A jar of marbles contains the following: two red marbles, three white marbles, five blue marbles, and seven green marbles.

What is the probability of selecting a red marble from a jar of marbles?

5/17

2/15

2/17

17/2

Answer: 0.50477

Step-by-step explanation:

Given : The sugar content of the syrup is canned peaches is normally distributed.

We assume the can is designed to have standard deviation [tex]\sigma=5[/tex] milligrams.

The sampling distribution of the sample variance is chi-square distribution.

Also,The data yields a sample standard deviation of [tex]s=4.8[/tex] milligrams.

Sample size : n= 10

Test statistic for chi-square :[tex]\chi^2=\dfrac{s^2(n-1)}{\sigma^2}[/tex]

i.e. [tex]\chi^2=\dfrac{(4.8)^2(10-1)}{(5)^2}=8.2944[/tex]

Now, P-value = [tex]P(\chi^2>8.2944)=0.50477[/tex]  [By using the chi-square distribution table for p-values.]

Hence, the chance of observing the sample standard deviation greater than 4.8 milligrams = 0.50477

Answer:

[tex]\frac{(x-8)^2}{8^2}-\frac{(y-0)^2}{6^2}=1[/tex]

Step-by-step explanation:

∵ The equation of a hyperbola along x-axis is,

[tex]\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1[/tex]

Where,

(h, k) is the center,

a = distance of vertex from the center,

b² = c² - a² ( c = distance of focus from the center ),

Here,

vertices are (0,0) and (16,0), ( i.e. hyperbola is along the x-axis )

So, the center of the hyperbola = midpoint of the vertices (0,0) and (16,0)

[tex]=(\frac{0+16}{2}, \frac{0+0}{2})[/tex]

= (8,0)

Thus, the distance of the vertex from the center, a = 8 unit

Now,  foci are (18,0) and (-2,0).

Also, the distance of the focus from the center, c =  18 - 8 = 10 units,

[tex]\implies b^2=10^2-8^2=100-64=36\implies b = 6[/tex]

( Note : b ≠ -6 because distance can not be negative )

Hence, the equation of the required hyperbola would be,

[tex]\frac{(x-8)^2}{8^2}-\frac{(y-0)^2}{6^2}=1[/tex]

Answer:

  { }

Step-by-step explanation:

There are no numbers that are both greater than 9 and less than 2. The expression describes the empty set.

Answer:

1/4 divided by 1/2 equals 1/2

Real-world problem:

A constructor official knows that he needs 1/2 sack of cement to produce 10 blocks of concrete for a wall. The official only has 1/4 of the sack left and want to know how many blocks he can produce with this material.

Step-by-step explanation:

Since you know that 1/2 of the sack is needed to make 10 blocks, you can use this information to find the number of blocks that 1/4 of a sack can make. The question you want to answer is:  

if [tex]\frac{1}{2}[/tex] of a sack produces 10 blocks, how may blocks [tex]\frac{1}{4}[/tex] of a sack can produce?

Using the Rule of Three you can solve

[tex]\frac{\frac{1}{4} }{\frac{1}{2}} =\frac{2}{4}=\frac{1}{2}[/tex]

Now you know that 1/4 of a sack can produce 1/2 the number of blocks that 1/2 of the sack can produces, this means that you can produce 5 blocks of concrete.

Answer:

if you have 1/4 of a rope and you need to give 7/16 to your friend how much rope did you give to your friend?

Step-by-step explanation:

Answer and Step-by-step explanation:

n > 0

n² divisible by 3 ⇒ n is divisible by 3.

Any number divisible by 3 has the sum of their components divisible by 3.

If n² is divisible by 3,  we can say that n² can be written as 3*x.

n² = 3x ⇒ n = √3x

As n is a positive integer √3x must be a integer and x has to have a 3 factor. (x = 3.a.b.c...)

This way, we can say that x = 3y and y is a exact root, because n is a integer.

n² = 3x ⇒ n = √3x ⇒ n = √3.3y ⇒ n = √3.3y ⇒ n = √3²y ⇒ n = 3√y

Which means that n is divisible by 3.

Answer:

You could order 16 different kinds of pizza.

Step-by-step explanation:

You have those following toppings:

-Pepperoni

-Sausage

-Mushrooms

-Anchovies

The order is not important. For example, if you choose Sausage and Mushrooms toppings, it is the same as Mushrooms and Sausage. So we have a combination problem.

Combination formula:

A formula for the number of possible combinations of r objects from a set of n objects is:

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

How many different kinds of pizza could you order?

The total T is given by

[tex]T = T_{0} + T_{1} + T_{2} + T_{3} + T_{4}[/tex]

[tex]T_{0}[/tex] is the number of pizzas in which there are no toppings. So [tex]T_{0} = 1[/tex]

[tex]T_{1}[/tex] is the number of pizzas in which there are one topping [tex]T_{1}[/tex] is a combination of 1 topping from a set of 4 toppings. So:

[tex]T_{1} = \frac{4!}{1!(4-1)!} = 4[/tex]

[tex]T_{2}[/tex] is the number of pizzas in which there are two toppings [tex]T_{2}[/tex] is a combination of 2 toppings from a set of 4 toppings. So:

[tex]T_{2} = \frac{4!}{2!(4-2)!} = 6[/tex]

[tex]T_{3}[/tex] is the number of pizzas in which there are three toppings [tex]T_{3}[/tex] is a combination of 3 toppings from a set of 4 toppings. So:

[tex]T_{3} = \frac{4!}{3!(4-3)!} = 4[/tex]

[tex]T_{0}[/tex] is the number of pizzas in which there are four toppings. So [tex]T_{4} = 1[/tex]

Replacing it in T

[tex]T = T_{0} + T_{1} + T_{2} + T_{3} + T_{4} = 1 + 4 + 6 + 4 + 1 = 16[/tex]

You could order 16 different kinds of pizza.

Answer:

0.138 hanks/lb

Step-by-step explanation:

Given:

Silver = 60 grain/yd

Now,

1 hank = 840 yd

or

1 yd = [tex]\frac{\textup{1}}{\textup{840}}[/tex] hank

And,

1 lb = 7000 grain.

or

1 grain = [tex]\frac{\textup{1}}{\textup{7000}}[/tex] lb

Thus,

60 grain/yd = [tex]\frac{60\times\frac{1}{7000}}{1\times\frac{1}{840}}[/tex] lb/hanks

or

60 grain/yd = 7.2 lb/ hanks

or

[tex]\frac{\textup{1}}{\textup{7.2}}[/tex] hanks/lb

or

0.138 hanks/lb

Answer:

False.

Step-by-step explanation:

Two angles are complementary when added up, they give a result of 90°.

So, to this question to be true we have to do:

Angle 1 + Angle 2 = 90

But if we resolve 56° + 124° = 180, so this means that this question is false, as the addition of both angles doesn't have a result of 90°.

Answer:

x=[15:-5:-25]'

Step-by-step explanation:

In order to create a vector you need to use this command:

x = [j:i:k]'

This creates a regularly-spaced vector x using i as the increment between elements. j is the initial value and k is the final value. Besides you need to add the character ' at the end in order to convert the arrow vector in a column vector

Answer: a) 390,625, b) 2916.

Step-by-step explanation:

Since we have given that

Number of possible grades = 5

a) Number of students = 8

Using the "Fundamental theorem of counting", we get that

[tex]5\times 5\times 5\times 5\times 5\times 5\times 5\times 5\\\\=5^8\\\\=390,625[/tex]

b) Number of students = 7

Number of students receive F = 0

Number of students receive A = 1

Number of remaining grades = 4

So, Using fundamental theorem of counting , we get that

[tex]4\times 3\times 3\times 3\times 3\times 3\times 3\\\\=4\times 3^6\\\\=2916[/tex]

Hence, a) 390,625, b) 2916.

Final answer:

There are 390,625 ways to assign grades to a class of eight students. Also, there are 4,096 ways to assign grades to a class of seven students if nobody receives an F and exactly one person receives an A.

Explanation:

(a)  In this case, each student can receive one of the five possible grades (A, B, C, D, or F). So, for each student, there are 5 choices. Since there are 8 students, we multiply the number of choices for each student together:

5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 = 58 = 390,625

Therefore, there are 390,625 ways to assign grades to the class of eight students.

(b)  In this case, the first student has only one choice, which is to receive an A. The remaining six students can receive one of the four possible grades (B, C, D, or F). So, for each of the remaining six students, there are 4 choices:

1 * 4 * 4 * 4 * 4 * 4 * 4 = 46 = 4,096

Therefore, there are 4,096 ways to assign grades to the class of seven students if nobody receives an F and exactly one person receives an A.

The mathematical proofs are useful to show that a mathematical statement is true. Generally a mathematical proof use other statements like theorems, or axioms. Also mathematical proofs are useful to know if the development of a theoretical process in other areas like physics is well done. Other thing that is useful of the proofs in mathematics is that it use a formal language  with symbols that minimize the ambiguity and make it universal.

Answer:

The principal square root of -4 is 2i.

Step-by-step explanation:

[tex]\sqrt{-4}[/tex] = 2i

We have the following steps to get the answer:

Applying radical rule [tex]\sqrt{-a} =\sqrt{-1} \sqrt{a}[/tex]

We get [tex]\sqrt{-4} =\sqrt{-1} \sqrt{4}[/tex]

As per imaginary rule we know that [tex]\sqrt{-1}=i[/tex]

= [tex]\sqrt{4} i[/tex]

Now [tex]\sqrt{4} =2[/tex]

Hence, the answer is 2i.

Answer:

a. No, Returned questionnaires can't be regarded as a random sample of households.

b. Non Response Bias

Step-by-step explanation:

a. Among the 500 households only 80 responses to the survey. This type of sample can't be regarded as a random sample. Because it is possible that the question asked to people contain any embarrassing information that peoples refuse to answer the questionnaire.

b. This type of bias is known as Non-Response Bias.

Further, Non Response bias can be considered as, In conducting a survey some people did not respond to our survey, this sometimes affects our survey result very much.

For Example: It can happen that some people may refuse to participate in a survey, as the question asked to people contain personal detail or illegal activities or asking any embarrassing information, so people refused to participate in the survey. This non-response causes the results of the survey to be biased.

Answer:

$68,500

Step-by-step explanation:

The following costs are included in the period costs:

Fixed selling expense = $3.10

Fixed administrative expense = $2.10

Sales commissions = $1.10

Variable administrative expense = $0.55

Hence,

the total period costs incurred

= Sum of the above expenses × Total number of  units sold

= ( $3.10 + $2.10 + $1.10 + $0.55 ) × 10,000

= $68,500

Answer: [tex](1,T), (2,T), (3,T), (4, T), (5,T), (6,T)\\(1,H), (2,H), (3,H), (4, H), (5,H), (6,H)[/tex]

Step-by-step explanation:

The total outcomes on a die = {1,2,3,4,5,6}=6

The total outcomes on a coin = {Tails  or Heads}=2

The number of possible outcomes =[tex]6\times2=12[/tex]

If you roll one die and flip one coin, then the possible outcomes are:  

[tex](1,T), (2,T), (3,T), (4, T), (5,T), (6,T)\\(1,H), (2,H), (3,H), (4, H), (5,H), (6,H)[/tex]

Here T denotes for Tails and H denotes for heads.

Answer:

[tex]P'(t) = 19P(1 - \frac{P}{8000}) - 9[/tex]

Step-by-step explanation:

The logistic equation is given by Equation 1):

1) [tex]\frac{dP}{dt} = rP(1 - \frac{P}{N})[/tex]

In which P represents the population, [tex]\frac{dP}{dt} = P'(t)[/tex] is the variation of the population in function of time, r is the growth rate of the population and N is the carrying capacity of the population.

Now for your system:

The problem states that the population has growth rate r=19.

The problem also states that the population has carrying capacity N=8000.

We can replace these values in Equation 1), so:

[tex]P'(t) = 19P(1 - \frac{P}{8000})[/tex]

However, beginning in 2000, 9 citizens of Cook Island have left every year to become a mathematician, never to return. So, we have to subtract these 9 citizens in the P'(t) equation. So:

[tex]P'(t) = 19P(1 - \frac{P}{8000}) - 9[/tex]

The correct differential equation modeling the population of Cook Island after 2000, taking into account the emigration of 9 citizens every year, is given by:[tex]\[ P' = \frac{P}{9}\left(1 - \frac{P}{8000}\right) - 9 \][/tex]

To derive this equation, we start with the standard logistic growth model, which is given by:

[tex]\[ P' = rP\left(1 - \frac{P}{K}\right) \][/tex]

where \( r \) is the intrinsic growth rate and [tex]\( K \)[/tex] is the carrying capacity of the environment. For the Cook Islands, we have [tex]\( r = 19 \) and \( K = 8000 \)[/tex].

However, since 9 citizens leave the island every year starting from 2000, we need to modify the logistic growth model to account for this emigration. The term representing the natural growth of the population remains the same, but we subtract 9 from the growth rate to represent the annual emigration:

[tex]\[ P' = rP\left(1 - \frac{P}{K}\right) - 9 \][/tex]

Substituting the given values of [tex]\( r \)[/tex] and [tex]\( K \)[/tex] into the equation, we get:

[tex]\[ P' = 19P\left(1 - \frac{P}{8000}\right) - 9 \][/tex]

 Now, we need to adjust the growth rate [tex]\( r \)[/tex] to reflect the fact that the population is also decreasing due to emigration. Since the population decreases by 9 every year, we divide the growth rate by 9 to account for this decrease:

[tex]\[ P' = \frac{19P}{9}\left(1 - \frac{P}{8000}\right) - 9 \][/tex]

However, the growth rate should not be divided by 9, as this would incorrectly alter the per capita growth rate. The correct adjustment is to subtract the constant rate of emigration from the overall growth rate:

[tex]\[ P' = 19P\left(1 - \frac{P}{8000}\right) - 9 \][/tex]

Upon reviewing the provided answer, we see that the growth rate [tex]\( r \)[/tex]has been incorrectly divided by 9. The correct differential equation should not have the growth rate divided by 9. Therefore, the correct differential equation modeling the population of the island [tex]\( P(t) \)[/tex] after 2000 is:

[tex]\[ P' = 19P\left(1 - \frac{P}{8000}\right) - 9 \][/tex]

Answer:

2x - 4x -7 = 4x -4x + 9

-2x -7 +7 = 9 + 7

-2x ÷ (- 2 ) = 16 ÷ (-2)

x = -8

Answer:

Cost of car = $31,210

Now we are given that  you put down 15% of the car’s cost.

So, Down payment = [tex]15\% \times 31210[/tex]

                                = [tex]\frac{15}{100} \times 31210[/tex]

                                = [tex]4681.5[/tex]

So, Amount of car loan =  Total cost - Down payment

Amount of car loan =$31210 - $4681.5

                                 =$26528.5

Thus Amount of car loan is $26528.5

Now To find the total amount of car

Principal = $26528.5

Rate of interest = 4.5%

Time = 5 years

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

[tex]A=26528.5(1+\frac{4.5}{100})^5[/tex]

[tex]A=33059.337533[/tex]

Total amount including down payment = $33059.337533+$4681.50 = $37740.837533

Hence  the total amount paid for the car (including the down payment) is $37740.83

Answer:

the unpaid balance after the 5 years will be 125400.

Step-by-step explanation:

Given,

Principal amount, P = 190,000

rate,r = 3%

total time,t = 25 years

So, the total interest after 25 years will be,

[tex]I\ =\ \dfrac{P\times r\times t}{100}[/tex]

   [tex]=\ \dfrac{190,000\times 3\times 25}{100}[/tex]

    = 142500

amount will be paid in 3 years with same interest rate can be given by

[tex]I_p\ =\ \dfrac{P\times r\times t}{100}[/tex]

       [tex]=\ \dfrac{190,000\times 3\times 3}{100}[/tex]

       = 17100

So, the amount of interest to be paid= 142500 - 17100

                                                             = 125400

so, the unpaid amount of interest after the 5 years will be 125400.

Answer:

☑ 30y²

☑ 30y² + x

Step-by-step explanation:

Polynomials contain indeterminates [variables] and operation performances, non-including negative exponents, fractional exponents, etcetera.

I am joyous to assist you anytime.

Answer:

x = 30

Step-by-step explanation:

9 + 22 = x + 1

9 + 22 = 31

31 = x + 1

-1          -1

30 = x

x = 30

Answer:

According to the Law of Absorption, these 2 expressions are equivalent:

p ∨ (p ∧ q) = p

Truth Table:

(see the image attached)

Step-by-step explanation:

To construct the Truth Table you can consider the 4 possible combinations of states that p and q could have, that is

1. p=T, q=T

2. p=T, q=F

3. p=F, q=T

4. p=F, q=F

Then you can calculate p ∨ (p ∧ q) = p for each combination

1. T ∨ (T ∧ T) = T

2. T ∨ (T ∧ F) = T

3. F ∨ (F ∧ T) = F

4. F ∨ (F ∧ F) = F

You can see that the previous values are the same states that p has, you can also see it in the table attached.

Answer:

  A ∩ B: 1. The set of all elements of 'Uʻ that are elements of both 'A' and 'B'.

  A ∪ B: 2. The set of all elements of 'U' that are elements of either 'A' or 'B'.

Step-by-step explanation:

1. The "intersection" symbol (∩) signifies the members that are in both sets. For example, {1, 2} ∩ {1, 3} = {1}.

__

2. The "union" symbol (∪) signifies the members that are in either set. For example, {1, 2} ∪ {1, 3} = {1, 2, 3}.

Answer:

Step-by-step explanation:

If we wish to model the data as a straight line, we need to use the straight line formula:

[tex]y(x)  = m * x + b[/tex]

where x is the years that have passed since the year 2000, m is the slope of the line and b the value of y when x=0, and y the numer (in millions) of employees.

For x=5 we know that y(5) = 14.4. So, we have:

[tex]y(5)  = m * 5 + b = 14.4 [/tex]

And for x=14 we know that y(14)= 18.8

[tex]y(14)  = m * 14 + b = 18.8 [/tex]

Subtracting the first equation from the second one:

[tex]y(14) - y(5) = m * 14 + b  - m * 5 - b = 18.8 -  14.4 [/tex]

[tex] m * (14  - 5 ) + b - b = 4.4[/tex]

[tex] m * 9  = 4.4[/tex]

[tex] m  = 4.4 / 9[/tex]

[tex] m  = 0.49 [/tex]

Putting this in the second equation

[tex]y(14)  = 0.49 * 14 + b = 18.8 [/tex]

[tex] 6.86 + b = 18.8 [/tex]

[tex]  b = 18.8 - 6.86 [/tex]

[tex]  b = 11.94 [/tex]

So, our equation will be:

[tex]y(x)  = 0.49 * x + 11.94 [/tex]

For 2011 the number of employees will be

[tex]y(11)  = 0.49 * 11 + 11.94 =17.33[/tex]

For 2011 the number of employees will be 17.33 millions.

The linear equation that best models the number of employees (in Millions) is  

Step-by-step explanation:

If we wish to model the data as a straight line, we need to use the straight line formula:

where x is the years that have passed since the year 2000, m is the slope of the line and b the value of y when x=0, and y the numer (in millions) of employees.

For x=5 we know that y(5) = 14.4. So, we have:

And for x=14 we know that y(14)= 18.8

Subtracting the first equation from the second one:

Putting this in the second equation

So, our equation will be:

For 2011 the number of employees will be

Answer:

  16.4 million viewers

Step-by-step explanation:

The number of viewers increased by 8 million from 10 to 18 million when the number of ads increased by 10 ads from 15 to 25. If 23 ads are run, that represents an increase of 8 ads from 15, so we expect 8/10 of the increase in viewers.

  8/10 × 8 million = 6.4 million

The number we expect with 23 ads is 6.4 million more viewers than 10 million viewers, so is 16.4 million.

_____

Alternate solution

We can write a linear equation in 2-point form for the number of viewers expected for a given number of ads:

  y = (18 -10)/(25 -15)(x -15) +10

  y = (8/10)(x -15) +10

  y = 0.8x -2 . . . . . million viewers for x ads

For 23 ads, this gives ...

  y = 0.8×23 -2 = 18.4 -2 = 16.4 . . . . million viewers, as above

_____

Comment on 8/10

I consider it coincidence that the number 23 is 8/10 of the difference between 25 and 15, and the slope of the line is 8/10. The point we're trying to interpolate has no relationship to the slope of the line, and vice versa.

Linear interpolation illustrates the use of linear equation of several points

The number of viewers is 16.4 million, if a 23 one-minute ads runs on Friday.

Linear interpolation is represented as:

[tex]\frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}[/tex]

Let:

[tex]x \to[/tex] Time

[tex]y \to[/tex] Viewers

So, we have:

[tex](x_1,y_1) = (15,10m)[/tex]

[tex](x_2,y_2) = (25,18m)[/tex]

[tex](x,y) = (23,y)[/tex]

Substitute the above points in:

[tex]\frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}[/tex]

So, we have:

[tex]\frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}[/tex]

[tex]\frac{18m - 10m}{25 -15} = \frac{y - 10m}{23 -15}[/tex]

[tex]\frac{8m}{10} = \frac{y - 10m}{8}[/tex]

Multiply both sides by 8

[tex]\frac{64m}{10} = y - 10m[/tex]

[tex]6.4m = y - 10m[/tex]

Collect like terms

[tex]y =10m + 6.4m[/tex]

[tex]y =16.4m[/tex]

Hence, the number of viewers is 16.4 million, if a 23 one-minute ads runs on Friday.

Read more about linear interpolation at:

https://brainly.com/question/4248868

Answer:

1.85 inches by 1.89 inches.

Step-by-step explanation:

We have been given that the dimensions of a nicotine transdermal patch system are 4.7 cm by 4.8 cm.

First of all, we will convert given dimensions into mm.

1 cm equals 10 mm.

4.7 cm equals 47 mm.

4.8 cm equals 48 mm.

We are told that 1 inch is equivalent to 25.4 mm, so to find new dimensions, we will divide each dimension by 25.4 as:

[tex]\frac{47\text{ mm}}{\frac{25.4\text{ mm}}{\text{inch}}}=\frac{47\text{ mm}}{25.4}\times \frac{\text{ inch}}{\text{mm}}=1.85039\text{ inch}\approx 1.85\text{ inch}[/tex]

[tex]\frac{48\text{ mm}}{\frac{25.4\text{ mm}}{\text{inch}}}=\frac{48\text{ mm}}{25.4}\times \frac{\text{ inch}}{\text{mm}}=1.8897\text{ inch}\approx 1.89\text{ inch}[/tex]

Therefore, the corresponding dimensions would be 1.85 inches by 1.89 inches.

Final answer:

To convert the dimensions of a nicotine transdermal patch from centimeters to inches, multiply the centimeter measurements by 10 to get millimeters, and then divide by 25.4 to get inches. The patch measures approximately 1.85 inches by 1.89 inches.

Explanation:

The student is asking to convert the dimensions of a nicotine transdermal patch system from centimeters to inches.

Given that 1 inch equals 25.4 millimeters (mm), this can be done by first converting the dimensions from centimeters (cm) to millimeters and then from millimeters to inches.

Since 1 cm equals 10 mm, the dimensions of the patch in millimeters are 47 mm by 48 mm. To convert these dimensions to inches, we would divide each by 25.4 (since there are 25.4 mm in an inch).

So, the dimension in inches for the patch's length would be 47 mm / 25.4 mm/inch ≈ 1.85 inches, and its width would be 48 mm / 25.4 mm/inch ≈ 1.89 inches.

Therefore, the nicotine patch measures approximately 1.85 inches by 1.89 inches.

Final answer:

Janae will reach the end of the 15-foot hallway in 56.8 seconds. She progresses 2 feet every 8 seconds, and in the last cycle, she only needs an additional 0.8 seconds to cover the final foot.

Explanation:

Calculating Janae's Time to Reach the End of Her Hallway

Janae is vacuuming by moving forwards and backwards in a consistent pattern. She moves 5 feet forwards in 4 seconds and then 3 feet backwards in the next 4 seconds. This means that every 8 seconds, Janae makes a net progress of 2 feet (5 feet - 3 feet = 2 feet).

To cover the entire 15-foot length of the hallway, we need to calculate how many 2-foot increments she can complete before reaching the end.

First, divide the total hallway length by Janae's net progress per cycle: 15 feet ÷ 2 feet per cycle = 7.5 cycles. Since Janae cannot complete half a cycle, she will have to complete a whole 8th cycle. Now, multiply the number of complete cycles by the time per cycle: 8 cycles × 8 seconds per cycle = 64 seconds.

However, in the last cycle, Janae only needs to make 1 extra foot instead of 2, since her total net progress after 7 cycles is 14 feet. Thus, during the 8th cycle, she moves forward 5 feet in 4 seconds, but as soon as she reaches the 15-foot mark, she stops.

This means that she won't need the full 8 seconds of the last cycle. We can calculate the extra time required to move the final foot by setting up a ratio. Since 5 feet take 4 seconds, 1 foot will take 4 seconds ÷ 5 = 0.8 seconds.

The total time Janae takes to reach the end of the hallway is the time for the 7 full cycles plus the time to move the last foot: (7 × 8 seconds) + 0.8 seconds = 56.8 seconds. This is the time required for Janae to reach the end of her 15-foot hallway.

Answer:

The area between the x-axis and the given curve equals 1/6 units.

Step-by-step explanation:

given any 2 functions f(x) and g(x) the area between the 2 figures is calculated as

[tex]A=\int_{x_1}^{x_2}(f(x)-g(x))dx[/tex]

The area needed is shown in the attached figure

The points of intersection of the given curve and x-axis are calculated as

[tex]x-x^2=0\\\\x(1-x)=0\\\\\therefore x=0,x=1[/tex]

hence the points of intersection are[tex](0,0),(1,0)[/tex]

The area thus equals

[tex]A=\int_{0}^{1}(x-x^2-0)dx\\\\A=\int_{0}^{1}xdx-\int_{0}^{1}x^2dx\\\\A=1/2-1/3\\\\A=1/6[/tex]

Answer:

(a) Profit function P(x) = 0.02x^2+60x-80

(b) Average profit P(x)/x = P/x = 0.02x+60-80/x

Marginal profit dP/dx = 0.04x+60

Step-by-step explanation:

Cost function: C(x) = -0.02x^2+40x+80

Price function: p(x) = 100

(a) The profit function P(x) = x*p(x)-C(x) can be expressed as:

[tex]P=x*p-C\\P=x*100-(-0.02x^{2} +40x+80)\\P=0.02x^{2}+60x-80[/tex]

(b)Average profit function: P(x)/x

[tex]P/x=(0.02x^{2}+60x-80)/x\\P/x = 0.02x+60-80/x[/tex]

Marginal profit function: dP/dx

[tex]P=0.02x^{2}+60x-80\\dP/dx=0.02*2*x+60+0\\dP/dx=0.04x+60[/tex]

Final answer:

The problem involves calculating the profit, average profit per item, and marginal profit for selling x items based on a given cost and price function. By subtracting the cost function from the revenue, we obtain the profit function P(x) = -0.02x² + 60x + 80. The average profit and marginal profit functions further analyze profitability.

Explanation:

To solve the problem given, we need to start by finding the profit function P(x), which is obtained by subtracting the cost function C(x) from the revenue function, where the revenue is the sale price per item times the number of items sold (xp(x)). Given C(x) = -0.02x² + 40x + 80 and p(x) = 100, the profit function can be determined.

Next, the average profit function is found by dividing the profit function by x, and the marginal profit function, dP/dx, is the derivative of the profit function with respect to x, which provides an approximation of the profit gained by selling one more item after x items have been sold.

Profit Function

Substituting p(x) = 100 into P(x) = xp(x) - C(x), we obtain:

P(x) = x(100) - (-0.02x² + 40x + 80)

P(x) = -0.02x² + 60x + 80

Average Profit Function

The average profit per item for x items sold is:

P(x)/x = (-0.02x² + 60x + 80) / x