There is a mound of g pounds of gravel in a quarry. Throughout the day, 300 pounds of gravel are added to the mound. Two orders of 700 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,500 pounds of gravel. Write the equation that describes the situation. Then solve for g.

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:

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

Keith is 600 pounds over the weight limit of his truck. Given that each bag of mulch weighs 40 pounds, Keith needs to remove 15 bags of mulch to be within the truck's weight capacity.

This is a straightforward math problem involving subtraction and division. First, let's find out how much weight is over the truck's capacity. Keith's truck is currently carrying 3600 pounds of mulch, but his truck's capacity is only 3000 pounds. So, he is over by 3600 - 3000 = 600 pounds.

Each bag of mulch weighs 40 pounds, so to find out how many bags Keith needs to remove, we simply divide the total excess weight by the weight of each bag: 600 / 40 = 15 bags. Therefore, Keith needs to remove 15 bags of mulch from his truck to meet the weight requirements.

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Keith needs to remove 15 bags of mulch to meet the weight requirements.

To find out how many bags Keith needs to remove, we need to determine the weight of one bag of mulch. If he has 40-pound bags and a total weight of 3600 pounds, we can divide the total weight by the weight of one bag:

Number of bags = Total weight / weight of one bag = 3600 pounds / 40 pounds = 90 bags

Since the truck's capacity is at most 3000 pounds, Keith needs to remove the excess weight:

Excess weight = Total weight - Truck's capacity = 3600 pounds - 3000 pounds = 600 pounds

Now, we can calculate how many bags he needs to remove using the weight of one bag:

Bags to remove = Excess weight / weight of one bag = 600 pounds / 40 pounds = 15 bags

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Step-by-step explanation:

We are asked to convert given percent to a decimal number.

We know to convert a number to decimal, we divide given percent by 100 as percent means per hundred.

We also know that to divide a number by hundred, we need to move decimal to two digits to left.

(15). [tex]7\%[/tex]

[tex]7\%=\frac{7}{100}=0.07[/tex]

(16). [tex]39\%[/tex]

[tex]39\%=\frac{39}{100}=0.39[/tex]

(17). [tex]5.15\%[/tex]

[tex]5.15\%=\frac{5.15}{100}=0.0515[/tex]

(18). [tex]0.75\%[/tex]

[tex]0.75\%=\frac{0.75}{100}=0.0075[/tex]

(19). [tex]\frac{1}{4}\%[/tex]

[tex]\frac{1}{4}\%=\frac{\frac{1}{4}}{100}=\frac{1}{4*100}=\frac{1}{400}=0.0025[/tex]

(20). [tex]\frac{3}{8}\%[/tex]

[tex]\frac{3}{8}\%=\frac{\frac{3}{8}}{100}=\frac{1}{8*100}=\frac{3}{800}=0.00375[/tex]

(21). [tex]135.9\%[/tex]

[tex]135.9\%=\frac{135.9}{100}=1.359[/tex]

(22). [tex]298.7\%[/tex]

[tex]298.7\%=\frac{298.7}{100}=2.987[/tex]

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:

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

-2x -7 +7 = 9 + 7

-2x ÷ (- 2 ) = 16 ÷ (-2)

x = -8

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: 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.

Answer:

A proof can be as follows:

Step-by-step explanation:

Remember that an odd interger is of the form [tex]2p+1[/tex] where [tex]p[/tex] is a integer and remember that two consecutive integer are two numbers of the form [tex]p, p+1[/tex]

[tex](\Rightarrow)[/tex] Suppose the [tex]n[/tex] is an odd integer.

Then [tex]n-1[/tex] must be an even integer and hence divisible by 2. Then we define

[tex]p=\dfrac{n-1}{2}\\q=\dfrac{n-1}{2}+1[/tex]

Then we have that

[tex]p+q=\dfrac{n-1}{2}+\dfrac{n-1}{2}+1=\frac{(n-1)+(n-1)}{2}+1=\frac{2(n-1)}{2}+1=n-1+1=n[/tex]

The converse is as follows:

[tex](\Leftarrow)[/tex] Let [tex]p[/tex] an integer, then[tex]p,p+1[/tex]  are two consecutive integers. Then

[tex]n=p+(p+1)=2p+1[/tex] is an odd integer.

Answer:

x = 30

Step-by-step explanation:

9 + 22 = x + 1

9 + 22 = 31

31 = x + 1

-1          -1

30 = x

x = 30

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

The probability that a randomly selected survivor was in second class is calculated by dividing the number of second class passengers who survived (118) by the total number of survivors (711), resulting in a probability of 16.6%.

The subject of this question is probability. To calculate the probability that a randomly selected survival was in the second class cabin, we need to consider the number of second class passengers who survived compared to the total number of survivors. From the data provided, we can see that 118 passengers in second class survived the incident. The total number of survivors is 711.

Therefore, the probability (P) is calculated as follows:
P = Number of successful outcomes / Total number of outcomes.
Hence, P = 118 / 711 = 0.166.

This means that there is a 16.6% chance that a passenger who survived was from the second-class cabin.

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The probability that a randomly selected survivor is from the second-class cabin is approximately 0.166, or 17% when rounded to the nearest percent.

Any calculation of probability involves dividing the number of favorable outcomes by the total number of outcomes. In this case, you want to find the probability that a randomly selected passenger who survived the Titanic was in a second-class cabin.

From the chart, we can see that 118 second-class passengers survived. The total number of survivors is 711. Hence, the probability of a survivor being from the second-class cabin is given by the formula:

Probability = favorable outcomes / total outcomes = number of second class survivors / total number of survivors.

Substituting these values into the formula we get: Probability = 118 / 711 = 0.166, or around 17% when rounded to the nearest percent.

So, if we randomly select a passenger who survived the Titanic, there is a 17% probability that this passenger is in a second-class cabin.

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The solution is [tex]\(\frac{x^2}{2} + \frac{y^3}{3} = C\)[/tex], where \(C\) is the constant of integration.

To solve the differential equation [tex]\( xdx + y^2 dy = 0 \),[/tex] we can separate the variables and integrate both sides:

[tex]\[ \int x \, dx + \int y^2 \, dy = 0 \][/tex]

Integrating each term separately:

[tex]\[ \frac{x^2}{2} + \frac{y^3}{3} = C \][/tex]

Where  C is the constant of integration.

Complete question : Solve the following differential equation xdx+y2dy=0.

Answer:

The remaining interior angles of this triangle are 140º and 10º

Step-by-step explanation:

The sum of the interior angles of a triangle is always 180º.

A triangle has 3 angles. In this problem, we have one of them, that i am going to call A1 = 30º.

The sum of a interior angle with it's respective exterior angle is also always 180º.

We have that one of the exterior angles is equal to 40°. So it's respective interior angle is

40º + A2 = 180º

A2 = 180º - 40º

A2 = 140º

Now we have two interior angles, and we know that the sum of the 3 interior angles is 180º. So:

A1 + A2 + A3 = 180º

A3 = 180º - A1 - A2

A3 = 180º - 30º - 140º

A3 = 180º - 170º

A3 = 10º

Answer:

140 and 10

Step-by-step explanation:

Answer:

1st question : 6564

2nd question : 6.6

Step-by-step explanation:

We have to subtract 8,878-2,314.

We will subtract the smaller number from greater number.

So, the answer will be = 6564

We also have to subtract 5.4 from 12 .

Hence, the difference will be = 6.6

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:

x= [tex]4\pm \sqrt{73}[/tex]

Step-by-step explanation:

First step. Solve the binomial product from the left side of the equation:

[tex](x-12)(x+4)=9[/tex]

[tex]x^2+4x-12x-48=9[/tex]

Second step. Simplify and move independent terms to the right side of the equation:

[tex]x^2-8x=57[/tex]

Third step. Find a number that multiplied by two gives -8, then square this number and sum it on both sides of the equation:

[tex]x^2-8x+16=57+16[/tex]

Fourth step. Write the left side of the equation as a squared binomial:

[tex](x-4)^2=73[/tex]

Fifth step. First, take the square root and then add 4 to both sides of the equation to solve for x:

[tex](x-4)=\pm \sqrt{73}[/tex]

[tex]x= 4\pm \sqrt{73}[/tex]

Answer:

274.576 + 8400 = $8674.576

Step-by-step explanation:

Here, Number of days = 21 + 31 +30 +31 +20 = 123

We know that,

[tex]Simple Interest = \frac{P\timesT\timesR}{100}[/tex]

where, P = Principle = 8400

T = time = 123 ÷ 365

R = Rate = 9.2

⇒ [tex]Simple Interest = \frac{8400\times123\times9.2}{365\times100}[/tex]

⇒ Simple Interest = 274.576

Thus, total amount Jane has on 20 August = 274.576 + 8400 = $8674.576

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:

(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

Answer:

Step-by-step explanation:

1. The change in value was 12.6% of $80.74, calculated as ...

  0.126×$80.74 = $10.17324 ≈ $10.17

__

2. The new price is lower than the price yesterday by that amount, so is ...

  $80.74 -10.17 = $70.57

_____

Note on percents

A percent should be no mystery. The word "per cent" literally means "per hundred", or "/100" in symbols. The symbol "%" is a shorthand way to write "/100". So 12.6% means 12.6/100 = 126/1000 = 0.126.

When written as a decimal, the units digit of a percent is placed in the hundredths digit of the decimal number, as you can see in the example above. (The "2" in "12.6" is in the hundredths place in 0.126.)

Generally percentages are used to express ratios. They are usually a "pure number" with no units attached. Since they are a ratio, they are generally useless unless you know what the numbers involved in the ratio are. Here, the denominator of the ratio, the "base" or "reference", is yesterday's stock price. The percentage is described as the drop in price since yesterday, so it is the ratio ...

  (drop in price since yesterday)/(yesterday's price)

Both of these values have units of dollars, so the numerator units cancel the denominator units and what is left is a pure number. The ratio is 0.126, so to express it as a percentage, we multiply it by 100%. Of course, 100% = 100/100 = 1, so we haven't changed the value; we've only changed the way it is presented. That is ...

  0.126 = 0.126 × 100% = 12.6%

_____

Additional note on decimals and percents

You may hear that to convert a decimal to a percent, you multiply it by 100. That is only partly right. Multiplying anything by 100 changes its value by that factor. When you change a decimal to a percentage, the goal is not to change the value, merely the form. What you actually want to do is multiply by 100% = 100/100. In the example we're using here, this looks like ...

  0.126 × 100/100 = 12.6/100 = 12.6% . . . . . . remember that % means /100.

Answer:

The potential energy associated with the given mass equals 1814.98 kilo Joules.

Step-by-step explanation:

We know that for a object of mass 'm' standing at a height of 'h' meters above the surface of earth the potential energy associated with the object is given by

[tex]P.E=mass\times g\times h[/tex]

where

'g' is acceleration due to gravity.

Since it is given that mass of 1 cubic meter of water is 1000 kilograms that the mass associated with given quantity of water is also 1000 kilograms since the volume is 1 cubic meter.

The height is given as 607 feet = [tex]{607}\times 0.3048=185.0136[/tex]meters

Applying the values in the above equation we get

[tex]P.E=1000\times 9.81\times 185.0136=1814.98kJ[/tex]  

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:

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:

  { }

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:

m + 1

Step-by-step explanation:

Given set,

B = {∅, {1}, {1, 2}, {1, 2,3}, · · · , {1, 2, · · · , m}},

Since, the elements of S are,

{} , {1}, {1, 2}, {1, 2, 3}....... {1, 2,.....m }

Thus, every next set contains one more succeeding natural number than the previous set.

So, if the last set contains m natural numbers.

Then there are 'm + 1' sets in B ( m sets included ∅ )

Hence, the number of elements in B is 'm+1'

I.e. Cardinality of B is m + 1.

Answer:

The required form is [tex]y=(x+7)^2-9[/tex].

Step-by-step explanation:

Consider the provided quadratic function.

[tex]y=x^2 + 14x + 40[/tex]

We need to put the equation into the form [tex]y = (x - h )^2 + k[/tex]

Add and subtract 49 in order to make the above function a perfect square.

[tex]y=x^2 + 14x+49-49 + 40[/tex]

[tex]y=x^2 + 14x+7^2-49 + 40[/tex]

[tex]y=(x+7)^2-49 + 40[/tex]

[tex]y=(x+7)^2-9[/tex]

Hence, the required form is [tex]y=(x+7)^2-9[/tex].

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]

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:

$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