f(x) = 2x + 10 g(x) = 1(1.01)x

x y

10 1.00

12 1.01

14 1.02

16 1.03

Which statement BEST compares the growth of the two functions as x approaches [infinity]?

Answer:

0.99 probability that at least one trip will last less than an hour

Step-by-step explanation:

For each time that the machine is used, there are only two possible outcomes. Either it lasts more than an hour, or it does not. The probability of a trip lasting more than an hour is independent of other trips. So the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

0.8 probability of lasting more than one hour.

So [tex]p = 0.8[/tex]

20 trips

So [tex]n = 20[/tex]

Assuming that each trip is equally likely to last for more than an hour, what is the probability that at least one trip will last less than an hour?

Either all the trips last for more than an hour, or at least one does not. The sum of the probabilities of these outcomes is decimal 1. So

[tex]P(X = 20) + P(X < 20) = 1[/tex]

We want P(X < 20). So

[tex]P(X < 20) = 1 - P(X = 20)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 20) = C_{20,20}.(0.8)^{20}.(0.2)^{0} = 0.0115[/tex]

[tex]P(X < 20) = 1 - P(X = 20) = 1 - 0.0115 = 0.9885[/tex]

0.9885 probability that at least one trip will last less than an hour

Rouding to the nearest hundreth.

0.99 probability that at least one trip will last less than an hour

Answer: The weight of each piece of fudge is 12.5 pounds

Step-by-step explanation: To find how much each piece of fudge weighs simply divide 75 by 6 to get the equal weight for 6 pieces.

75/6 = 12.5 pounds per piece.

Answer:

The correct answer is x [tex]\geq[/tex] 106, where x is the marks score by Jose in his remaining test, n.

Step-by-step explanation:

Jose scored 544 points in his math test.

Jose needs minimum of 650 points to get an A.

Let Jose scores x in the remaining test, n.

Jose needs to score an A. So, to reach 650 points, Jose need to score a minimum of 650 - 544 = 106 points.

Thus the value of x must be greater than of equal to 106, in his remaining test, n, to ensure that Jose gets an A.

⇒ x [tex]\geq[/tex] 106.

The error interval for y is 7.8 ≤ y < 7.9.

The error interval is determined by the decimal digits that will determine the value of y. Since the calculator display shows 7.8, y will be between 7.8 and the next smallest number, which will be 7.9. Therefore, the error interval for y is:

7.8 ≤ y < 7.9

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The error interval for y is [7.8, 7.9).

To find the error interval for y, we need to consider the decimal place immediately after the given value on Sarah's calculator display. Since the given value is 7.8, we look at the digit after the decimal point, which is 0. If this digit is 5 or greater, we round up the previous digit. In this case, 0 is not greater than 5, so we do not round up. Therefore, the error interval for y is [7.8, 7.9).

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Answer:

Net worth is defined as:

Assets - Liabilities.

Assets are what you own.

Liabilities are what you have to pay.

So, home + truck + investments are assets.

Likewise, mortgage + student loans + credit cards are liabilities.

Doing the math, Susan's net worth is

Assets - Liabilities

$143,392.90 - $98,077.12 =

$45,315.78

Susan's net worth is $45,315.78

Good luck!

Susan's net worth is calculated by subtracting her total liabilities (mortgage, student loans, and credit card debt) from her total assets (home, truck, and investments). Therefore, Susan's net worth is $45,315.78.

The question is about calculating Susan's net worth. Net worth is calculated by subtracting the total liabilities from total assets. In Susan's case, her assets include her house, truck, and investments, which total to $143,392.90 ($128,318.74 for the house, $10,615.00 for the truck, and $4,459.16 for her investments). Her liabilities include her mortgage, student loans, and credit card debt, which sums up to $98,077.12 ($93,987.74 for the mortgage, $2,874.46 for the student loans, and $1,214.92 for the credit card debt). To calculate Susan's net worth, subtract total liabilities from total assets. Therefore, Susan's net worth is $45,315.78 ($143,392.90 - $98,077.12).

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Answer:

Correct answers is

[tex] \frac{1}{216} [/tex]

Step-by-step explanation:

Given: The chances of striking the bull's-eye each time of throw =\frac{1}{6}

Since, if we throe dart on dart board , each event will be independent of the other.

Then, If we throw 3 times, what is the probability that we will strike the bull's-eye all 3 times is given by :-

[tex]p(strike \: bulls \: eye \: ) = \frac{1}{6} x \frac{1}{6} x \frac{1}{6} = \frac{1}{216} [/tex]

Hence, the probability that we will strike the bull's-eye all 3 times is 1/216

Answer:

1/216

Step-by-step explanation:

To determine the possible length of the third side in a triangle with sides measuring 5 in. and 12 in., we need to check if the sum of the given sides is greater than the length of the third side.

In a triangle, the length of any side must be less than the sum of the lengths of the other two sides. Therefore, to determine the possible length of the third side, we need to check if the sum of the given sides is greater than the length of the third side.

Let the third side be denoted as 'x'.

For a triangle with sides measuring 5 in. and 12 in., the possible length of the third side must satisfy the inequality:

5 + 12 > x

17 > x

Therefore, any length less than 17 in. is a valid possibility for the third side.

Final answer:

The length of the third side of the triangle must be greater than 7 inches but less than 17 inches, following the triangle inequality theorem. If considering a right triangle, the hypotenuse would measure exactly 13 inches according to the Pythagorean theorem.

Explanation:

Based on the question about the lengths of the sides of a triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given that two sides of the triangle measure 5 inches and 12 inches, the length of the third side must be greater than 7 inches (12 - 5) but less than 17 inches (12 + 5).

This is because the third side must be long enough to reach between the ends of the other two sides to close the triangle, but can't be so long that it would stretch beyond both ends if laid out straight. In the context of a right triangle, such as described by the Pythagorean theorem, we could consider that if the triangle were right-angled, then using the given sides as legs, the length of the hypotenuse would actually be exactly 13 inches, as 5² + 12² = 13².

Answer:

[tex]A=60^0, B=30^0, C=90^0\\a=3.46, b=2, c=4[/tex]

Step-by-step explanation:

In the diagram below:

First, we determine the value of [tex]\beta[/tex]

[tex]\alpha+\beta=90^0 $ (Other Angles of a Right Triangle)$\\60+\beta=90^0\\\beta=90^0-60^0=30^0[/tex]

To determine the value of side a, we apply the Sine rule

[tex]\dfrac{c}{Sin C} =\dfrac{a}{Sin \alpha} \\\dfrac{4}{Sin 90}=\dfrac{a}{Sin 60}\\ a=\dfrac{4*sin60}{sin 90}\\a=3.46[/tex]

Similarly, to determine the value of side b, we apply the Sine rule

[tex]\dfrac{c}{Sin C} =\dfrac{b}{Sin \beta} \\\dfrac{4}{Sin 90}=\dfrac{b}{Sin 30}\\ b=\dfrac{4*sin30}{sin 90}\\b=2[/tex]

Therefore:

[tex]A=60^0, B=30^0, C=90^0\\a=3.46, b=2, c=4[/tex]

This problem involving a right triangle with given values can be solved using the Pythagorean theorem and trigonometric identities. The final values for a, b and β are A=2√3, B=2 and Co = 30°.

In the given question, we are dealing with a right triangle where α=60°, β is the other acute angle and c=4. We can use trigonometric ratios and identities to solve for the unknowns.

Since sin(α)=a/c, we can find side 'a' by substituting α as 60° and c as 4. This gives us a=4sin(60°)=2√3.

Using the Pythagorean theorem a² + b² = c², we can substitute the values we know to solve for 'b'. This gives us b=√[c² - a²]=√[16 - (2√3)²] = √4 = 2.

For the angle β, since it is an acute angle in the right triangle and the sum of angles in a triangle is 180°, we have β= 180° - 90° - α = 180° - 90° - 60° = 30°.

Therefore, the values corresponding to a, b and β would be A=√3 (since a= 2√3), B=2 (since b=2) and Co = 30° (since β = 30°).

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Answer:

193 shoppers

Step-by-step explanation:

2nd  120 +12

3rd  132+13.2

4rd 145.2+ 14.52

5th 159.72+15.972

6th 175.692 +17.5692

7th 193.2612

Answer:

[tex]z=\frac{0.639 -0.65}{\sqrt{\frac{0.65(1-0.65)}{180}}}=-0.309[/tex]  

[tex]p_v =P(z<-0.309)=0.379[/tex]  

So the p value obtained was a very high value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults with the medication was effective is not significantly less than 0.65

Step-by-step explanation:

Data given and notation

n=180 represent the random sample taken

X=115 represent the adults with the medication was effective

[tex]\hat p=\frac{115}{180}=0.639[/tex] estimated proportion of adults with the medication was effective

[tex]p_o=0.65[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that true proportion is less than 0.65.:  

Null hypothesis:[tex]p \geq 0.65[/tex]  

Alternative hypothesis:[tex]p < 0.65[/tex]  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.639 -0.65}{\sqrt{\frac{0.65(1-0.65)}{180}}}=-0.309[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a left tailed test the p value would be:  

[tex]p_v =P(z<-0.309)=0.379[/tex]  

So the p value obtained was a very high value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults with the medication was effective is not significantly less than 0.65

Answer:

H0: p = 0.65

Ha: p < 0.65

Sample proportion  = 115 / 180 = 0.6389

Test statistics

z =  - p / sqrt( p( 1 -p ) / n)

= 0.6389 - 0.65 / sqrt ( 0.65 * 0.35 / 180)

= -0.31

Critical value at 0.05 level = -1.645

Since test statistics falls in non-rejection region, do not reject H0.

We conclude at 0.05 level that we fail to support the claim.

Step-by-step explanation:

Answer:

the answer is 9

Step-by-step explanation:

Answer:

4320 ways.

Step-by-step explanation:

Question asked:

Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that (i) the vowels (A, E, O) are together and the consonants (C, G, H, N, P) are together,

Solution:

By using Permutation formula:

[tex]^{n} P_{r} \ =\frac{n!}{(n-r)!}[/tex]

[tex]''n'' \ is\ the\ number\ of\ letters\ taking\''r'' at\ a\ time.[/tex]

CGHNP AEO EN

Total number of letters  = 10

Let consonant (CGHNP) = C

And vowel (AEO) = V

Now we have only four letters CVEN

We can arrange this 4 letters in = [tex]^{4} P_{4} \ ways\\ \\[/tex]

                                                     [tex]=\frac{4!}{(4-4!)} \\ \\ =\frac{4!}{(0!)}\\ \\ =4\times3\times2\times1=24\ ways[/tex]

Consonants having 5 letters arrange themselves in = [tex]^{5} P_{5} \ ways\\ \\[/tex]

                                                                                      [tex]=\frac{5!}{(5-5)!} \\ \\ =\frac{5\times4\times3\times2\times1}{0!} \\ \\ =120\ ways[/tex]

Vowels having 3 letters arrange themselves in  = [tex]^{3} P_{3} \ ways\\ \\[/tex]

                                                                               = [tex]=\frac{3!}{(3-3)!} \\ \\ 3\times2\times1=6 \ ways[/tex]  

Repeated letter :-

E = 2 times in [tex]^{2} P_{2} \ ways=2\ ways[/tex]

N = 2 times in 2  ways

Total arrangements of repeated letters = 2 [tex]\times[/tex] 2 = 4 ways

Total number of ways = [tex]\frac{24\times120\times6}{Repated\ letters\ arrangements}[/tex]

                                    = [tex]\frac{17280}{4} =4320\ ways[/tex]

Therefore, the number of ways all 10 letters of the word can be arranged in 4320 ways.

Final answer:

To answer the question, there are 1440 different ways to arrange the letters of the word COPENHAGEN with the vowels and consonants grouped together, by factoring in permutations of each subset and the complete grouping.

Explanation:

The student has asked us to find the number of ways to arrange the letters of the word COPENHAGEN so that the vowels and consonants are grouped together. To solve this, we can use combinatorial mathematics to calculate permutations.

Step 1: Grouping the vowels together

We have three vowels: A, E, and O. We will consider them as a single entity for now. Therefore, the group of vowels can be arranged in 3! (three-factorial) ways, which means 3 times 2 times 1 = 6 ways.

Step 2: Grouping the consonants together

We have five consonants: C, G, H, N, and P. They can be arranged in 5! (five-factorial) ways, which equals 5 times 4 times 3 times 2 times 1 = 120 ways.

Step 3: Arranging the two groups

Now, our word is represented as a combination of two groups: the vowel group and the consonant group. These two groups can be arranged in 2! ways, which is 2 times 1 = 2 ways.

Step 4: Calculating the total arrangements

To find the total number of arrangements, we multiply the permutations from each step together:
6 (vowels) times 120 (consonants) times 2 (groups) = 1440.

Therefore, there are 1440 different ways to arrange the letters of the word COPENHAGEN with the vowels and consonants grouped together.

Answer:

3/20 of the students play soccer

Step-by-step explanation:

1. Divide the amount of students that play sports, 3/4 by the amount of those that play soccer, 1/5

2. 3/4 divided by 1/5= 3/20

Answer:

3/20

Step-by-step explanation:

Answer:

A=Bh

the area of a parallelogram is base times height

Answer:

The system of equations are

x+y=557

[tex]\frac35x+\frac56 y=407[/tex]

The total pages in shorter book is 245

The total pages in longer book is 312

Step-by-step explanation:

Given that,

Scott has read 407 of the total number of 557 pages, which [tex]\frac35[/tex] of the shorter book and [tex]\frac56[/tex] of longer book.

Total number of pages of shorter book be x and longer book be y.

Then,

[tex]\frac35[/tex] page of the shorter book  [tex]=\frac35 x[/tex]

[tex]\frac56[/tex] pages of the longer book = [tex]\frac56y[/tex]

So, he has read [tex]=\frac35x+\frac 56y[/tex]

Total number of pages of both book is = x+y

According to the problem,

x+y=557.........(1)

[tex]\frac35x+\frac56 y=407[/tex].......(2)

We can write equation (2) as

[tex]\frac35x+\frac56 y=407[/tex]

[tex]\Rightarrow \frac{18x+25y}{30}=407[/tex]

[tex]\Rightarrow {18x+25y}=407\times 30[/tex]

[tex]\Rightarrow {18x+25y}=12,210[/tex]......(3)

Now 18 times of equation (1) subtract from equation (3)

   18x+25y=12210

   18x+18y=10026

-        -        -

_______________

        25y-18y=12,210-10,026

     ⇒7y=2,184

     [tex]\Rightarrow y=\frac{2184}{7}[/tex]

     ⇒y= 312

Plug y=312 in equation (1)

x+312=557

⇒x=557-312

⇒x=245

The total pages in shorter book is 245

The total pages in longer book is 312

The system of equation that can be used to determine the total number of pages in the shorter book x, and the total number of pages in the longer book, y is

3 / 5 x + 5 / 6 y = 407

x + y = 557

Scott has read 407 of the total number of 557 pages, which is  3 / 5 of the shorter book and 5 / 6 of longer book.

x = number of pages in the shorter book

y = number of pages in the longer book

Therefore, the system of equation that can be used to determine the total number of pages in the shorter book x, and the total number of pages in the longer book, y is as follows;

3 / 5 x + 5 / 6 y = 407

x + y = 557

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1 is union so all all the numbers inside the circles:

{8,9,14,15,16,17}

2 is intersection so the numbers where the circles cross each other

{14,17}

3. A’ is any number not included with A

There are 10 total numbers.

7 are not associated with A

Probability would be 7/10 = 0.70

Answer:

The required amount of minutes is 5796.05 minutes

Step-by-step explanation:

Here, since we have that the sick leave is given as 1 minute per hour worked per month and

The amount of unused sick leave is uniformly distributed between 0 and 480

Therefore, there are 481 employees, counting from 0 to 480 with 0 included

Where the company wishes no more than 5% of all employees get cash payment then we have

Total number of minutes  = 1 to 481 = 115921  minutes

Therefore, we have 5% of 115921 = 5796.05 minutes  

The balance amount of minutes = 115921  minutes - 5796.05 minutes  

= 110124.95 minutes.

Answer:

About 7,500 registered voters are expected to vote for the new dog park.

Step-by-step explanation:

- A town has 15,000 registered voters.

- A random sample of 200 voters shows that 100 voters are in favour of a new dog park.

How many registered voters are likely to vote for the dog park?

The laws of probability allows us to extrapolate and use the proportion of randomly sampled registered voters that vote for a new dog park to calculate the actual number of registered voters that will vote for a new dog park.

Proportion of randomly sampled registered voters that voted for a new dog park

= (100/200) = 0.50

Proportion of overall registered voters that will vote for a new dog park will also be 0.50.

Number of likely registers voters that'll vote for a new dog park = 0.50 × 15,000 = 7,500

Hope this Helps!!!

Approximately 7,500 registered voters are likely to vote for the new dog park

To determine how many of the 15,000 registered voters in a town are likely to vote for a new dog park based on a random sample, follow these steps:

Therefore, approximately 7,500 registered voters are likely to vote for the new dog park.

Answer:

4

Step-by-step explanation:

Hope this helped :)

Answer:

it is 9

Step-by-step explanation:

because 2+2=4+2=6+2=9

Answer:

[tex]11[/tex]

Step-by-step explanation:

[tex] \frac{90}{360} \times 2 \times \pi \times 7 \\ \frac{1}{4} \times 2 \times \frac{22}{7} \times 7 \\ = 11[/tex]

Answer:

11

Step-by-step explanation:

[tex] = \frac{1}{2} \pi \: r \\ = \frac{1}{2} \times \frac{22}{7} \times 7 \\ = \frac{1}{2} \times 22 \\ = 11[/tex]

Answer:

3.49

Step-by-step explanation:

5-1.51=3.49

Answer:

3.49 meters long

Step-by-step explanation:

5-1.51 = 3.49

- If you need this further explained please let me know. I would be glad to help.

Answer:

5

Step-by-step explanation:

Figuring the short side, it becomes a nice 3, 4, 5 triangle

so the short side is 5 units.

Hope this helped

:)

Final answer:

The length of the shortest side of the triangle is the length of side BC, which is calculated to be 5 units using the distance formula.

Explanation:

To find the length of the shortest side of the triangle with the given vertices, we will calculate the length of each side using the distance formula √((x2-x1)² + (y2-y1)²) and then determine the shortest one.

Therefore, the length of the shortest side is the length of side BC, which is 5 units.

Answer:

0.75 is the probability of selecting a CD that is not comedy.

Step-by-step explanation:

We are given the following in the question:

Number of comedy CD = 4

Number of science CD = 5

Number of adventure CD = 7

Total number of CD,n = 16

We have to find the probability for selecting a random CD that is not comedy.

Formula:

[tex]\text{Probability} = \displaystyle\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}[/tex]

P( Non-Comedy CD) =

[tex]=\dfrac{\text{n(Non-Comedy)}}{n}\\\\=\dfrac{5+7}{16} = \dfrac{12}{16}=\dfrac{3}{4}= 0.75 = 75\%[/tex]

Thus, 0.75 is the probability of selecting a CD that is not comedy.

Answer:

r=1/π

Step-by-step explanation:

Area of the circle is defined as:

Area = πr²

Derivating both sides

[tex]\frac{dA}{dr}[/tex]=2πr

[tex]\frac{dA}{dt}[/tex]  =  [tex]\frac{dA}{dr}[/tex] x [tex]\frac{dr}{dt}[/tex]  =  2πr[tex]\frac{dr}{dt}[/tex]

If area of an expanding circle is increasing twice as fast as its radius in linear units. then we have : [tex]\frac{dA}{dt}[/tex]  =2[tex]\frac{dr}{dt}[/tex]

Therefore,

2πr [tex]\frac{dr}{dt}[/tex]  =  2  [tex]\frac{dr}{dt}[/tex]

r=1/π

Answer:

r = 1/π

Step-by-step explanation:

Here we have

Area of a circle given as

Area = πr²

Where:

r = Radius of the circle

When the area of the circle is expanding twice as fast s the radius we have

[tex]\frac{dA}{dt} =2 \times \frac{dr}{dt}[/tex]

However,

[tex]\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}[/tex] and

[tex]\frac{dA}{dr} = \frac{d\pi r^2}{dr} = 2\pi r[/tex]

Therefore, we have

[tex]\frac{dA}{dt} =2 \times \frac{dr}{dt} = 2\pi r \times \frac{dr}{dt}[/tex]

Cancelling like terms

[tex]1= \pi r[/tex]

Therefore, [tex]r = \frac{1}{\pi }[/tex].

Answer: The equilibrium point is where; Quantity supplied = 100 and Quantity demanded = 100

Step-by-step explanation: The equilibrium point on a demand and supply graph is the point at which demand equals supply. Better put, it is the point where the demand curve intersects the supply curve.

The supply function is given as

S(q) = (q + 6)^2

The demand function is given as

D(q) = 1000/(q + 6)

The equilibrium point therefore would be derived as

(q + 6)^2 = 1000/(q + 6)

Cross multiply and you have

(q + 6)^2 x (q + 6) = 1000

(q + 6 )^3 = 1000

Add the cube root sign to both sides of the equation

q + 6 = 10

Subtract 6 from both sides of the equation

q = 4

Therefore when q = 4, supply would be

S(q) = (4 + 6)^2

S(q) = 10^2

S(q) = 100

Also when q = 4, demand would be

D(q) = 1000/(4 + 6)

D(q) = 1000/10

D(q) = 100

Hence at the point of equilibrium the quantity demanded and quantity supplied would be 100 units.

A. The point at which supply and demand are in equilibrium is [tex]q=4[/tex].

B. The consumer's surplus is 178.16 .

C. The producer's surplus is 66.6 .

Given,

The supply function for a certain item is,

[tex]S(q)= (q+6)^2[/tex]

The demand function is,

[tex]D(q)= \dfrac{1000}{ (q+6)}[/tex]

Now we know that the supply and demand are in equilibrium where the supply and demand functions are equal.

So for equilibrium,

[tex]S(q)= D(q)[/tex]

[tex](q+6)^2=\dfrac{1000}{q+6}[/tex]

[tex](q+6)^3=1000[/tex]

[tex]q+6=\sqrt[3]{1000}[/tex]

[tex]q+6=10[/tex]

[tex]q=4[/tex]

Hence the point is [tex]q=4[/tex], at this point supply and demand are in equilibrium.

At equilibrium the supply is [tex](4+6)^2=100[/tex] and demand is also 100.

so, [tex](q^*,p^*)[/tex] is [tex](4,100)[/tex]

Now, the consumer's surplus will be,

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=\int\limits^4_0 {\dfrac{1000}{q+6} } \, dq -4\times 100[/tex]

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000[log10-log6]-400[/tex]

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000[1-0.778]-400[/tex]

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=1000\times 0.22184-400[/tex]

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=221.84-400[/tex]

[tex]\int\limits^{q^*}_0 {D(q)} \, dq-p^*q^*=178.16[/tex]

Now, the producer's surplus will be,

[tex]p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=400-\int\limits^{4}_0(q+6)^2dq[/tex]

[tex]p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=400-\frac{1}{3} [1000-0][/tex]

[tex]p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=\dfrac{200}{3}[/tex]

[tex]p^*q^*-\int\limits^{q^*}_0 {s(q)} \, dq=66.66[/tex]

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Answer: They must hire 12 Counselors and 18 Aides

Step-by-step explanation: The most important factor here is the fact that they need to minimize hiring costs. The math camp can afford to hire up to 45 members of staff and can as well run properly with 30 members of staff.

The camp must have at least 10 aides (simply put, any number above 10 will do). Also they may have up to 3 aides for every 2 counselors, that is the ratio of counselor to aide has been given as;

2 : 3

If the math camp decides to hire the minimum required in order to minimize cost, they would be hiring 30, and that would be divided according to the following ratio;

(Counselor):

30 x 2/5 = 12

(Aide):

30 x 3/5 = 18

Hence they would be hiring 12 counselors at a cost of

2400 x 12 = 28800

And 18 Aides at a cost of

1100 x 18 = 19800

Therefore the total (minimum) cost of hiring staff is $48,600

The problem is a mathematical optimization problem, solved using systems of inequalities. The camp's optimal hiring strategy can be obtained by satisfying a set of constraints and minimizing a cost function.

This problem can be solved using systems of inequalities in mathematics. Let's let C represent the number of counselors and A represent the number of aides. Here are the constraints you need to consider:

Also, keep in mind that the camp wants to minimize costs, and the average monthly salary is $2400 for counselors and $1100 for aides, so you need to minimize the total cost function: 2400C + 1100A. By solving this system of inequalities and minimizing the cost function, the camp will have an optimal hiring strategy.

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Answer:

Step-by-step explanation:

Using "s" and "o" to represent the prices of packs of sugar and oatmeal cookies, respectively, we can write the revenue equations as ...

  10s +2o = 56

  9s +3o = 60

In standard form (with common factors removed), these equations are ...

  5s + o = 28

  3s + o = 20

Subtracting the second equation from the first, we find ...

  (5s +o) -(3s +o) = (28) -(20)

  2s = 8 . . . . simplify

  s = 4

Substituting into the first equation, we have ...

  5·4 +o = 28

  o = 8 . . . . . . subtract 20

The price of a pack of sugar cookies is $4; of oatmeal cookies, $8.

The price of a pack of sugar cookies is $4 and the price of a pack of oatmeal cookies is $8.

Let's denote the price of a pack of sugar cookies as s and the price of a pack of oatmeal cookies as o

From Amy's sales, we have the equation:

 10s + 2o = 56 (Equation 1)

 From Adam's sales, we have the equation:

9s + 3o = 60 (Equation 2)

Let's use the elimination method.

3(10s + 2o) = 3(56)

30s + 6o = 168 (Equation 3)

 2(9s + 3o) = 2(60)

18s + 6o = 120 (Equation 4)

 Now we subtract Equation 4 from Equation 3 to eliminate o

(30s + 6o) - (18s + 6o) = 168 - 120

12s = 48

s = 4

 Now that we have the price of a pack of sugar cookies s=4  we can substitute this value back into either Equation 1 or Equation 2 to find the price of a pack of oatmeal cookies.

Let's use Equation 1:

10(4) + 2o = 56

o = 8

Answer:

There is no way. Its 6.

Step-by-step explanation:

2 x 2 is 4

4 +2 is 6

Answer:

Step-by-step explanation:

105.80 + 64.20 = 170 +57 = 227

= $46