santa has nine reindeer that pull his sleigh on christmas eve what is the ratio of legs ears as it relates to santad reindeer?​

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]

Answer:

y = -3x + 14

Step-by-step explanation:

Para levantar 500 toneladas de mineral a una altura de 90 pies en una hora, se requieren aproximadamente 90.91 caballos de fuerza (HP), calculados utilizando la fórmula de trabajo y potencia.

Para calcular la potencia requerida en caballos de fuerza (HP) para levantar 500 toneladas de mineral hasta una altura de 90 pies en una hora, necesitamos utilizar la fórmula de trabajo y la potencia. La potencia se mide en unidades de trabajo por unidad de tiempo.

Primero, convirtamos la altura a pies a la misma unidad que la tonelada, que es la tonelada-pie (ton-ft). 1 tonelada-pie es igual al trabajo necesario para levantar una tonelada a una altura de un pie.

500 toneladas * 90 ft = 45,000 ton-pie

El trabajo total necesario es de 45,000 toneladas-pie.

Dado que el trabajo se realiza en una hora (3600 segundos), podemos usar la siguiente fórmula para calcular la potencia en HP:

[tex]\[Potencia (HP) = \frac{Trabajo (ft-lbf)}{Tiempo (s)} \times \frac{1}{550}.\][/tex]

Donde 1 HP es igual a 550 ft-lbf/s.

Sustituyendo los valores conocidos:

[tex]\[Potencia (HP) = \frac{45,000 \text{ ton-pie} \times 2,000 \text{ lbf/ton} \times 1 \text{ ft}}{3600 \text{ s} \times 550 \text{ ft-lbf/s}} = \frac{180,000,000 \text{ lbf-ft}}{1,980,000 \text{ ft-lbf/s}} \approx 90.91 HP.\][/tex]

Por lo tanto, se requieren aproximadamente 90.91 caballos de fuerza (HP) para realizar este trabajo en una hora.

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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:

77

Step-by-step explanation:

Simplify the following:

52 + 7×3 + 4

7×3 = 21:

52 + 21 + 4

| 5 | 2

| 2 | 1

+ | | 4

| 7 | 7:

Answer: 77

To evaluate the expression 52 + 7 ⋅ 3 + 4, follow the order of operations (PEMDAS). First, multiply 7 by 3 to get 21, then add 52 and 4 to get the final result of 77.

Evaluation of the Expression:

To evaluate the expression 52 + 7 ⋅ 3 + 4, it is important to follow the order of operations (PEMDAS/BODMAS):

Using these rules, solve the expression step-by-step:

Therefore, the correct answer is:

77

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.

The cost of the camera if Sharon paid $78 sales tax at a rate of 6.5% is $1200

What was the cost of the camera?

Amount of sales tax = $78

Sales tax rate = 6.5%

Cost of the camera = x

.

Amount of sales tax = Sales tax rate × Cost of the camera

78 = 6.5% × x

78 = 0.065 × x

78 = 0.065x

Divide both sides by 0.065

x = 78/0.065

x = 1200

Therefore, the camera cost $1200

You will place the sticker [tex]\frac{5}{8}[/tex] inches from the edge of the switch board

Step-by-step explanation:

Given data:

Sticker’s length =  [tex]2 \frac{1}{2}[/tex]

Breadth of switch box = [tex]3 \frac{3}{4}[/tex]

Subtracting the above value, we get

    [tex]\text { Switch Box's total area }=3 \frac{3}{4}-2 \frac{1}{2}=\frac{15}{4}-\frac{5}{2}=\frac{15-10}{4}=\frac{5}{4}[/tex]

Divided [tex]\frac{5}{4}[/tex] by 2 gives you the distance from the edge if you put the sticker in the centre). Therefore,

    [tex]\frac{\left(\frac{5}{4}\right)}{\left(\frac{2}{2}\right)}=\frac{5}{8}[/tex]

So, will place the sticker [tex]\frac{5}{8}[/tex] inches far from the edge of the switch board.

Answer:

For a )    y = 6x - 3

[tex]slope = m = 6\\y-intercept = c = -3\\[/tex]

For b )     y = -2x - 10

[tex]slope = m = -2\\y-intercept = c = -10\\[/tex]

For c )    y = -4x + 4

[tex]slope = m = -4\\y-intercept = c = 4\\[/tex]

Step-by-step explanation:

Given:

a. y = 6x - 3

b. y = -2 (x + 5)

   y = -2x - 10

c. y = 4 (-x + 1)

  y = -4x + 4

To Find:

y-intercept and the slope for each equation = ?

Solution:

Slope-intercept Formula is given by

[tex]y=mx+c[/tex]

Where,

m = slope

c = y-intercept

So on comparing the Given equations with the above Equation we get

For a )   y = 6x - 3

[tex]slope = m = 6\\y-intercept = c = -3\\[/tex]

For b )    y = -2x - 10

[tex]slope = m = -2\\y-intercept = c = -10\\[/tex]

For c )    y = -4x + 4

[tex]slope = m = -4\\y-intercept = c = 4\\[/tex]

Answer:

The water she need is [tex]7\frac{7}{8} \ cups.[/tex] and strawberry syrup [tex]11\frac{13}{16}\ cups[/tex].

Step-by-step explanation:

Given:

Dina wants to make 15 3/4 cups of strawberry drink by mixing water and strawberry syrup with a ratio of 2 1/4 cup of water for every 3/4 cup of syrup.

Now, to find the quantity of water and syrup she need to use.

As given in question ratio so:

Strawberry syrup = 2 1/4 = 9/4.

Water = 3/4.

Total cups of strawberry drink = 15 3/4 = 63/4.

Let the strawberry syrup be [tex]\frac{9}{4} x[/tex].

And let the water be [tex]\frac{3}{4} x[/tex].

According to question:

[tex]\frac{9x}{4} + \frac{3x}{4}=\frac{63}{4}[/tex].

On adding the fractions:

⇒[tex]\frac{9x+3x}{4} =\frac{63}{4}[/tex]

⇒[tex]\frac{12x}{4} =\frac{63}{4}[/tex]

Multiplying both sides by 4 we get:

⇒[tex]12x=63[/tex]

Dividing both sides by 12 we get:

⇒[tex]x=\frac{63}{12}[/tex]

Dividing numerator and denominator by 3 on R.H.S we get:

⇒[tex]x=\frac{21}{4}[/tex]

Now, putting the value of [tex]x[/tex] on ratios:

Strawberry syrup =  [tex]\frac{9}{4}\times x=\frac{9}{4}\times\frac{21}{4}[/tex]

                             =  [tex]\frac{189}{16}[/tex]

                             =  [tex]11\frac{13}{16}\ cups[/tex]  

Water = [tex]\frac{3}{4}\times x =\frac{3}{4} \times\frac{21}{4}[/tex]

          = [tex]\frac{63}{8}[/tex]

          = [tex]7\frac{7}{8} \ cups.[/tex]

Therefore, the water she need is [tex]7\frac{7}{8} \ cups.[/tex] and strawberry syrup [tex]11\frac{13}{16}\ cups[/tex].

Answer:

Step-by-step explanation:

2 + (-3) + 7 = 2 - 3 + 7 = 9 - 3 = 6 <==

a positive multiplied by a negative will be negative

Answer: the answer is correct

Step-by-step explanation: edge 2020 trust me

Answer:

[tex]\frac{5}{7}-(-\frac{1}{7})=\frac{6}{7}[/tex]

Step-by-step explanation:

To evaluate :

[tex]\frac{5}{7}-(-\frac{1}{7})[/tex]

Solution:

Two negatives multiply to become a positive.

Thus, we can remove parenthesis by reversing the signs of the fraction by multiplying the negative outside.

⇒ [tex]\frac{5}{7}+\frac{1}{7}[/tex]

Since the denominators are same for both fractions, so we simply add the numerators.

⇒ [tex]\frac{5+1}{7}[/tex]

⇒ [tex]\frac{6}{7}[/tex]  (Answer)

There is no way to determine the rate of change [slope], so it is impossible to answer this question. I apologise.

Answer:

5x(5x-7)+y(7-y)

Step-by-step explanation:

25x^2-35x+7y-y^2

5x(5x-7)+y(7-y)

Answer:

(5x-7y) (5x-y)

Step-by-step explanation:

Answer:

Apple Produce pays employees  38.5 for per bushels of apples picked.

Step-by-step explanation:

Given:

Apple produce pays it's employees by the formula;

[tex]P(b) = \frac{7}{2}b+35[/tex]

[tex]P(b)[/tex] ⇒ Employee's Total daily pay

[tex]b[/tex] ⇒ Number of bushels of apples

We need to find the Rate employees are paid per bushels of apples picked.

Rate employees are paid per bushels of apples picked can be calculated by substituting the value of "b = 1" in above formula.

Substituting the value of b = 1 in above formula we get;

[tex]P(1) = \frac{7}{2} \times 1+35\\[/tex]

Now We will take LCM to make the Denominator common.

[tex]P(1) = \frac{7}{2} +\frac{35\times 2}{2} = \frac{7}{2} +\frac{70}{2} = \frac{77}{2}= 38.5[/tex]

Hence Apple Produce pays employees  38.5 for per bushels of apples picked.

Answer:

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

Step-by-step explanation:

We have to solve for x from the logarithmic equation as follows:

[tex]\log x^{2} + \log 25 = 3[/tex]

⇒ [tex]\log 25x^{2}  = 3[/tex]  

{Since, using logarithmic property [tex]\log A + \log B = \log AB[/tex]}

Now, converting this logarithmic equation above into exponential equation we get,

[tex]25x^{2}  = 10^{3}[/tex]  

{Since we know that if [tex]\log_{10}a = b[/tex] then, we can write [tex]a = 10^{b}[/tex]}

⇒ [tex]25x^{2}  = 1000[/tex]

⇒ [tex]x^{2}  = 40[/tex]

⇒ [tex]x = 2\sqrt{10}[/tex] (Answer)

Answer:

y = 30 + 1.5x and y = 2x

Step-by-step explanation:

Website 1 has a plan for a yearly fee of $30 and $1.5 for each download.

Therefore, if x is the number of downloads for a year and y is the total cost for the year, then we can model the conditions as  

y = 30 + 1.5x ......... (1)  

Website 2 has a plan of $2 for each download.

Therefore, we can models the condition as  

y = 2x ........ (2)

Therefore, equations (1) and (2) represent the costs for one year. (Answer)

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:

[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:

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

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

Hope this helped! (Plz mark me brainliest)

Answer:

- [tex]\frac{24}{25}[/tex]

Step-by-step explanation:

Given 630 < A > 720 then A is in the fourth quadrant where

cosA > 0 and sinA < 0

Given

cosA = [tex]\frac{3}{5}[/tex] = [tex]\frac{adjacent}{hypotenuse}[/tex]

Then the triangle is a 3- 4 - 5 with opposite side 4, thus

sinA = - [tex]\frac{opposite}{hypotenuse}[/tex] = - [tex]\frac{4}{5}[/tex]

Using the trigonometric identity

sin2A = 2sinAcosA

         = 2 × - [tex]\frac{4}{5}[/tex] × [tex]\frac{3}{5}[/tex]

         = [tex]\frac{2(-4)(3)}{5(5)}[/tex] = - [tex]\frac{24}{25}[/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:

There is not solution to that it is just 12 because there is no variable

Step-by-step explanation:

Answer:

I would say one because the solution is 12.

12 = 12.

[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:

$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.