Given the graph of the function f(x) below what happens to f x when x is a very small negative number?

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: 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:

8 degrees

Step-by-step explanation:

6° - (-2°) = 8°

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:

  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:

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:

The area of the triangle, as a function of the angle between the two given sides, is: [tex]A(Q) = 10sin(Q)\ m^{2}[/tex]

Step-by-step explanation:

We know that the area of a triangle is given by the formula A = b*h/2, where b stands for the base and h for the height.

In our problem, we can choose anyone of them as the base. Let us choose, for example, b = 5 m. Now that we know the value of the base, we can use the value of the other side (4 m) and the angle between these two sides (Q) to calculate the height:

[tex]h = (4 m)sin(Q)[/tex]

Therefore, the are of the triangle, as a function of the angle between these two sides is:

[tex]A(Q) = b*h/2 =5*4*sin(Q)/2\ m^{2} = 10sin(Q)\ m^{2}[/tex]

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:

The additive identity of [tex]V[/tex], denoted here by [tex]0_{V}[/tex], must be an element of [tex]U[/tex]. With this in mind and the provided properties you can prove it as follows.

Step-by-step explanation:

In order to a set be a vector space it is required that the set has two operations, the sum and scalar multiplication,  and the following properties are also required:

Now, if you have that [tex]V[/tex] is a vector space over a field [tex]\mathbb{K}[/tex]  and [tex]U\subset V[/tex] is a subset that contains the additive identity [tex]e=0_{V}[/tex] then [tex]U[/tex] and [tex]cv+w \in U[/tex] provided that [tex]u,v\in U, c\in \mathbb{K}[/tex], then [tex]U[/tex] is a closed set under the operations of sum and scalar multiplicattion, then it is a vector space since the properties listed above are inherited from V since the elements of [tex]U[/tex] are elements of V. Then [tex]U[/tex] is a subspace of [tex]V[/tex].

Now if we know that [tex]U[/tex] is a subspace of [tex]V[/tex] then [tex]U[/tex] is a vector space, and clearly it satisfies the properties [tex]cv+w\in U[/tex] whenever [tex]v,w\in U, c\in \mathbb{K}[/tex] and [tex]0_{V}\in U[/tex].

This is an useful criteria to determine whether a given set is subspace of a vector space.

Answer:

Each child will receive 0.125 (or 12.5%) of the estate.

Step-by-step explanation:

If the surviving spouse gets one half of the estate, the other half have to be divided among the four surviving children.

So its 0,5 divided among the 4 surviving children. That is 0.125 or 12.5% of the estate.

Each child will receive 0.125 of the estate.

Answer:

receive 0.125 (or 12.5%) of the estate.

Step-by-step explanation:

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:

Mass=1068gr

Step-by-step explanation:

Volume= 7.5 cm x 2.53 cm x 7.15 cm=135.7cm^3

Density=7.87gr/cm^3

Mass=Density*Volume=135.7*7.87=1068gr

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:

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:

angle made by the vector with positive x axis,

[tex]\theta\ =\ 49.51^o[/tex]

the angle by the positive direction of y axis,

[tex]\alpha\ =\ 40.48^o[/tex]

unit vector in the direction of the given vector,

[tex]\hat{n}\ =\ \dfrac{(-35)i+(-41)j}{53.9}[/tex]

Step-by-step explanation:

Given vector is

[tex]\vec{V}=\ (-35)i\ +\ (-41)j[/tex]

we have to calculate the angle made by the vector with positive x and y axis,

The angle made by the vector with positive x axis can be given by,

[tex]tan\theta\ =\ \dfrac{-41}{-35}[/tex]

[tex]=>\ \theta\ =\ tan^{-1}\dfrac{-41}{-35}[/tex]

[tex]=>\ \theta\ =\ 49.51^o[/tex]

And the angle by the positive direction of y axis can be given by

[tex]\alpha\ =\ 90^o-\theta[/tex]

           [tex]=\ 90^o-49.51^o[/tex]

            [tex]=\ 40.48^o[/tex]

Now, we will calculate the unit vector in the direction of the given vector.

So,

[tex]\hat{n}\ =\ \dfrac{\vec{A}}{|\vec{A}|}[/tex]

            [tex]=\ \dfrac{(-35i)+(-41)j}{\sqrt{(-35)^2+(-41)^2}}[/tex]

            [tex]=\ \dfrac{(-35)i+(-41)j}{53.9}[/tex]

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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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: 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:

Step-by-step explanation:

So what multiplied by what is equal to 0.64? Well you know that 8*8 is equal to positive 64, and since 0.64 is just 64 moved down two decimal spaces, you do the same with 8. So For k, it's 0.8

For m, you do the same. So what multiplied by what is equal to 0.25? Well you know that 5*5 is equal to positive 25, and since 0.25 is just 25 moved down two decimal spaces, you do the same with 5. So For m, it's 0.5.

Answer:

  (a)  f(4) = 11

  (b)  f(-1) = 211

  (c)  f(a) = 5a² -55a +151

  (d)  f(2/m) = (151m² -110m +20)/m²

  (e)  x = 5 or x = 6

Step-by-step explanation:

A graphing calculator can help with function evaluation. Sometimes numerical evaluation is easier if the function is written in Horner Form:

  f(x) = (5x -55)x +151

(a) f(4) = (5·4 -55)4 +151 = -35·4 +151 = -140 +151 = 11

__

(b) f(-1) = (5(-1)-55)(-1) +151 = 60 +151 = 211

__

(c)  Replace x with a:

  f(a) = 5a² -55a +151

__

(d) Replace x with 2/m; simplify.

  f(2/m) = 5(2/m)² -55(2/m) +151 = 20/m² -110m +151

Factoring out 1/m², we have ...

  f(2/m) = (151m² -110m +20)/m²

__

(e) Solving for x when f(x) = 1, we have ...

  5x² -55x +151 = 1

  5x² -55x +150 = 0 . . . . subtract 1

  x² -11x +30 = 0 . . . . . . . divide by 5

  (x -5)(x -6) = 0 . . . . . . . . factor

Values of x that make the factors (and their product) zero are ...

  x = 5, x = 6 . . . . values of x such that f(x) = 1

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:

a) The degree of vertex P is 2.

b) The degree of vertex O is 0.

c) The graph has 2 components.

Step-by-step explanation:

a) The edges that have P as a vertice are {M,P} and {P,R}.

b) There is no edge with extreme point O.

c) One of the components is the one with the only vertex as O and has no edges. The other component is the one with the rest of the vertices and all the edges described.

The file has a realization of the graph.

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:

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.

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:

[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

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:

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:

(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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