A basic cellular phone plan costs $4 per month for 70 calling minutes. Additional time costs $0.10 per minute. The formula C= 4+0.10(x-70) gives the monthly cost for this plan, C, for x calling minutes, where x>70. How many calling minutes are possible for a monthly cost of at least $7 and at most $8?

Answer:

You should make 200 quarts of Creamy Vanilla and 50 quarts of Continental Mocha to earn the largest profit.

Step-by-step explanation:

You are going to earn the largest profit when you manage to use all the eggs and cups of cream that you have in stock.

This problem can be solved by a first order equation

I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha.

The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 450 eggs in stock, so:

2x + y = 450.

The problem also states that each quart of Creamy Vanilla uses 3 cups of cream and that each quart of Continental Mocha uses 3 cups of cream. There are 750 cups of cream in stock, so:

3x + 3y = 750

Now we have to solve the following system of equations

1) 2x + y = 450

2) 3x + 3y = 750

The first thing i am going to do is simplify the equation 2) by 3

2) 3(x+y)/3 = 750/3

2) x+y = 250

So now, we have the following system

1) 2x + y = 450

2) x + y = 250

I am going to write y as a function of x in 2), and replace in 1)

y = 250 - x

Replacing in 1)

2x + 250 - x = 450

2x - x = 450 - 250

x = 200

You should make 200 quarts of Creamy Vanilla

From 2), we have

y = 250 - x = 250 - 200 = 50

You should make 50 quarts of Continental Mocha

Answer:

Part 1) The distance is [tex]d=7.3\ units[/tex]

Part 2) The measure of angle 2 is 121°

Part 3) The coordinates of endpoint V are (7,-27)

Part 4) The value of x is 10

Step-by-step explanation:

Part 1) Find the distance between M(6,16) and Z(-1,14)

we know that

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

substitute the given values in the formula

[tex]d=\sqrt{(14-16)^{2}+(-1-6)^{2}}[/tex]

[tex]d=\sqrt{(-2)^{2}+(-7)^{2}}[/tex]

[tex]d=\sqrt{53}\ units[/tex]

[tex]d=7.3\ units[/tex]

Part 2) Find the measure m∠2

we know that

If two angles are supplementary, then their sum is equal to 180 degrees

In this problem we have

m∠1+m∠2=180°

substitute the given values

[tex](4y+7)\°+(9y+4)\°= 180\°[/tex]

Solve for y

[tex](13y+11)\°= 180\°[/tex]

[tex]13y= 180-11[/tex]

[tex]13y=169[/tex]

[tex]y=13[/tex]

Find the measure of  m∠2

[tex](9y+4)\°[/tex]

substitute the value of y

[tex](9(13)+4)=121\°[/tex]

Part 3) The midpoint of UV is (5,-11). The coordinates of one endpoint are U(3,5) Find the coordinates of endpoint V

we know that

The formula to calculate the midpoint between two points is

[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]

we have

[tex]M(5,-11)[/tex]

[tex](x1,y1)=(3,5)[/tex]

substitute and solve for (x2,y2)

[tex](5,-11)=(\frac{3+x2}{2},\frac{5+y2}{2})[/tex]

so

Equation 1

[tex]5=(3+x2)/2[/tex]

[tex]10=3+x2[/tex]

[tex]x2=7[/tex]

Equation 2

[tex]-11=(5+y2)/2[/tex]

[tex]-22=(5+y2)[/tex]

[tex]y2=-27[/tex]

therefore

The coordinates of endpoint V are (7,-27)

Part 4) GI bisects ∠DGH so that ∠DGI is (x-3) and ∠IGH is (2x-13) Find the value of x

we know that

If GI bisects ∠DGH

then

∠DGI=∠IGH

Remember that bisects means, divide into two equal parts

substitute the given values

[tex]x-3=2x-13[/tex]

solve for x

[tex]2x-x=-3+13[/tex]

[tex]x=10[/tex]

Answer:

The present value of K is, [tex]K=251.35[/tex]

Step-by-step explanation:

Hi

First of all, we need to construct an equation system, so

[tex](1)K=\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}[/tex]

[tex](2)K=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]

Then we equalize both of them so we can find [tex]i[/tex]

[tex](3)\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]

To solve it we can multiply [tex](3)*(1+i)^{4}[/tex] to obtain [tex](1+i)^{4}*(\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}})[/tex], then we have [tex]225(1+i)^{2}+225=169(1+i)^{3}+169(1+i)^{2}[/tex].

This leads to a third-grade polynomial [tex]169i^{3}+451i^{2}+395i-112=0[/tex], after computing this expression, we find only one real root [tex]i=0.2224[/tex].

Finally, we replace it in (1) or (2), let's do it in (1) [tex]K=\frac{169}{(1+0.2224)} +\frac{169}{(1+0.2224)^{2}}\\\\K=251.35[/tex]

Answer:

139,600.96

Step-by-step explanation:

We use the payment of a loan formula:

[tex] \displaystyle PMT = \frac{P \left(\displaystyle \frac{r}{n}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{r}{n}\right)^{-nt} \right]} [/tex]

P is the principal: $200,000. t is the number of years: 30, n is 12 since it is compounded monthly. And r is 0.039 which is 3.9% in decimal form (3.9/100)

So the formula becomes:  

[tex] \displaystyle PMT = \frac{200000 \left(\displaystyle \frac{0.039}{12}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{0.039}{12}\right)^{-12(30)} \right]} [/tex]

And using our calculator we get: PMT = $943.336

Then the total amount of money paid in the mortgage is:  

PMT*n*t = $943.336(12)(30) = $339,600.96

Therefore, the interest paid is:  

$339,600.96 - $200,000 = $139,600.96

You have to enter it without $ and rounded to the nearest cent so: 139,600.96

If the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

First, let's convert the 1 hour dosage rate of 15 mg to minutes. Since there are 60 minutes in an hour, divide 15 mg by 60 to find the dosage rate per minute.

15 mg / 60 min = 0.25 mg/min

Next, convert the dosage rate from minutes to half minutes. Since there are 2 half minutes in a minute, multiply the dosage rate per minute by 2 to find the dosage rate per half minute.

0.25 mg/min * 2 = 0.5 mg/half min

Therefore, if the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

Know more about unit conversion,

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If 15 mg of medication is delivered per hour, the rate per half-minute would be 0.125 mg. This is calculated by converting 15 mg/hour to 0.25 mg/minute, and then halving to find the dose per half-minute.

The student's question pertains to the calculation of dosage rates in medical treatment, specifically how much medication is delivered per half-minute given a rate of 15 mg of medication per hour. To figure this out, it's a simple conversion from hours to minutes and then to half-minutes.

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Answer:

To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients and subtract their exponents. In either case, the answer must be converted to scientific notation.

Step-by-step explanation:

Answer:

10 growth per 1000.

Step-by-step explanation:

Given,

Rate of birth = 30 births per 1000

Rate of death = 20 deaths per 1000

As the growth in population is the difference in the number of the child take birth and the person die.

As we are calculating the rate of birth and rate of growth in per thousands of members, so the growth rate will be also in per thousands.

As we can see on every one thousand people,

total birth = 30

total death = 20

so, total growth = total birth - total growth

                           = 30 - 20

                           = 10

As at every 1000 persons, there are 10 persons survive, so the rate of growth will be 10 growth per 1000.

Answer:

[tex]t = 9.05[/tex] weeks

Step-by-step explanation:

The mass of this particular substance can be modeled by the following exponential function:

[tex]m(t) = m(0)*e^{rt}[/tex]

In which [tex]m(t)[/tex] is the mass in function of time, [tex]m(0)[/tex] is the initial mass and r, in decimal, is the growth rate of the mass.

The problem states that:

The mass of a particular substance is known to grow exponentially at a rate of 17% per week. Its initial mass was 12 grams and, after t weeks, it weighed 56 grams. So:

[tex]r = 17% = 0.17[/tex]

[tex]m(0) = 12[/tex]

[tex]m(t) = 56[/tex]

We have to solve this equation for t. So:

[tex]m(t) = m(0)*e^{rt}[/tex]

[tex]56 = 12*e^{0.17t}[/tex]

[tex]e^{0.17t} = \frac{56}{12}[/tex]

[tex]e^{0.17t} = 4.67[/tex]

To solve for [tex]t[/tex], we put ln in both sides

[tex]ln e^{0.17t} = ln 4.67[/tex]

[tex]0.17t = 1.54[/tex]

[tex]t = \frac{1.54}{0.17}[/tex]

[tex]t = 9.05[/tex] weeks

Answer:

$37685.56

Step-by-step explanation:

Given,

Total amount we want to accumulate,A = $500,000

Total time, we have,t = 30 years

Interest rate,r = 9%

We are asked to calculate how much money we should deposit to get the required amount after a certain time period.

So, according to compound interest formula,

[tex]A\ =\ P(1+r)^t[/tex]

Where, P = amount of money we need to deposit

[tex]=>\ 500,000\ =\ P(1+0.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P(1.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P\times 13.267[/tex]

[tex]=>\ \dfrac{500,000}{13.267}\ =\ P[/tex]

[tex]=>\ P\ =\ 37685.568[/tex]

So, we need to deposit total amount of $37,685.56.

Answer:

Functions with straight lines have this sort of equation: Y = mx + b where the m is called slope and b, the y-intercept. As you don't have the slope you have to find out. In this case, the slope if 10, so when u have the m, just replace with one of your points to get the b, (y-intercept). The final equation for this function is, Y= 10x+35

Step-by-step explanation:

Answer:

The average sales per week for the first 16 weeks is $477 million dollars per week.

Step-by-step explanation:

Here, you have to find the average value of a continuous function over an interval.

Suppose you have a function [tex]f(x)[/tex] over an interval from a to b. The average of the function in this interval is given by:

[tex]\frac{1}{b-a}\int\limits^b_a {f(x)} \, dx[/tex]

Solution:

In this problem, the function is given by:

[tex]S(t) = 59t + 5[/tex]

The problem asks the average value for the first 16 weeks. It means that our interval goes from 0 to 16. So [tex]a = 0, b = 16[/tex].

The average value is given by the following integral:

[tex]A = \frac{1}{16}\int\limits^{16}_{0} {(59t + 5)} \, dt[/tex]

[tex]A = \frac{59t^{2}}{32} + \frac{5t}{16}, 0 \leq t \leq 16[/tex]

[tex]A =\frac{59*(16)^{2}}{32} + \frac{5*16}{16}[/tex]

[tex]A = $477[/tex]

The average sales per week for the first 16 weeks is $477 million dollars per week.

Answer:

the dimension called by the specification is larger

Step-by-step explanation:

Data provided in the question:

micrometer reading = 7.5 in

Dimension called by the specifications = 7.59 in

Now,

Subtracting the Micrometer reading from the dimension called by specification , we get

 7.59

- 7.5

--------

0.09

since, the result is positive thus,

the dimension called by the specification is larger

The specification of 7.59 inches is larger than the micrometer reading of 7.57 inches. Therefore, the part does not meet the required specification.

To determine whether the micrometer reading or the specification is larger, we need to compare the two values. The micrometer reading is 7.57 inches, and the specification calls for a dimension of 7.59 inches.

When comparing these two measurements:

Clearly, 7.59 inches is larger than 7.57 inches.

This comparison shows that the specification is larger than the micrometer reading.

1. Y-axis. 2. X-axis. 3. X-axis

Step-by-step explanation:

1. Look at the shapes, is it reflecting vertically (y axis) or horizontally (x axis)? Take the shape, and mentally flip it over over each axis. If it lines up with the reflection provided, the answer is the axis you flipped over.

Answer:

a) MAX--> PC (R,P) = 0,3R+ 0,5P

b) Optimal solution: 40.000 units of R and 10.000 of PC = $17.000

c) Slack variables: S3=1000, is the unattended demand of P, the others are 0, that means the restrictions are at the limit.

d) Binding Constaints:

1. 0.3 R+0.6 P ≤ 18.000

2. R+P ≤ 50.000

3. P ≤ 20.000

4. R ≥ 0

5. P ≥ 0

Step-by-step explanation:

I will solve it using the graphic method:

First, we have to define the variables:

R : Regular Gasoline

P: Premium Gasoline

We also call:

PC: Profit contributions

A: Grade A crude oil

• R--> PC: $0,3 --> 0,3 A

• P--> PC: $0,5 --> 0,6 A

So the ecuation to maximize is:

MAX--> PC (R,P) = 0,3R+ 0,5P

The restrictions would be:

1. 18.000 A availabe (R=0,3 A ; P 0,6 A)

2. 50.000 capacity

3. Demand of P: No more than 20.000

4. Both P and R 0 or more.

Translated to formulas:

Answer d)

1. 0.3 R+0.6 P ≤ 18.000

2. R+P ≤ 50.000

3. P ≤ 20.000

4. R ≥ 0

5. P ≥ 0

To know the optimal solution it is better to graph all the restrictions, once you have the graphic, the theory says that the solution is on one of the vertices.

So we define the vertices: (you can see on the graphic, or calculate them with the intersection of the ecuations)

V:(R;P)

• V1: (0;0)

• V2: (0; 20.000)

• V3: (20.000;20.000)

• V4: (40.000; 10.000)

• V5:(50.000;0)

We check each one in the profit ecuation:

MAX--> PC (R,P) = 0,3R+ 0,5P

• V1: 0

• V2: 10.000

• V3: 16.000

• V4: 17.000

• V5: 15.000

As we can see, the optimal solution is  

V4: 40.000 units of regular and 10.000 of premium.

To have the slack variables you have to check in each restriction how much you have to add (or substract) to get to de exact (=) result.  

To maximize the total profit contribution, a linear programming model is formulated with decision variables, objective function, and constraints. The model is: Maximize 0.30x + 0.50y, Subject to: 0.3x + 0.6y ≤ 18,000, x + y ≤ 50,000, y ≤ 20,000, x, y ≥ 0.

To formulate a linear programming model, we need to define the decision variables, objective function, and constraints. Let's assume that x represents the number of gallons of regular gasoline produced and y represents the number of gallons of premium gasoline produced.

The objective function is to maximize the total profit contribution, which can be expressed as: 0.30x + 0.50y.

The constraints are:

So, the linear programming model can be formulated as:

Maximize 0.30x + 0.50y

Subject to:

0.3x + 0.6y ≤ 18,000

x + y ≤ 50,000

y ≤ 20,000

x, y ≥ 0

Answer:

Mean = 86.067

Median = 85

Step-by-step explanation:

The given data set is

{ 93, 94, 95, 89, 85, 82, 87, 85, 84, 80, 78, 78, 84, 87, 90 }

Number of observations = 15

Formula for mean:

[tex]Mean=\frac{\sum x}{n}[/tex]

where, n is number of observations.

Using the above formula, we get

[tex]Mean=\frac{93+94+95+89+85+82+87+85+84+80+78+78+84+87+90}{15}[/tex]

[tex]Mean=\frac{1291}{15}[/tex]

[tex]Mean\approx 86.067[/tex]

Therefore the mean of the given data set is 86.067.

Arrange the given data set in ascending order.

{ 78, 78, 80, 82, 84, 84, 85, 85, 87, 87, 89, 90, 93, 94, 95 }

Number of observation is 15 which is an odd term. So

[tex]Median=(\frac{n+1}{2})\text{th term}[/tex]

[tex]Median=(\frac{15+1}{2})\text{th term}[/tex]

[tex]Median=8\text{th term}[/tex]

[tex]Median=85[/tex]

Therefore the median of the data set is 85.

The distribution function for the length of the time that the first to arrive has to wait for the other is given by a triangular shape. This comes from the continuous uniform distribution nature of their arrival time. More probable waiting times are represented by the peak of the triangle.

This problem relates to the mathematical concept of a continuous uniform distribution. In the meeting scenario of Mr. Warren Buffet and Mr. Zhao Danyang, each person arrives between 12 pm and 1 pm at random with a uniform probability. This means there is equal likelihood for each time interval within the hour for them to arrive.

We are interested in the length of time the first person to arrive has to wait for the second. Let's denote the time of arrival of Mr. Buffet by X and that of Mr. Zhao by Y, both from 0 (12:00 pm) to 1 (1:00 pm). The time the first to arrive has to wait is |X - Y|. The distribution function of this waiting time (W = |X - Y|) is a triangle with its peak at W = 0 (no waiting time). The two ends of the base of the triangle fall at -1 and 1. This triangular shape comes from the fact that shorter waiting times (near the peak) are more probable because there is more overlap between X and Y.

Therefore, the cumulative distribution function for the waiting time, W, is given by:

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Answer:

50%

Step-by-step explanation:

29 right and 29 wrong. 29+29=58

Divide 29 by 58 and you get .5

Move the decimal over two to the right, and there’s the 50%.

You could also multiply .5 by 100 and you’ll still get 50!

Answer:

Step-by-step explanation:

1. Find the total number of problems, incorrect and correct. 29 incorrect problems + 29 correct problems is 58 total problems.

2. Find the total number of correct problems divided by the total number of problems. 29 incorrect problems divided by 58 total problems. This is simplified to 1/2.

3. Finally, convert 1/2 into a fraction. We can do this by multiplying the fraction by 50/50 to get a fraction of 50/100, and we know that 50/100 50 percent.

Answer:

. . . False

Step-by-step explanation:

5 is not among the elements of the given set.

__

The symbol ∈ means "is an element of ...".

Answer:

c. a strawberry or a red fruit

Step-by-step explanation:

Since, Banana and apple, banana and kiwi are different fruits. Also, it is given that kiwi is brown, thus kiwi or a red fruit are different sets. So they are mutually exclusive.

Since strawberry is red fruit. Thus a strawberry or a red fruit is not mutually exclusive.

Further, If two sets A and B is Mutually Exclusive, then any units of Set A is not a member of Set B and vice-versa.

Answer:

(1) [tex]y=(\frac{2}{a+bx})^3[/tex]

By differentiating w.r.t. x,

[tex]\frac{dy}{dx}=3(\frac{2}{a+bx})^2\times \frac{d}{dt}(\frac{2}{a+bx})[/tex]

[tex]=3(\frac{2}{a+bx})^2\times (-\frac{2}{(a+bx)^2})[/tex]

[tex]=-\frac{24}{(a+bx)^4}[/tex]

(2) [tex]y=(at^3-3bt)^3[/tex]

By differentiating w.r.t. t,

[tex]\frac{dy}{dt}=3(at^3-3bt)^2\times \frac{d}{dt}(at^3-3bt)[/tex]

[tex]=3(at^3-3bt)^2 (3at^2-3b)[/tex]

[tex]=9t^2(at^2-3b)^2(at^2-b)[/tex]

(3) [tex]y=(t^b)(e^\frac{b}{t})[/tex]

Differentiating w.r.t. t,

[tex]\frac{dy}{dt}=t^b\times \frac{d}{dt}(e^\frac{b}{t})+\frac{d}{dt}(t^b)\times e^\frac{b}{t}[/tex]

[tex]=t^b(e^\frac{b}{t})\times \frac{d}{dt}(\frac{b}{t}) + bt^{b-1}(e^\frac{b}{t})[/tex]

[tex]=t^be^\frac{b}{t}(-\frac{b}{t^2})+bt^{b-1}e^{\frac{b}{t}}[/tex]

(4) [tex]z = ax^2.sin (4x)[/tex]

Differentiating w.r.t. x,

[tex]\frac{dz}{dt}=ax^2\times \frac{d}{dx}(sin (4x))+sin (4x)\times \frac{d}{dx}(ax^2)[/tex]

[tex]=ax^2\times cos(4x).4+sin (4x)(2ax)[/tex]

[tex]=4ax^2cos (4x)+2ax sin (4x)[/tex]

Answer:

a) 0.0202

b)The seller's claim is not correct.

c) If the seller's claim is true we cannot have individual length of 72.15 inch.

Step-by-step explanation:

In the question it is given that

population mean, μ = 72 inch

Population standard deviation, σ = 0.5 inch

Sample size, n = 47

Sample mean, x =72.15 inch

a) z score = [tex]\frac{x-\mu}{\sigma/\sqrtr{n}}[/tex] = [tex]\frac{72.15-72}{0.5/\sqrt{47}}[/tex] = 2.0567

P(x ≥ 72.15) = P(z ≥ 2.0567) = 0.5 - 0.4798 = 0.0202

We calculated the probability with the help of standard normal table.

b) The sellers claim that the machine cuts the lumber with a mean length of 72 inch is not correct as we obtained a very low probability.

c)If we assume that the seller's claim is true that is the machine cuts the lumber into mean length of 72 inch then we cannot have an individual length 72.15.

Standard error = [tex]\frac{\sigma}{\sqrt{n} }[/tex] = 0.0729

Because it does not lie within the range of two standard errors that is (μ±2 standard error) = (71.8541,72.1458)

Answer:

Maximum profits are earned when x = 64 that is when 64 units are sold.

Maximum Profit = P(64) = 2,08,490.666667$

Step-by-step explanation:

We are given the following information:[tex]P(x) = 6400x - 18x^2 - \frac{x^3}{3} - 40000[/tex], where P(x) is the profit function.

We will use double derivative test to find maximum profit.

Differentiating P(x) with respect to x and equating to zero, we get,

[tex]\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2[/tex]

Equating it to zero we get,

[tex]x^2 + 36x - 6400 = 0[/tex]

We use the quadratic formula to find the values of x:

[tex]x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex], where a, b and c are coefficients of [tex]x^2, x^1 , x^0[/tex] respectively.

Putting these value we get x = -100, 64

Now, again differentiating

[tex]\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x[/tex]

At x = 64,  [tex]\displaystyle\frac{d^2(P(x))}{dx^2} < 0[/tex]

Hence, maxima occurs at x = 64.

Therefore, maximum profits are earned when x = 64 that is when 64 units are sold.

Maximum Profit = P(64) = 2,08,490.666667$

Answer:

8 ten-thousands is 80 thousands.

Step-by-step explanation:

To find : How do I write 8 ten-thousands?

Solution :

We have to write 8 ten thousand,

We know that, according to numeric system

[tex]\text{Thousand is 1000}[/tex]

[tex]\text{Ten-Thousand is 10,000}[/tex]

[tex]\text{8 ten thousand is}\ 8\times 10000[/tex]

[tex]\text{8 ten thousand is 80000}[/tex]

Therefore, 8 ten-thousands is 80 thousands.

Answer:

The graph of given inequality is shown below.

Step-by-step explanation:

The given linear inequality is

[tex]x>7[/tex]

We need to graph the region of solution of the given linear inequality.

The related equation of given linear inequality is

[tex]x=7[/tex]

We know that x=a is a vertical line which passes through the point (a,0).

Here, a =7. So x=7 is a vertical line which passes through the point (7,0).

Relate line is a dotted line because the sign of inequality is >. It means the points on the line are not included in the solution set.

Shaded region is right side of the related line because the solution set contains all possible values of x which are greater than 7.

Answer:

At x = 2 and 10.

Step-by-step explanation:

Given : The cost of producing x hundred items is given by the equation [tex]C(x) = x^2-3x + 7[/tex]

The revenue generated from sales of x hundred units is given by the equation [tex]R(x) = -x^2 + 21x-33[/tex]

To Find :What values of x will the company break even?

Solution:

Cost function : [tex]C(x) = x^2-3x + 7[/tex]

Revenue function : [tex]R(x) = -x^2 + 21x-33[/tex]

Now to find the company break even :

[tex]-x^2 + 21x-33= x^2-3x + 7[/tex]

[tex]24x= 2x^2+40[/tex]

[tex]12x= x^2+20[/tex]

[tex]x^2-12x+20=0[/tex]

[tex]x^2-10x-2x+20=0[/tex]

[tex]x(x-10)-2(x-10)=0[/tex]

[tex](x-2)(x-10)=0[/tex]

So, x = 2,10

Hence the company break even at x = 2 and 10.

Answer:

Infinite circles but with values (h,k,r)=(h,3h-3,√(10h²-30h+25))

Step-by-step explanation:

You are correct, if we only have two points, then we will have infinite possibilities, but all those circles will have to be attached to those two points. We have to find a condition to have infinite circles which pass through those points.

Let h be the x coordinate of the center of the circle

Let k be the y coordinate of the center of the circle

Let r be the radius of the circle. Having the circle equation:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

And now expand every binomial and substitute the values of the points:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}\\\\(0-h)^{2}+(2-k)^{2}=r^{2}\\\\(3-h)^{2}+(1-k)^{2}=r^{2}[/tex]

[tex]x^{2}-2xh+h^{2}+y^{2}-2yk+k^{2}=r^{2}\\\\h^{2}+4-4k+k^{2}=r^{2} (I)\\\\9-6h+h^{2}+1-2k+k^{2}=r^{2}[/tex]

Now substract the first and secod equation, and then the first and the third one:

[tex]x^{2}-2xh+y^{2}-4-2yk+4k=0\\\\x^{2}-9-2xh+6h+y^{2}-1-2yk+2k=0[/tex]

Now we substract the second equation from the first one:

[tex]-4+4k+9-6h+1-2k=0[/tex]

Reduce terms and we will have a linear equation, clear k:

[tex]2k-6h+6=0\\\\k=3h-3[/tex]

The last equation let us know the value of k if we asign any value to h.

Now we substitute h and k=3h-3 in I:

[tex]h^{2}+4-4(3h-3)+(3h-3)^{2}=r^{2}\\\\10h^{2}-30h+25=r^{2}\\\\r=\sqrt{10h^{2}-30h+25}[/tex]

The last equation let us know the value of r if we asign any value to h.

These equations (in fact functions), will let us know every set for h, k and r to form a circle (which will pass through the mentioned points)

[tex](h,k,r)=(h,3h-3,\sqrt{10h^{2}-30h+25})[/tex]

Where the indepent variable is h for any real, k is any real and the minimum r is √2.5:

[tex]h:R\\k:R\\r:(\sqrt{2.5},infinite)[/tex]

Answer:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

Step-by-step explanation:

The matrix representation of the system of linear equations is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\11&13&9&\vdots&1\\3&5&7&\vdots&-5\\8&8&2&\vdots&6\\7&5&2&\vdots&2\end{array}\right)[/tex]

First apply the following row operations:

[tex]\begin{array}{c}R_{2}\to R_{2}+(-\frac{11}{15})R_{1}\\R_{3}\to R_{3}+(-\frac{1}{5})R_{1}\\R_{4}\to R_{4}+(-\frac{8}{15})R_{1}\\R_{5}\to R_{5}+(-\frac{7}{15})R_{1}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&\frac{12}{5}&\frac{31}{5}&\vdots&-\frac{33}{5}\\0&\frac{16}{15}&-\frac{2}{15}&\vdots&\frac{26}{15}\\0&-\frac{16}{15}&\frac{2}{15}&\vdots&-\frac{26}{15}\end{array}\right)[/tex]

Then apply  the row operations:

[tex]\begin{array}{c}R_{3}\to R_{3}+(-\frac{12}{5}\cdot \frac{15}{52})R_{2}\\R_{4}\to R_{4}+(-\frac{-16}{15}\cdot \frac{15}{52})R_{2}\\R_{5}\to R_{5}+(\frac{16}{15}\cdot \frac{15}{52})R_{2}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&-2&\vdots&\frac{42}{13}\\0&0&2&\vdots&-\frac{42}{13}\end{array}\right)[/tex]

Now apply the row operations:

[tex]\begin{array}{c} R_{4}\to R_{4}+R_{3}\\R_{5}\to R_{5}+(-1)R_{3}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&0&\vdots&0\\0&0&0&\vdots&0\end{array} \right)[/tex]

The equivalent linear system associated to this matrix is

[tex]\begin{cases}15x+13y+4z=8\\\frac{52}{15}y+\frac{91}{15}z=-\frac{73}{15}\\2z=-\frac{42}{13}\end{cases}[/tex]

To Solve this last system is very simple by substitution. The solutions are:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

Answer: If we have a set defined as the difference of two sets, this is

A = B - C, then A = B - B∩C

So this is defined as, if you subtract the set C to the set B, then you are subtracting the common elements between C and B, from the set B.

so if A = {a,b,c} - {a,b,d}, the common elements are a and b, so: A = {c}

Answer:

17.21 × 10^7

Step-by-step explanation:

When writting in scientific notation, you only place the decimal as 17.21. Then, you count how many zeros are after of before the numbers. If it's before 17.21, then it would be 17.21^-7, if it is after 17.21, then it would be 17.21^7.

Answer:

[tex]1.721 \cdot 10^{10}[/tex]

Step-by-step explanation:

17,210,000,000 written in scientific notation

To write the number in scientific notation we move the decimal point after the first number.

To move the decimal point after 1, move 10 places to the left

So the number becomes

1.7210000000

Ignore all the zeros. we moved 10 places to the left so multiply with 10^10

[tex]1.721 \cdot 10^{10}[/tex]

Answer:

The number of solutions of a system is given by the number of different variables in the system, this number has to be the same as the number of independent equations. The coefficients and the augmented matrix of the system show these values in a matrix form. A system has a unique solution when the rank of both matrixes and the number o variables in the system are the same.

Step-by-step explanation:

For example, the following system has 2 different variables, x and y.

[tex]x+y=1\\x+2y=5[/tex]

In order to find a unique solution to the system, the number of independent equations and variables in the system must be the same In the previous example, you have 2 independent equations and 2 variables, then the solution of the system is unique.  

The rank of a matrix is the dimension of the vector generated by the columns, in other words, the rank is the number of independent columns of the matrix.  

According to Rouché-Capelli Theorem, a system of equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. The inconsistency of the system is because you can't find a combination of the variables that will solve the system.