x and the output value y , and we say y is a function of x , or y=f(x) y = f ( x ) when the function is named f .

Step-by-step explanation:

Answer:

8 or -4

Step-by-step explanation:

The error is they believed x2 meant x^2 OR they forgot that x could also equal to -4.

x^2 = 16 has two solutions. They are 4 and -4. So the value of x could also be -4.

Answer:

Ans => 4, 5

Step-by-step explanation:

let the integers be x and y

if they are consecutive, that means the larger integer is one greater than the smaller

=> y = x+1

2 = 3x - 2y

3x - 2(x+1) = 2

3x - 2x - 2 = 2

x = 4

y = 4+1 = 5

From Patrick's gains, tax rates, and the inflation rate, the following are true: $346.25, $1,731.25, $145.43 and $4,702.08.

State tax on initial profit can be calculated by;

= (17,000 - 10,000 - 25 - 25 - 25) x 5%

= $346.25

The federal tax is:

= 25% x (17,000 - 10,000 - 25 - 25 - 25) x 25%

= $1,731.25

Inflation after taxes is:

= (17,000 - 7,000 - 25 - 25 - 25 - 346.25 - 1,731.25) * (1% + 1% + 1%)

= $145.43

Real value in profit is:

= 17,000 - 7,000 - 25 - 25 - 25 - 346.25 - 1,731.25 - 145.43

= $4,702.08

Find out more on federal taxes at ;

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Patrick's initial profit is $7,000, and after state and federal taxes, he has $4,900. When adjusted for a 1% annual inflation rate over three years, the real value of his profit is approximately $4,805.03.

Patrick's profit from selling his stock is calculated by subtracting the initial investment from the current value, $17,000 - $10,000 = $7,000. The state tax he must pay on this profit is 5% of $7,000, which equals $350. Similarly, the federal tax on the profit is 25% of $7,000, equaling $1,750. After state and federal taxes, the remaining amount is $7,000 - $350 - $1,750 = $4,900. Lastly, the inflation effect over three years at 1% per year diminishes the purchasing power of the remaining amount. The cumulative effect of inflation can be calculated by the formula P = P0 * ((1 - r)^n), where P is the future value, P0 is the present value, r is the inflation rate, and n is the number of years. Thus, Patrick's remaining profit adjusted for inflation would be $4,900 * ((1 - 0.01)^3), which gives a result of approximately $4,805.03. This is the real value of Patrick’s profit after taxes and inflation adjustments.

Answer:

If Discriminant,[tex]b^{2} -4ac >0[/tex]

Then it has Two Real Solutions.

Step-by-step explanation:

To Find:

If discriminant (b^2 -4ac>0) how many real solutions

Solution:

Consider a Quadratic Equation in General Form as

[tex]ax^{2} +bx+c=0[/tex]

then,

[tex]b^{2} -4ac[/tex] is called as Discriminant.

So,

If Discriminant,[tex]b^{2} -4ac >0[/tex]

Then it has Two Real Solutions.

If Discriminant,[tex]b^{2} -4ac < 0[/tex]

Then it has Two Imaginary Solutions.

If Discriminant,[tex]b^{2} -4ac=0[/tex]

Then it has Two Equal and Real Solutions.

Answer:

Step-by-step explanation:

[tex]f(x)=\dfrac{12}{4x+2}\\\\f(-1)-\text{put}\ x=-1\ \text{to}\ f(x):\\\\f(-1)=\dfrac{12}{4(-1)+2}=\dfrac{12}{-4+2}=\dfrac{12}{-2}=-6[/tex]

Answer:

f(-1) = -6

Step-by-step explanation:

substitute -1 into the equation and solve, you'll get -6

Answer:

$2,444.95

Step-by-step explanation:

A = P (1 + r)^(nt)

where A is the final amount,

P is the principal,

r is the rate,

n is the compoundings per year,

and t is the number of years.

500,000 = P (1 + 0.03)^(12 × 15)

500,000 = P (1.03)^180

P = 500,00 (1.03)^-180

P ≈ 2,444.95

Final answer:

To determine the principal amount that must be invested at a rate of 3%, compounded monthly, to have $500,000 available for retirement in 15 years, the formula for compound interest can be used. The principal that must be invested is approximately $310,334.84.

Explanation:

To determine the principal amount that must be invested at a rate of 3%, compounded monthly, to have $500,000 available for retirement in 15 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the future value, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

Plugging in the given values, we have:

A = $500,000, r = 0.03 (3%), n = 12 (compounded monthly), and t = 15.

Substituting these values into the formula, we get:

$500,000 = P(1 + 0.03/12)^(12*15)

Solving for P gives:

P = $500,000 / (1 + 0.03/12)^(12*15)

Calculating this expression gives P ≈ $310,334.84. Therefore, the principal that must be invested is approximately $310,334.84.