If the interest rate is 3% and a total of $4,370.91 will be paid to you at the end of 3 years, what is the present value of the sum

Answer:

8 degrees

Step-by-step explanation:

6° - (-2°) = 8°

Check the picture below.

all we need to get the equation of the line is two points on it, in this case those would be (-3,2) and (1,1),

[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{1}-\stackrel{y1}{2}}}{\underset{run} {\underset{x_2}{1}-\underset{x_1}{(-3)}}}\implies \cfrac{-1}{1+3}\implies -\cfrac{1}{4}[/tex]

[tex]\bf \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{-\cfrac{1}{4}}[x-\stackrel{x_1}{(-3)}]\implies y-2=-\cfrac{1}{4}(x+3) \\\\\\ y-2=-\cfrac{1}{4}x-\cfrac{3}{4}\implies y=-\cfrac{1}{4}x-\cfrac{3}{4}+2\implies y=-\cfrac{1}{4}x+\cfrac{5}{4}[/tex]

The equation of line is 4y + x = 4.

The slope of the line is defined as the angle of the line. It is denoted by m

Slope m = (y₂ - y₁)/(x₂ -x₁ )

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁,y₁) and Point 2 has coordinates (x₂, y₂)

We have been given that Line b passes through the points  (-3,2) and (1,1),

Let

x₁ = -3, y₁ = 2

x₂ = 1, y₂ = 1

∵ (y - y₁) = {(y₂ - y₁)/(x₂ -x₁ )}(x -x₁  )

Substitute values in the formula

(y - 2) = {(1 - 2)/(1 - (-3))}(x -(-3))

(y - 2)  = {(-1)/(1+3)}(x+4)

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

4y - 8 = -x - 4

4y + x = 4

Hence, the equation of line is 4y + x = 4.

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Final answer:

To find the average sale, add up all the sales ($105.78) and divide by the total number of sales (6), resulting in an average sale of $17.63.

Explanation:

To calculate the average sale made by the salesperson, we first need to add up all the sales and then divide by the total number of sales.

The sales were: $15.50, $18.98, $16.80, $14.00, $18.50, and $22.00.

First, let's find the total:

$15.50 + $18.98 + $16.80 + $14.00 + $18.50 + $22.00 = $105.78

Next, we divide this total by the number of sales to find the average. There were 6 sales in total.

Average Sale = Total Sales / Number of Sales

Average Sale = $105.78 / 6 = $17.63

Therefore, the average sale made by the salesperson was $17.63.

Answer:

78 mg/hr

Step-by-step explanation:

Data provided in the question;

Amount of dopamine contained in solution = 208 mg

Volume of solution = 32 mL

Dosage = 12 mL/h

Concentration of dopamine in solution = [tex]\frac{\textup{Amount of dopamine}}{\textup{Volume of solution}}[/tex]

or

Concentration of dopamine in solution = [tex]\frac{\textup{208 mg}}{\textup{32 mL}}[/tex]

or

Concentration of dopamine in solution = 6.5 mg/mL

Now,

The flow rate = Concentration × Dose

or

The flow rate = ( 6.5 mg/mL ) × ( 12 mL/hr )

or

The flow rate = 78 mg/hr

Answer:

Rate of flow of dopamine = 78 mght

Step-by-step explanation:

Given,

total amount of solution = 32 ml

total amount of dopamine in 32 ml solution = 208 mg

[tex]=>\textrm{total amount of dopamine in 1 ml solution }= \dfrac{208}{32}[/tex]

                                                                      [tex]=\ \dfrac{13}{2}\ mg[/tex]

[tex]=>\ \textrm{ amount of dopamine in 12 ml solution }=\ \dfrac{208}{32}\times 12[/tex]

                                                                               [tex]=\ \dfrac{13}{2}\times 12\ mg[/tex]

                                                                               = 78 mg

Since, the rate of flow of solution = 12 mlht

That means 12 ml of solution is flowing in 1 unit time and 12 ml of solution contains 78 mg of dopamine, so the rate of flow of dopamine will be 78 mght.

Answer: If the ratio is 1:78, a) the Cutty Sark is 3510 inches long or 292.5 ft; b) the Cutty Sark is 2028 inches tall or 169 ft according to this model

Step-by-step explanation: The ratio indicates that for every inch of the model, it corresponds to 78 inches of the actual size. If the length is 45 inches for the model, it would be an equivalent of 45*78 of the actual size = 3510 inches. The same can be applied to the height. Multiplying 26 x 78, the actual size should have a height of 2028 inches.

Answer:

1) There were 34 $3,000 investors and 26 $6,000 investors.

2) There are 26 nickels and 44 dimes in the jar.

3) 1600 sodas and 1400 hot dogs were sold.

Step-by-step explanation:

1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $258,000, then how many investors contributed $3,000 and how many contributed $6,000?

x is the number of investors that contributed 3,000.

y is the number of investors that contributed 6,000.

Building the system:

There are 60 investors. So:

[tex]x + y = 60[/tex]

In all, the partnership raised $258,000. So:

[tex]3000x + 6000y = 258000[/tex]

Simplifying by 3000, we have:

[tex]x + 2y = 86[/tex]

Solving the system:

The elimination method is a method in which we can transform the system such that one variable can be canceled by addition. So:

[tex]1)x + y = 60[/tex]

[tex]2)x + 2y = 86[/tex]

I am going to multiply 1) by -1, then add 1) and 2), so x is canceled.

[tex]1) - x - y = -60[/tex]

[tex]2) x + 2y = 86[/tex]

[tex]-x + x -y + 2y = -60 +86[/tex]

[tex]y = 26[/tex]

Now we get back to equation 1), and find x

[tex]x + y = 60[/tex]

[tex]x = 60-y = 60-26 = 34[/tex]

There were 34 $3,000 investors and 26 $6,000 investors.

2) A jar contains 70 nickels and dimes worth $5.70. How many of each kind of coin are in the jar?

I am going to say that x is the number of nickels and y is the number of dimes.

Each nickel is worth 5 cents and each dime is worth 10 cents.

Building the system:

There are 70 coins. So:

[tex]x + y = 70[/tex]

They are worth $5.70. So:

[tex]0.05x + 0.10y = 5.70[/tex]

Solving the system:

[tex]1) x+y = 70[/tex]

[tex]2) 0.05x + 0.10y = 5.70[/tex]

I am going to divide 1) by -10, so we can add and cancel y:

[tex]1) -0.1x -0.1y = -7[/tex]

[tex]2) 0.05x + 0.1y = 5.70[/tex]

[tex] -0.1x + 0.05x -0.1y + 0.1y = -1.3[/tex]

[tex]-0.05x = -1.3[/tex] *(-100)

[tex]5x = 130[/tex]

[tex]x = \frac{130}{5}[/tex]

[tex]x = 26[/tex]

Now:

[tex]x+y = 70[/tex]

[tex]y = 70 - x = 70 - 26 = 44[/tex]

There are 26 nickels and 44 dimes in the jar.

3) The concession stand at an ice hockey rink had receipts of $7400 from selling a total of 3000 sodas and hot dogs. If each soda sold for $2 and each hot dog sold for $3, how many of each were sold?

x is the nuber of sodas and y is the number of hot dogs.

Building the system:

3000 items were sold. So:

[tex]x + y = 3000[/tex]

$7,4000 was the total price of these items. So:

[tex]2x + 3y = 7400[/tex]

Solving the system:

[tex]1)x + y = 3000[/tex]

[tex]2)2x + 3y = 7400[/tex]

I am going to multiply 1) by -2, so we can cancel x

[tex]1) -2x -2y = -6000[/tex]

[tex]2) 2x + 3y = 7400[/tex]

[tex]-2x + 2x -2y + 3y = -6000 + 7400[/tex]

[tex]y = 1400[/tex]

Now, going back to 1)

[tex]x + y = 3000[/tex]

[tex]x = 3000-y = 3000-1400 = 1600[/tex]

1600 sodas and 1400 hot dogs were sold.

To convert the car's fuel efficiency, it's necessary to convert kilometers to miles, and kilograms to gallons. Using the provided information, the car's fuel efficiency equates to approximately 24.4 miles per gallon.

To convert the car's fuel efficiency from kilometers per kilogram to miles per gallon, we will use the given conversions:

We start with the given efficiency of 7.6 km/kg and convert km to miles:

7.6 km/kg * (1 mile / 1.609 km) = 4.721 miles/kg

Now we convert kg to gallons using the density of gasoline:

4.721 miles/kg * (1 kg / 0.729 liters) * (3.785 liters / 1 gallon) = 24.4 miles/gallon.

Therefore, the car's fuel efficiency is approximately 24.4 miles per gallon.

Answer:  The evaluations are done below.

Step-by-step explanation:  We are given the following function :

[tex]f(x)=3x^2-5x+7.[/tex]

We are to find the value of the following expressions :

[tex](A)~f(x+h)\\\\(B)~f(x+h)-f(x)\\\\(C)~\dfrac{f(x+h)-f(x)}{h}[/tex]

To find the above expressions, we must use the given value of f(x) as follows :

[tex](A)~\textup{We have}\\\\f(x+h)\\\\=3(x+h)^2-5(x+h)+7\\\\=3(x^2+2xh+h^2)-5x-5h+7\\\\=3x^2+6xh+3h^2-5x-5h+7.[/tex]

[tex](B)~\textup{We have}\\\\f(x+h)-f(x)\\\\=(3x^2+6xh+3h^2-5x-5h+7)-(3x^2-5x+7)\\\\=6xh+3h^2-5h.[/tex]

[tex](C)~\textup{We have}\\\\\dfrac{f(x+h)-f(x)}{h}\\\\\\=\dfrac{6xh+3h^2-5h}{h}\\\\\\=\dfrac{h(6x+3h-5)}{h}\\\\=6x+3h-5.[/tex]

Thus, all the expressions are evaluated.

Answer:

26-7i

Step-by-step explanation:

1. expand it you will get : 20-15i+8i+6 ( notice that i^2= -1 )

2. simplify it: 26-7i

Using complex numbers, it is found that the result of the multiplication is: 26 - 7i

In this problem, the multiplication is: [tex](5 + 2i)(4 - 3i)[/tex].

Applying the distributive property:

[tex](5 + 2i)(4 - 3i) = 20 - 15i + 8i -6i^2 = 20 - 7i + 6 = 26 - 7i[/tex]

The result is: 26 - 7i

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Answer:

½ = x

Step-by-step explanation:

There is no illustration, but I can show you how to solve for x:

6x + 9 = 12

- 9 - 9

____________

6x = 3

__ __

6 6

x = ½

I am joyous to assist you anytime.

Answer:

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

Step-by-step explanation:

We are given that an equation

[tex]6x+9=12[/tex]

We have to find the equation which is equivalent to given equation.

[tex]6x+9=12[/tex]

[tex]6x=12-9[/tex]

Subtraction property of equality

[tex]6x=3[/tex]

[tex]x=\frac{3}{6}[/tex]

Division property of equality

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

Answer:[tex]x=\frac{1}{2}[/tex]

Answer:

Part 1) The system has infinite solutions. Is a DEPENDENT system

Part 2) The solution of the system is the point (5,7)

Part 3) The solution of the system is the point (4,3)

Part 4) The number of investor that contributed with $4,000 was 33 and the number of investor that contributed with $8,000 was 27

Step-by-step explanation:

Part 1) we have

[tex]x+y=7[/tex] ------> equation A

[tex]-x-y=-7[/tex] ------> equation B

Solve the system by graphing

Remember that the solution is the intersection point both graphs

using a graphing tool

The system has infinity solutions (both lines are identical)

see the attached figure  

Is a DEPENDENT system

Part 2) we have

[tex]x+y=12[/tex] ------> equation A

[tex]2x+3y=31[/tex] ------> equation B

Solve the system by the elimination method

Multiply equation A by -2 both sides

[tex]-2(x+y)=12(-2)[/tex]

[tex]-2x-2y=-24[/tex] ------> equation C

Adds equation B and C and solve for y

[tex]2x+3y=31\\-2x-2y=-24\\---------\\3y-2y=31-24\\y=7[/tex]

Find the value of x

substitute the value of y in the equation A (or B or C) and solve for x

[tex]x+(7)=12[/tex]

[tex]x=5[/tex]

The solution is the point (5,7)

Part 3) we have

[tex]3x+y=15[/tex] ------> equation A

[tex]x+2y=10[/tex] ------> equation B

Solve the system by the elimination method

Multiply equation A by -2 both sides

[tex]-2(3x+y)=15(-2)[/tex]

[tex]-6x-2y=-30[/tex] -----> equation C

Adds equation B and equation C

[tex]x+2y=10\\-6x-2y=-30\\---------\\x-6x=10-30\\-5x=-20\\x=4[/tex]

Find the value of y

substitute the value of x in the equation A (or B or C) and solve for y

[tex]3(4)+y=15[/tex]

[tex]12+y=15[/tex]

[tex]y=3[/tex]

therefore

The solution is the point (4,3)

Part 4) Formulate the situation as a system of two linear equations in two variables

Let

x ----> the number of investor that contributed with $4,000

y ----> the number of investor that contributed with $8,000

we have that

The system of equations is

[tex]x+y=60[/tex] ------> equation A

[tex]4,000x+8,000y=348,000[/tex] -----> equation B

Solve the system by elimination method

Multiply by -4,000 both sides equation A

[tex]-4,000(x+y)=60(-4,000)[/tex]

[tex]-4,000x-4,000y=-240,000[/tex] -----> equation C

Adds equation B and equation C and solve for y

[tex]4,000x+8,000y=348,000\\-4,000x-4,000y=-240,000\\-----------\\8,000y-4,000y=348,000-240,000\\4,000y=108,000\\y=27[/tex]

Find the value of x

Substitute the value of x in the equation A ( or equation B or equation C) and solve for x

[tex]x+27=60[/tex]

[tex]x=33[/tex]

so

The solution of the system is the point (33,27)

therefore

The number of investor that contributed with $4,000 was 33 and the number of investor that contributed with $8,000 was 27

Answer:

125 text messages.

Step-by-step explanation:

Let x represent number of text messages.

We have been given that the Call First cell phone company charges $35 per month and an additional $0.16 for each text message sent during the month.

The cost of sending x text messages using call first would be [tex]0.16x[/tex].

The total cost of sending x text messages using call first would be [tex]0.16x+35[/tex].

Cellular Plus, charges $45 per month and an additional $0.08 for each text message sent during the month.

The cost of sending x text messages using cellular plus would be [tex]0.08x[/tex].

The total cost of sending x text messages using cellular plus would be [tex]0.08x+45[/tex].

Now, we will equate both expressions to solve for x as:

[tex]0.16x+35=0.08x+45[/tex]

[tex]0.16x-0.08x+35=0.08x-0.08x+45[/tex]

[tex]0.08x+35=45[/tex]

[tex]0.08x+35-35=45-35[/tex]

[tex]0.08x=10[/tex]

[tex]\frac{0.08x}{0.08}=\frac{10}{0.08}[/tex]

[tex]x=125[/tex]

Therefore, 125 text messages would have to be sent in a month to make both plans cost the same.

Answer:

10,932,240

Step-by-step explanation:

According to the Naspa World list American english have 16,564 6-letters words. Now about the special characters we have the next list  !"#$%&'()*+,-./:;<=>?@[\]^_`{|}~ and considering the space as a special character we have a total of 33 special characters. For numbers we have a total of 10 digits.

Then to know how many possibles exists we have to find how many possibles are for the last two characters then.

[tex]33\cdot10=330[/tex]

That is the amount os possibles if always the special character go before de number, but as the number could be before the special character we have to multiply this quantity by 2.

Then we have 16,564 words for the first 6 characters and 660 options for the last two. To know the total amount of possibilities we just need to multiply this numbers, then:

[tex]16,564\cdot660=10,932,240[/tex]

Final answer:

To find the number of possible passwords, calculate the number of 6-letter words possible with 26 letters, then multiply by the number of digits (10), the number of special characters (32), and account for the two possible orders of digit and special character, leading to the formula 26⁶ * 10 * 32 * 2.

Explanation:

The question involves calculating the number of possible passwords that can be formed by using an English word of exactly 6 letters, followed by a digit and a special character in any order. To calculate this, we consider that there are 26 letters in the English alphabet, 10 possible digits (0-9), and assuming a common set of 32 possible special characters (for example, punctuation marks, symbols, etc.).

First, calculate the number of 6-letter English words that can be formed. Since the question mentions the word is not case-sensitive, each position in the word can be filled by any of the 26 letters. Therefore, the number of 6-letter words is 26⁶.

Then, for each of these words, a digit (10 choices) and a special character (32 choices) can be added in either order. Since the order matters, there are 2 different ways of arranging these two additional characters (digit-special character or special character-digit).

Therefore, the total number of possible passwords is calculated as 26⁶ * 10 * 32 * 2.

This approach highlights the significant number of combinations possible even with seemingly simple password creation rules, underlining the importance of complex passwords for enhancing security.

Answer:

  Use the Rational Root Theorem.

Step-by-step explanation:

Any rational roots will be factors of the ratio of the constant (=p(0)) to the leading coefficient of the polynomial p(x). In the general case, that ratio is a rational number and the roots have numerator that is a factor of its numerator, and a denominator that is a factor of its denominator.

__

To see how this works, consider the polynomial with rational roots b/a and d/c. Factors of it will be ...

  p(x) = (ax -b)(cx -d)( other factors if p(x) is of higher degree )

The leading coefficient here is ac; the constant term is bd. The rational root theorem says any rational roots are factors of (bd)/(ac), which b/a and d/c are.

Answer:

[tex]0.1\frac{mg}{kg*min}=100  \frac{\mu g}{kg*min}[/tex]

Step-by-step explanation:

First write the weight of the client in kg, considering that 1 pound is 0.45 kg:

[tex]w=176lb* \frac{0.45kg}{1lb}=79.83kg[/tex]

Then, transform the infusion flow to ml/min, considering that 1 hour is 60 minutes:

[tex]48ml/h*\frac{1h}{60min}=0.8ml/min[/tex]

Now it is possible to calculate the to total mass flow injected to the client per minute:

[tex]0.8\frac{ml}{min}*10\frac{mg}{ml}  =8mg/min[/tex]

To find the mass flow of Brevibloc injected by unit of weight of the pacient, just divide the total mass flow by the weight of the client:

[tex]8\frac{mg}{min}*\frac{1}{79.83kg}=0.1\frac{mg}{kg*min}[/tex]

In the question is not clear, but if you need the answer in micrograms/kg/minute just multiply by 1000:

[tex]0.1\frac{mg}{kg*min}*\frac{1000\mu g}{1mg}=100  \frac{\mu g}{kg*min}[/tex]

Answer:

(x,y)=(0,-4)

Step-by-step explanation:

Given : [tex]\frac{x}{5}- \frac{y}{4} = 1\\\\\frac{x}{6}+ y = -4[/tex]

To Find : (x,y)

Solution :

Equation 1 ) [tex]\frac{x}{5}- \frac{y}{4} = 1[/tex]

[tex]\frac{4x-5y}{20}= 1[/tex]

[tex]4x-5y= 20[/tex]  ---A

Equation 2)  [tex]\frac{x}{6}+ y = -4[/tex]

[tex]\frac{x+6y}{6} = -4[/tex]

[tex]x+6y = -24[/tex]  ---B

Solve A  and B by substitution

Substitute the value of x from B in A

[tex]4(-24-6y)-5y= 20[/tex]

[tex]-96-24y-5y= 20[/tex]

[tex]-96-29y= 20[/tex]

[tex]-96-20= 29y[/tex]

[tex]-116= 29y[/tex]

[tex]\frac{-116}{29}= y[/tex]

[tex]-4= y[/tex]

Substitute the value of y in B to get value of x

[tex]x+6(-4) = -24[/tex]  

[tex]x-24= -24[/tex]  

[tex]x=0[/tex]  

So,(x,y)=(0,-4)

Check graphically

Plot the lines A and B on graph

[tex]x+6y = -24[/tex] -- Black line

[tex]4x-5y= 20[/tex] -- Purple line

Intersection point gives the solution

So, by graph intersection point is (0,-4)

Hence verified

So, (x,y)=(0,-4)

The solutions to the system of equations are (x, y) = (-16, -4). The equations are multiplied by factors to eliminate fractions and then solved using the method of substitution. The solution is checked graphically by plotting the lines and finding the intersection point.

The subject of this question is a system of equations. We're asked to find all solutions to a given system of equations, and then to check our answer graphically. The equations given are x/5 - y/4 = 1 and x/6 + y = -4.

The first step is to eliminate fractions by multiplying each equation by a factor that will eliminate the fraction. For the first equation, this factor is 20, and for the second equation, it's 6, hence: 4x - 5y = 20 and x + 6y = -24.

Next, we can solve the system of equations using a method of our choice, for example, substitution or addition/subtraction. In this case, let's use substitution. We rearrange the first equation for x: x = (5y + 20) / 4. Substituting this into the second equation gives ((5y + 20) / 4) + 6y = -24. Solving for y, we find y = -4.

Then we substitute y = -4 into the first equation and find x. Hence, we get the solutions (x, y) = (-16, -4). In order to graphically check our solution, plot the system of lines representing the equations and find the point where they intersect. This intersection point corresponds to the solution of the system and should match our algebraic solution.

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Answer:

i think it is E the last one

Step-by-step explanation:

Answer:

[tex]\displaystyle c = -\frac{35}{4} = -8.75[/tex].

Step-by-step explanation:

Let the smaller root to this equation be [tex]m[/tex]. The larger one will equal [tex]m + 6[/tex].

By the factor theorem, this equation is equivalent to

[tex]a(x - m)(x - (m+6))= 0[/tex], where [tex]a \ne 0[/tex].

Expand this expression:

[tex]a\cdot x^{2} - a(2m + 6)\cdot x + a(m^{2} + 6m) =0[/tex].

This equation and the one in the question shall differ only by the multiple of a non-zero constant. It will be helpful if that constant is equal to [tex]1[/tex]. That way, all constants in the two equations will be equal; [tex](m^{2} + 6m)[/tex] will  be equal to [tex]c[/tex].

Compare this equation and the one in the question:

The coefficient of [tex]x^{2}[/tex] in the question is [tex]1[/tex] (which is omitted.) The coefficient of [tex]x^{2}[/tex] in this equation is [tex]a[/tex]. If all corresponding coefficients in the two equations are equal to each other, these two coefficients shall also be equal to each other. Therefore [tex]a = 1[/tex].

This equation will become:

[tex]x^{2} - (2m + 6)\cdot x + (m^{2} + 6m) =0[/tex].

Similarly, for the coefficient of [tex]x[/tex],

[tex]\displaystyle -(2m +6) = 1[/tex].

[tex]\displaystyle m = -\frac{7}{2}[/tex].

This equation will become:

[tex]x^{2} + x + \underbrace{\left(-\frac{35}{4}\right)}_{c} =0[/tex].

[tex]c[/tex] is the value of the constant term of this quadratic equation.

Answer: C= 35/4

Step-by-step explanation: As per Vieta's Theorem, when a polynomial is [tex]ax^2+bx+c =0[/tex] then two roots of the equation p & q are

Given [tex]x^2+x+c =0\\[/tex], a & b are 1 here, and p-q= 6

Therefore, p+q= -b/a= -1/1 = -1..............(Equation 1)

Also given p-q= 6............... (Equation 2)

Solving equation 1 & 2

2q = -7

q = -7/2 (value of one root q)

Putting the value of q in equation 2 we can get

p + 7/2 = 6

p = 6- 7/2

p = 5/2 ( Value of 2nd root p)

Again, as per the formula p.q = c/a, here p.q= c as a= 1

p.q = (-7/2 ) (5/2) = -35/4

So, The value of c is -35/4.

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This is a mathematical problem where the student needs to allocate $5000 between two investments, Investment A and B, fitting certain conditions. By developing a series of equations based on the conditions given, it is possible to determine the appropriate allocations.

The subject of this question pertains to the allocation of funds in two investments, a process which involves applying principles of mathematics and financial planning. The person wants to invest $5000, with a certain percentage in Investment A (yielding 5%) and the rest in Investment B (yielding 8%), as per the stipulated conditions. To adhere to these requirements, let's denominate the investment in A as 'x' and that in B as 'y'. The restrictions provided, i.e., x needs to be at least 25% of $5000 (i.e., $1250) and y should not be more than 50% of $5000 (i.e., $2500), and x should be half the investment in y, lead us to the equation x = y/2. If you solve this system of equations, the allocations into A and B can be found. For instance, one feasible solution might be $2000 in A and $3000 in B. This ensures that A is at least 25%, B is at most 50%, and A is half of B, which abides by all the stipulations provided.

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The individual should allocate $1250 to Investment A and $3750 to Investment B.

Let's denote the amount invested in Investment A as[tex]\( x[/tex]  and the amount invested in Investment B as[tex]\( y \)[/tex] . The individual has a total of $5000 to invest, so we have our first equation:\[ x + y = 5000 \]The individual wants to invest at least 25% of the total amount in Investment A, which gives us the second equation[tex]:\[ x \geq 0.25 \times 5000 \][ x \geq 1250 \][/tex]. The individual also wants to invest at most 50% of the total amount in Investment B, which gives us the third equation:[tex]\[ y \leq 0.50 \times 5000 \][ y \leq 2500 \[/tex]]. Additionally, the investment in A should be at least half the investment in B, leading to the fourth equation:[tex]\[ x \geq \frac{1}{2} y \][/tex] Now, let's solve these equations. From the first equation, we can express[tex]\( y \) in terms of \( x \):[ y = 5000 - x ]Substituting \( y \) into the inequality from the third equation, we get:[ 5000 - x \leq 2500 \][ x \geq 5000 - 2500 \][ x \geq 2500 \]This satisfies the condition from the second equation \( x \geq 1250 \).Now, we substitute \( y \) into the fourth equation:\[ x \geq \frac{1}{2} (5000 - x) \] 2x \geq 5000 - x \] 3x \geq 5000 \][ x \geq \frac{5000}{3} \][ x \geq 1666.\overline{6} Since \( x \)[/tex] must be a whole number of dollars, the smallest whole number that satisfies[tex]\( x \geq 1666.\overline{6} \) is \( x = 1667 \)[/tex] . However, we must also ensure that \( y \) is within the allowed range. Let's calculate [tex]\( y \) using \( x = 1667 \):\[ y = 5000 - x \]\[ y = 5000 - 1667 \]\[ y = 3333 \][/tex]

This allocation does not satisfy the condition that[tex]\( y \)[/tex] must be at most $2500. Therefore, we need to find the maximum value of \[tex]( x[/tex]  that satisfies both [tex]\( x \geq 1666.\overline{6} \) and \( y \leq 2500 \).Since \( x \) must be at least half of \( y \), and \( y \) must be at most $2500, we can set \( x \) to half of $2500, which is $1250:\[ x = \frac{1}{2} \times 2500 x = 1250 \]Now, let's check if \( y \) is within the allowed range:[ y = 5000 - x \][ y = 5000 - 1250 \][ y = 3750 \][/tex]This allocation satisfies all the conditions:-[tex]\( x = 1250 \)[/tex]  is more than 25% of the total investment.- [tex]\( y = 3750 \[/tex] ) is less than 50% of the total investment.- [tex]\( x \)[/tex]  is half o[tex]f \( y \).[/tex] Therefore, the individual should allocate $1250 to Investment A and $3750 to Investment B.

Answer:

679.1N

Step-by-step explanation:

Assuming equation 3 is Newton's universal law of gravity:

[tex]F_g = G\frac{m_1m_2}{d^2}[/tex]

Where G is the universal gravity constant:

[tex]G=6.673 x10^{-11}\frac{Nm^2}{kg^2}[/tex]

You need to express the radius of earth in m:

[tex]3959mi*\frac{1609m}{1mi}=6.370*10^6m[/tex]

If you weight 70Kg, just replace the values in the equation:

[tex]F_g = 6.673 *10^{-11}*\frac{70*5.900*10^{24}}{(6.370*10^6)^2}= 6.673 *10^{-11}*\frac{4.13*10^{26}}{4.058*10^{13}}=\\6.673*10^{-11}*1.018*10^{13}= 679.1N[/tex]

Answer:

The amount of the repayment check was $21862.29.

Step-by-step explanation:

Principal P = $19221

Rate r = 9.7% = 0.097

Time t = 17 months = [tex]17/12= 1.41667[/tex] years

[tex]I= p\times r\times t[/tex]

[tex]I= 19221\times0.097\times1.41667[/tex] = $2641.29

The loan repayment is the original principal plus the interest.

= [tex]19221+2641.29=21862.29[/tex] dollars

The amount of the repayment check was $21862.29.

Answer: 2.83 units

Step-by-step explanation:

The distance between the two points (a,b) and (c,d) on the coordinate system is given by :-

[tex]D=\sqrt{(d-b)^2+(c-a)^2}[/tex]

Given : A certain corner of a room is selected as the origin (0,0) of a rectangular coordinate system.

If  a fly is crawling on an adjacent wall at a point having coordinates (2.1, 1.9), then the distance of the fly from the corner (0,0) of the room will be :-

[tex]D=\sqrt{(2.1-0)^2+(1.9-0)^2}\\\\\Rightarrow\ D=\sqrt{4.41+3.61}\\\\\Rightarrow\ D=\sqrt{8.02}\\\\\Rightarrow\ D=2.8319604517\approx2.83\text{ units}[/tex]

Hence, the distance of the fly from the corner of the room = 2.83 units.

Final answer:

The distance of the fly from the corner of the room, given its coordinates on an adjacent wall are (2.1, 1.9), is approximately 2.83 meters. This distance is calculated using the Pythagorean theorem.

Explanation:

To find the distance of the fly from the corner of the room, given it is crawling on an adjacent wall at coordinates (2.1, 1.9) meters in a rectangular coordinate system, we use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this scenario, the two sides of the right-angled triangle are represented by the x-coordinate (2.1 meters) and the y-coordinate (1.9 meters) of the fly’s position.

To calculate the distance (d), we use the formula:

Plug the coordinates into the Pythagorean theorem equation: d^2 = 2.1^2 + 1.9^2.

Calculate the squares: 4.41 (2.1^2) + 3.61 (1.9^2).

Sum the results: 4.41 + 3.61 = 8.02.

Take the square root of the sum to find the distance: √8.02 ≈ 2.83 meters.

Therefore, the distance of the fly from the corner of the room is approximately 2.83 meters.

Answer: Elements of J = {-9,7,39,87}

Step-by-step explanation:

Since we have given that

S={1,3,5,7}

Define of set J is given by

[tex]J=\{2j^2-11:j\epsilon S\}[/tex]

Put j = 1

[tex]2j^2-11\\\\=2-11\\\\=-9[/tex]

Put j = 3

[tex]2(3)^2-11\\\\=2\times 9-11\\\\=18-11\\\\=7[/tex]

Put j = 5

[tex]2(5)^2-11\\\\=2\times 25-11\\\\=50-11\\\\=39[/tex]

Put j = 7

[tex]2(7)^2-11\\\\=2\times 49-11\\\\=98-11\\\\=87[/tex]

Hence, elements of J = {-9,7,39,87}

Answer:

The point (10,40) is on the graph of g

Step-by-step explanation:

If the point (8,12) is on the graph of f(x) means that

f(8)=12

So, if you choose x=10, then x-2 = 8 and

g(10) = 3f(10-2)+4 = 3f(8)+4

But f(8) = 12, so

3f(8)+4 = 36+4 = 40

Hence g(10) = 40

Which means that the point (10,40) is on the graph of g

Answer: [tex]\pm0.1706[/tex]

Step-by-step explanation:

Given : Sample size : n= 33

Critical value for significance level of [tex]\alpha:0.05[/tex] : [tex]z_{\alpha/2}= 1.96[/tex]

Sample mean : [tex]\overline{x}=2.5[/tex]

Standard deviation : [tex]\sigma= 0.5[/tex]

We assume that this is a normal distribution.

Margin of error : [tex]E=\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]

i.e. [tex]E=\pm (1.96)\dfrac{0.5}{\sqrt{33}}=\pm0.170596102837\approx\pm0.1706[/tex]

Hence, the  margin of error is [tex]\pm0.1706[/tex]

Answer:

tex]a^2 - 4b \neq 2[/tex]

Step-by-step explanation:

We are given that a and b are integers, then we need to show that [tex]a^2 - 4b \neq 2[/tex]

Let  [tex]a^2 - 4b = 2[/tex]

If a is an even integer, then it can be written as [tex]a = 2c[/tex], then,

[tex]a^2 - 4b = 2\\(2c)^2 - 4b =2\\4(c^2 -b) = 2\\(c^2 -b) =\frac{1}{2}[/tex]

RHS is a fraction but LHS can never be a fraction, thus it is impossible.

If a is an odd integer, then it can be written as [tex]a = 2c+1[/tex], then,

[tex]a^2 - 4b = 2\\(2c+1)^2 - 4b =2\\4(c^2+c-b) = 2\\(c^2+c-b) =\frac{1}{4}[/tex]

RHS is a fraction but LHS can never be a fraction, thus it is impossible.

Thus, our assumption was wrong and [tex]a^2 - 4b \neq 2[/tex].

To demonstrate that a2 - 4b cannot equal 2 for integers a and b, we can argue based on the discriminant of a quadratic equation, which should be non-positive for the equation to have one or no real roots.

In mathematics, particularly algebra, understanding the properties of polynomial equations is fundamental. When we consider the quadratic equation X^2 + aX + b = 0, it can have either one or no real roots, which is determined by its discriminant, denoted as Det = a^2 - 4b. Now, the condition for a quadratic equation to have a single (degenerate) real root or no real roots at all is that the discriminant must be non-positive.

To prove that a22 - 4b

qq 2 for all integers a and b, we can reason that if a2 - 4b were equal to 2, the quadratic equation would have two distinct real roots, which contradicts the earlier statement that the discriminant must be non-positive for it to have one or no real roots. Therefore, this proves that a2 - 4b cannot be equal to 2; hence a2 - 4b

nn2 for all integers a and b.

Answer:

The reuired probability is 0.756

Step-by-step explanation:

Let the number of trucks be 'N'

1) Trucks on interstate highway N'= 76% of N =0.76N

2) Truck on intra-state highway N''= 24% of N = 0.24N

i) Number of trucks flagged on intrastate highway  = 3.4% of N'' = [tex]\frac{3.4}{100}\times 0.24N=0.00816N[/tex]

ii)  Number of trucks flagged on interstate highway  = 0.7% of N' = [tex]\frac{0.7}{100}\times 0.76N=0.00532N[/tex]

Part a)

The probability that the truck is an interstate truck and is not flagged for safety is [tex]P(E)=P_{1}\times (1-P_{2})[/tex]

where

[tex]P_{1}[/tex] is the probability that the truck chosen is on interstate

[tex]P_{2}[/tex] is the probability that the truck chosen on interstate is flagged

[tex]\therefore P(E)=0.76\times (1-0.00532)=0.756[/tex]

Answer:

[tex]C(x)=4000+0.3x[/tex]

[tex]R(x)=1.75x[/tex]

[tex]Profit= 1.45x-4000[/tex]

Step-by-step explanation:

We are given that A rose garden can be planted for $4000.

The marginal cost of growing a rose is estimated to $0.30,

Let x be the number of roses

So, Marginal cost of growing x roses = [tex]0.3x[/tex]

Total cost = [tex]4000+0.3x[/tex]

So, Cost function : [tex]C(x)=4000+0.3x[/tex] ---A

Now we are given that the total revenue from selling 500 roses is estimated to $875

So, Marginal revenue = [tex]\frac{\text{Total revenue}}{\text{No. of roses}}[/tex]

Marginal revenue = [tex]\frac{875}{500}[/tex]

Marginal revenue = [tex]1.75[/tex]

Marginal revenue for x roses  = [tex]1.75x[/tex]

So, Revenue function =  [tex]R(x)=1.75x[/tex] ----B

Profit = Revenue - Cost

[tex]Profit= 1.75x-4000-0.3x[/tex]

[tex]Profit= 1.45x-4000[/tex]  ---C

Now Plot A , B and C on Graph

[tex]C(x)=4000+0.3x[/tex]  -- Green

[tex]R(x)=1.75x[/tex]  -- Purple

[tex]Profit= 1.45x-4000[/tex]  --- Black

Refer the attached graph