1. The volume of a cube is increasing at a rate of 1200 cm/min at the moment when the lengths of the sides are 20cm. How fast are the lengths of the sides increasing at that [10] moment?

Answer:  13%

Step-by-step explanation:

We know that the formula to find the simple interest :

[tex]I=Prt[/tex], where P is the principal amount , r is rate (in decimal )and t is the time period (in years).

Given : Albert borrowed $19,100 for 4 years. The simple interest is $9932.00. Find the rate.

i.e. P = $19,100 and  t= 4 years and I = $9932.00

Now, Substitute all the values in the formula , we get

[tex]9932=(19100)r(4)\\\\\Rightarrow\ r=\dfrac{9932}{19100\times4}\\\\\Rightarrow\ r=0.13\ \ \ \text{[On simplifying]}[/tex]

In percent, [tex]r=0.13\times100\%=13\%[/tex]

Hence, the rate of interest = 13%

Answer:

A 95% confidence interval for the mean number of months elapsed since the last visit to a dentist, is [tex][5.79063, 28.20937][/tex]

Step-by-step explanation:

We will build a 95% confidence interval for the mean number of months elapsed since the last visit to a dentist. A (1 - [tex] \alpha [/tex]) x100% confidence interval for the mean number of months elapsed since the last visit to a dentist with unknown variance and is given by:

[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]

[tex]\bar X = 17[/tex]

[tex]n = 5[/tex]

[tex]\alpha = 0.05[/tex]

[tex]T_{(n-1,\frac{\alpha}{2})}=2.7764[/tex]

[tex]S = 9.0277[/tex]

[tex][17 -2.7764 \frac{9.0277}{\sqrt{5}}, 17 +2.77644 \frac{9.0277}{\sqrt{5}}][/tex]

A 95% confidence interval for the mean number of months elapsed since the last visit to a dentist, is    [tex][5.79063, 28.20937][/tex]

Answer:

(a) 0.01029 (b) 4.167 customers (c) 0.4167 hours or 25 minutes (d) 0.5 hours or 30 minutes

Step-by-step explanation:

With an arrival time of 6 minutes, λ=10 clients/hour.

With a service time of 5 minutes per transaction, μ=12 transactions/hour.

(a) The probability of 3 or fewer customers arriving in one hour is

P(C<=3)=P(1)+P(2)+P(3)

[tex]P(X)=\lambda^{X}*e^{-\lambda}  /X![/tex]

[tex]P(1)=10^{1}*e^{-10}  /1!=0.00045\\P(3)=10^{3}*e^{-10}  /3!=0,00227\\P(2)=10^{2}*e^{-10}  /2!=0,00757\\P(x\leq3)=0.00045+0.00227+0.00757 = 0.01029[/tex]

(b) The average number of customers waiting at any point in time (Lq) can be calculated as

[tex]L_{q}=p*L=\frac{\lambda}{\mu}*\frac{\lambda}{\mu-\lambda}\\L_{q}=\frac{10}{12}*\frac{10}{12-10}\\\\L_{q}=100/24=4.167[/tex]

(c) The average time waiting in the system (Wq) can be calculated as

[tex]W_{q}=p*W=\frac{\lambda}{\mu}*\frac{1}{\mu-\lambda}\\W_{q}=(10/12)*(1/(12-10))\\W_{q}=(10/24)=0.4167[/tex]

(d) The average time in the system (W), waiting and service, can be calculated as

[tex]W=\frac{1}{\mu-\lambda}\\W=\frac{1}{12-10}=0.5[/tex]

Answer:

Yes, a Morpho is an insect.

Step-by-step explanation:

As per the question,

We have been provided the information that

I. All butterflies are insects.

II. A Morpho is a butterfly.

By drawing the Venn-diagram of given conditions, drawn as below.

Now,

We can conclude that a Morpho comes in the circle of butterfly and the circle of butterflies comes under the circle of Insects.

All butterflies = Insects

A morpho = A butterfly

∴ A morpho = An insect

Therefore if a morpho is a butterfly than it is definitely an insect because all butterflies are insects.

Hence, the given statement is true.

Answer:

Angle 1 = 3 . 43.5+6 =136.5°

Step-by-step explanation:

To angles are supplementary when added up, the result is 180°.

So:

Angle 1 + Angle 2 = 180°

We know that angle 1 = 3x+6 and angle 2= x

So we get:

[tex]3x+6+x=180\\4x+6=180\\4x=180-6\\4x=174\\x=174 : 4\\x=43.5[/tex]

If we know now that x=43.5, we can calculate the value of angle 1.

Angle 1 = 3 . 43.5+6 =136.5°

Answer:

[tex]\frac{1}{6}[/tex]

Step-by-step explanation:

y = x     .....(1)

[tex]y=\sqrt{x}[/tex]     .... (2)

By solving equation (1) and equation (2)

[tex]x = \sqrt{x}[/tex]

[tex]\sqrt{x}\left ( \sqrt{x}-1 \right )=0[/tex]

[tex]\sqrt{x}=0[/tex] or [tex]\left ( \sqrt{x}-1 \right )=0[/tex]

x = 0, x = 1

y = 0, y = 1

A = [tex]\int_{0}^{1}ydx(curve)-\int_{0}^{1}ydx(line)[/tex]

A = [tex]\int_{0}^{1}\sqrt{x}dx-\int_{0}^{1}xdx[/tex]

A = [tex]\frac{2}{3}[x^\frac{3}{2}]_{0}^{1}-\frac{1}{2}[x^2]_{0}^{1}[/tex]

A = [tex]\frac{2}{3}-\frac{1}{2}[/tex]

A = [tex]\frac{1}{6}[/tex]

Answer:

The speed of the slower car is: 51 miles/hour and the speed of the faster car is 54 miles/hour.

Step-by-step explanation:

First, we can easily find the speed of the slower car, since we know that the speed is given by the formula v=d/t, where d stands for the traveled distance and t for the elapsed time.

[tex]v_{s}=\frac{d_{s}}{t}=\frac{153\ miles}{3\ hours} = 51\ miles/hour[/tex]

Next, for the faster car, we know that the distance traveled is 153+9 miles in the same time, therefore, its speed is given by:

[tex]v_{f}=\frac{d_{f}}{t}=\frac{153+9\ miles}{3\ hours} = 54\ miles/hour[/tex]

Answer:

The sample size to meet the 95% confidence and a maximum error of 4200 in the salary estimate, is 71

Step-by-step explanation:

You want to estimate the average salary of production managers with more than 15 years of experience, for this you will use a 95% confidence interval with a maximum estimation error of 4200.

[tex]\bar X = 71000\\\\\alpha = 0.05\\\\Z_{\frac{\alpha}{2}}=1.95996\\\\\sigma = 18000\\\\\epsilon=4200[/tex]

The expression for the calculation of the sample size is given by:

[tex]n= (\frac{\sigma Z_{\frac{\alpha}{2}}}{\epsilon})^2=(\frac{18000*1.95996}{4200})^2=70.5571[/tex]

You can use the margin of error formula to get  the needed sample size.

The needed sample size for the given condition is 71

Suppose that we have:

Then the margin of error(MOE) is obtained as

[tex]MOE = Z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}[/tex]

[tex]MOE = Z_{\alpha /2}\dfrac{s}{\sqrt{n}}[/tex]

where [tex]Z_{\alpha/2}[/tex] is critical value of the test statistic at level of significance  [tex]\alpha[/tex]

Since the limit of error is $4200, thus,

MOE = $4200

The level of significance is [tex]\alpha[/tex] = 100 - 95% = 5% = 0.05

Critical value  [tex]Z_{\alpha/2}[/tex] at 0.05 level of significance for two tailed test is 1.96

The standard deviation estimate(sample standard deviation) is s= $18000, thus, we have:

[tex]MOE = Z_{\alpha /2}\dfrac{s}{\sqrt{n}}[/tex]

[tex]4200 = 1.96 \times \dfrac{18000}{\sqrt{n}}\\\\n = (\dfrac{35280}{4200})^2 = (8.4)^2 = 70.56[/tex]

Since MOE is inversely proportional to root of n, thus, we will take sample as 71 which will make MOE to be less than (or say within, as asked in problem) $4200.

Thus,

The needed sample size for the given condition is 71

Learn more about margin or error here:

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Answer:

x = 37

Step-by-step explanation:

2x + 9 = 83

-9          -9

2x = 74

---    ----

2     2

x = 37

Hey!

----------------------------------------------------

Solution:

2x + 9 = 83

~Subtract 9 to both sides

2x + 9 - 9 = 83 - 9

~Simplify

2x = 74

~Divide 2 to both sides

2x/2 = 74

~Simplify

x = 37

----------------------------------------------------

Answer:

x = 37

----------------------------------------------------

Hope This Helped! Good Luck!

Final answer:

Amy could have answered approximately 89.29% correctly, assuming the test had the minimum number of questions allowing for all conditions in the problem to be satisfied.

Explanation:

The question involves determining the greatest possible percent of questions Amy could have answered correctly on a true/false test, given specific details about how Amy and Scott answered their questions.

To find the maximum percentage Amy could have gotten correct, we must assume that the test had the smallest number of questions that fulfill all the given conditions. This would be when the question before the last question, which Scott got wrong, aligns with one of the first three questions that Amy got wrong. Since they were both incorrect only once on the same question, Amy's third question is the same as Scott's second-to-last question. Thus, the test would have a minimum of 28 questions (the 26 Scott got right, plus the two he got wrong).

Amy answered the first three questions wrong and all others correctly, so she got 25 out of 28 questions correct. To find the percentage, we divide 25 by 28 and multiply by 100. Therefore, Amy answered approximately 89.29% of the questions correctly. Amy could not have a higher percentage because having more questions on the test would only lower her score.

Answer:

Step-by-step explanation:

[tex] p \Rightarrow q \equiv (\neg p \vee q) [/tex] [logical equivalence]

[tex] \neg (q \vee r) \equiv (\neg q \wedge \neg r) [/tex] [morgan laws]

if [tex] (\neg q \wedge \neg r) [/tex] is true, then [tex] \neg q [/tex] is true and [tex] \neg r [/tex] is too.

with [tex] \neg q [/tex] true, then [tex] q [/tex] is false [double denial]

In the first equivalence it follows that [tex] \neg p [/tex] is true [identity law]

Then it can be concluded that [tex] \neg p [/tex]

Answer:

A and B

Step-by-step explanation:

It has exactly one solution if the slope is different on both sides. A and B has different slopes on both sides. There will only be one intersection thus only one solution.

If the slope is same on both sides then it either has infinite solutions if the equation can be reduced to true equation (e.g. 1 = 1). Or no solution if it can be reduced to false equation (e.g. 1 = 0)

Answer:

See answer below

Step-by-step explanation:

In inductive reasoning we arrive to a general conclusion based on particular observations.

For example:

"My friends Peter and Mary study at XYZ University. Peter and Mary are brilliant.

Yesterday I met Joe in a party. Joe also studies at XYZ University and he happens to be brilliant, too.

I conclude that all the students of XYZ University are brilliant"

Notice that this kind of reasoning might lead to a false conclusion.

In deductive reasoning, we arrive to a particular conclusion based on general observations. So, deductive reasoning is the opposite of inductive reasoning.

An example of deductive reasoning could be the following:

"To be accepted in the XYZ University you must pass a test with a score greater than 70%.

My friend Peter studies at XYZ University, so he passed the test with a score greater than 70%"

In deductive reasoning you may arrive to a false conclusion if your general assumption is false.

Answer:

but did now am the am

Step-by-step explanation:

Answer:

$34.14

Step-by-step explanation:

We are given $34.13673. We have to round off this amount of money to cents. Now, we know that 100 Cents =  1 Dollar($)

Thus, we need to round of this number to nearest hundredths or rounding of to nearest 0.01 that is the number should have to decimal places after rounding off.

The hundredth place in this number is 3. The next smaller digit to 3 is 6 which is greater than 5, so we simply remove the number ahead and  thus our rounded of number becomes 34.14.

Thus the rounded off number is $34.14

Answer:  b. number of trees

Step-by-step explanation:

The concept of geometric probability is basically use when we have continuous data .

Since it is impossible to count continuous data , but geometrically ( in form of length, area etc) we can count the outcomes in general to calculate the required probability.

Therefore, from the given options , Option b. "number of trees" would not be used for geometric probability because among all it is the only discrete case which is countable.

Rest of items ( a. area of a rug , c. length of time , d. length of a field) would be used for geometric probability,

Answer:

The patient would receive 500mg from the intravenous infusion of a liter of the solution

Step-by-step explanation:

The problem states that an intravenous solution contains 500 μg of a drug substance in each milliliter. And asks how many milligrams of the drug would a patient receive from the intravenous infusion of a liter of the solution.

First step:

Conversion of 500ug to mg.

Each mg has 1,000ug. So

1mg - 1,000ug

xmg - 500ug

1,000x = 500

[tex]x = \frac{500}{1,000}[/tex]

x = 0.5mg

Each milliliter has 0.5mg. The problem asks how many milligrams of the drug would a patient receive from the intravenous infusion of a liter of the solution.

Each liter has 1000 milliliters. So, the problem asks how many miligrams of the drug would a patient receive from the intravenous infusion of 1000 mililiters of the solution. So

1mL - 0.5mg

1,000mL - xg

x = 0.5*1,000

x = 500mg

The patient would receive 500mg from the intravenous infusion of a liter of the solution

The patient receives 500 milligrams of the drug from an intravenous infusion of 1 liter, by converting 500 micrograms to milligrams and multiplying by 1000 milliliters.

To determine how many milligrams (mg) of the drug a patient would receive from an intravenous infusion of 1 liter (1000 milliliters) of the solution, we start with the given information that there are 500 micrograms (μg) of the drug per milliliter (mL) of the solution.

First, convert the amount of drug per milliliter from micrograms to milligrams.

Note that 1 milligram (mg) = 1000 micrograms (μg):

Now, calculate the total amount of drug in 1000 milliliters (since 1 liter = 1000 mL):

Thus, the patient would receive 500 mg of the drug from the intravenous infusion of 1 liter of the solution.

Answer:

[tex] 0.0078[/tex]

Step-by-step explanation:

To compute the probability of a rivet being defective we can do the following:

The seam won't need reworking if the 30 rivets are working as intended. Since there's a 21% chance of the seam needing reworking, we then know that there's a 79% chance of the seam not needing reworking, which means that there's a 79% chance of having the 30 rivets working as intended. Now, each rivet is either defective with a probability p, or NOT defective with a probability 1-p. Since rivets are defective independently from one another, the probability of the 30 rivets working as intended is [tex] (1-p)^{30}[/tex], and since we know this has a chance of happening of 79%, we get the equation:

[tex] 0.79=(1-p)^{30}[/tex]

Solving for p, we get:

[tex] 0.79^{1/30}=1-p[/tex]

[tex] p=1-0.79^{1/30}\approx 0.0078[/tex]

Answer with explanation:

The motion of the bird is shown in the attached figure

part a)

In the ordinate axis as shown in the attached figure we can see that the tree is located at a point [tex](9cos(60),9sin(60)=(4.5km,7.794km)[/tex] with respect to origin O

Part b)

The location of the telephone pole as seen from the co-ordiante axis is at

[tex]x=4.5+20cos(45)=18.64km\\\\y=7.794-20sin(45)=-6.348km[/tex]

The points are located as shown in the figure

Answer:

Option B is right

Step-by-step explanation:

[tex]H_0: x bar =2\\H_a: x bar <2[/tex]

(One tailed test at 5% significance level)

n =16 and x bar =1.97

s =0.1

Std error of mean = [tex]\frac{s}{\sqrt{n} } \\=0.025[/tex]

Mean diff = [tex]1.97-2 =-0.03[/tex]

t statistic =Mean diff/se =-1.2

df =16-1=15

p value =0.124375

(B) do not reject the null because the test statistic (-1.2) is > the critical value (-1.7531).

The null hypothesis should not be rejected.

The question is asking whether to reject or not reject the null hypothesis based on the given information. The null hypothesis states that the house cleaning service can clean a four bedroom house in less than 2 hours. To test this hypothesis, a sample of 16 houses was taken, with a sample mean of 1.97 hours and a sample standard deviation of 0.1 hours. Using a significance level of 0.05, we compare the test statistic to the critical value to make the decision.

The correct conclusion in this case is to do not reject the null hypothesis, because the test statistic of -1.2 is greater than the critical value of -1.7531. Therefore, there is insufficient evidence to conclude that the cleaning service takes more than 2 hours to clean a four bedroom house.

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Answer with Step-by-step explanation:

Let S be  a semi-group.

Semi group: It is that set of elements which satisfied closed property and associative property.

We have to prove that S is  a group if and only if the following condition hold'

1.S has  left identity -There exist an element [tex]e\in S[/tex] such that ea=a for all [tex]a\in S[/tex]

2.Each element of S has a left inverse - for each [tex]a\inS[/tex], there exist an element [tex]a^{-1}\in S[/tex] such that [tex]a^{-1}a=e[/tex]

1.If there exist an element [tex]e\in S[/tex] such that ea=a for all [tex]a\in S[/tex]

Then , identity exist in S.

2.If there exist left inverse for each [tex]a\inS[/tex]

Then, inverse  exist for every element in S.

All properties of group satisfied .Hence, S is a group.

Conversely, if S is a group

Then, identity exist and inverse of every element exist in S.

Identity exist: ea=ae=a

Inverse exist:[tex]aa^{-1}=a^{-1}a=e[/tex]

Hence, S has  a left identity for every element [tex]a\in S[/tex]

and each element of S has a left  inverse .

Hence, proved.

I'm pretty sure b is your answer

Final answer:

The false statement is (A), which claims that the intersection of connected sets is connected. This is not always true, as connected sets can intersect in a way that creates disjoint, non-connected subsets. So the correct option is A.

Explanation:

The student is asked to identify which statement is false among those given related to the properties of connected and compact sets in the context of mathematical analysis. In these options, (A) claims that the intersection of connected sets is connected, (B) states that the union of two connected sets with a non-empty intersection is connected, (C) states that the intersection of any number of compact sets is compact, (D) asserts that the continuous image of a compact set is compact, and (E) maintains that the continuous image of a connected set is connected.

The false statement among those provided is (A). It is not true that the intersection of connected sets is necessarily connected. For example, consider two overlapping connected sets such that their intersection is not connected, like two overlapping annuli where the intersection creates separate disjoint areas. However, options (B), (C), (D), and (E) are correct under the definitions of connected and compact sets in topology.

Answer:

11.196%

Step-by-step explanation:

Given:

Buying cost of the investment or the principle amount = $1000

Time, n = 5 years

Selling cost of investment or amount received = $1700

Now,

the formula for compound interest is given as:

[tex]\textup{Amount}=\textup{Principle}(1+r)^n[/tex]

here, r is the rate of interest

on substituting the respective values, we get

[tex]\textup{1700}=\textup{1000}(1+r)^5[/tex]

or

(1 + r)⁵ = 1.7

or

1 + r = 1.11196

or

r = 0.11196

or

r = 0.11196 × 100% = 11.196%

Answer:

a) There is an 8% probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

b) There is a 42% probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby.

Step-by-step explanation:

In this problem, we have these following probabilities:

-A 40% probability that a world class tennis player is over 6 feet tall.

-A 60% probability that a world class tennis player is not over 6 feet tall.

-A 20% probability that a world class tennis player over 6 feet tall was a premature baby. This also means that there is a 80% probability that a world class tennis player over 6 feet tall was not a premature baby

-A 30% probability that a world class tennis player who is not over 6 feet tall was a premature baby. This also means that there is a 70% probability that a world class tennis player who is not over 6 feet tall was not a premature baby.

a. Calculate the probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

[tex]P = P_{1}*P_{2}[/tex]

[tex]P_{1}[/tex] is the probability that a random selected world class tennis player is over 6 feet tall. So

[tex]P_{1} = 0.4[/tex]

[tex]P_{2}[/tex] is the probability that a world class tennis player over 6 feet tall was a premature baby. So

[tex]P_{2} = 0.2[/tex]

[tex]P = P_{1}*P_{2} = 0.4*0.2 = 0.08[/tex]

There is an 8% probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

b. Calculate the probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby

[tex]P = P_{1}*P_{2}[/tex]

[tex]P_{1}[/tex] is the probability that a random selected world class tennis player is not over 6 feet tall. So

[tex]P_{1} = 0.6[/tex]

[tex]P_{2}[/tex] is the probability that a world class tennis player under 6 feet tall was not a premature baby. So

[tex]P_{2} = 0.7[/tex]

[tex]P = P_{1}*P_{2} = 0.6*0.7 = 0.42[/tex]

There is a 42% probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby.

Answer:

Te correct answer is c) 0.750

Step-by-step explanation:

Lets call:

A = {Allan wins the election}

B = {Barnes wins the election}

MA = {the model predicts that Allan wins}

MB = {the model predicts Barnes wins}

We know that the model has a 50:50 chance of correctly predicting the election winner when there are two candidates. Then:

P(MA | A) = 0.5 = P(MA | B)

P(MB | B) = 0.5 = P(MB | A)

The prior probability P(A) given by the election researcher is 0.75

We must find the posterior probability P(A | MB)

We use Bayes theorem:

[tex]P(A|MB) = \frac{P(MB|A)P(A)}{P(MB)} = \frac{0.5*0.75}{0.5} = 0.75[/tex]

We used the result:

[tex]P(MB) = P(MB|A)P(A) + P(MB|B)P(B) = 0.5*0.75+0.5*0.25=0.5[/tex]

Answer:  [tex]\bold{y=-\dfrac{3}{2}x+\dfrac{15}{2}}[/tex]

Step-by-step explanation:

First, use the slope formula: [tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

Then input the slope (m) and ONE of the points (x₁, y₁) into the

Point-Slope formula: y - y₁ = m(x - x₁)

[tex]m=\dfrac{3-6}{3-1}\quad =\dfrac{-3}{2}\quad \implies \quad m=-\dfrac{3}{2}\\\\\\\\y-3=-\dfrac{3}{2}(x-3)\\\\\\y-3=-\dfrac{3}{2}x+\dfrac{9}{2}\\\\\\y\quad =-\dfrac{3}{2}x+\dfrac{15}{2}[/tex]

Answer:

The amount is $23200.08.

The interest is $6300.08.

Step-by-step explanation:

To find : Complete the following using compound future value ?

Solution :

The compound future value formula is given by,

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]

Where, A is the amount (future value)

P is the principal (present value) P=$16,900

r is the annual interest rate r=8%=0.08

n is the number of compounding periods per year n=4

t is the time in years t=4

Substitute the value in the formula,

[tex]A=16900(1+\frac{0.08}{4})^{4\times 4}[/tex]

[tex]A=16900(1+0.02)^{16}[/tex]

[tex]A=16900(1.02)^{16}[/tex]

[tex]A=16900\times 1.37278570509[/tex]

[tex]A=23200.078416[/tex]

Round to nearest cent 23200.078416=23200.08.

The amount is $23200.08.

The interest formula is

[tex]I=A-P[/tex]

Substitute the values,

[tex]I=23200.08-16900[/tex]

[tex]I=6300.08[/tex]

The interest is $6300.08.

To find the future value with compound interest for a principal of $16,900 at an 8% annual rate compounded quarterly over 4 years, use the compound interest formula. The future value minus the original principal gives the total compound interest, rounding the final values to the nearest cent.

To calculate the future value with compound interest, use the formula:

Future Value = Principal × (1 + interest rate/n)n × t

Where:

For our scenario:

Principal = $16,900

Rate = 8% or 0.08 annually

Compounded Quarterly (4 times per year)

Time = 4 years

Future Value = $16,900 × (1 + 0.08/4)4 × 4

Calculate the amount within the parentheses and the exponent, and then compute the future value.

Once we have the future value, we can find the compound interest by subtracting the Principal from the Future Value:

Compound interest = Future Value - Principal

After doing the calculations, we can round the results to the nearest cent as requested for the answer.

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Answer: Weight(lbf) = 4830

Step-by-step explanation:

Given :

Mass of the roommate = 150 lbm

Since we know that g = [tex]32.2 feet/s^{2}[/tex]

Therefore, we can calculate his/her weight in lbf using the following formula:

Weight(lbf) = Mass × g

Weight(lbf) = 150 × 32.2

Weight(lbf) = 4830

Answer:

a) [tex]10^{14}[/tex]

b) 0.044

Step-by-step explanation:

Part a)

A Sonet is a 14-line poem. Raymond Queneau published a book containing just 10 sonnets, each on different pages. This means, on each page the writer wrote a 14-line poem. We have to find how many sonnets can be created from the 10 sonnets in the book.

Since, the first line of the new sonnet can be the first line of any of the 10 sonnets. So, there are 10 ways to select the first line. Similarly, the second line of the new sonnet can be the second line of any of the 10 sonnets. So, there are 10 ways to select the second line. Same goes for all the 14 lines i.e. there are 10 ways to select each of the line.

According to the fundamental principle of counting, the total number of possible sonnets would be the product of all the possibilities of all 14 lines.

So,

The number of sonnets that can be created from the book = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = [tex]10^{14}[/tex] sonnets

So, [tex]10^{14}[/tex] can be created from the 10 in the book.

Part b)

First we need to find how many such sonnets can be created that have none of their lines from 1st and last page. Since, total number of pages is 10, if we are NOT to select from 1st and last page, this will leave us with 8 pages (8 sonnets).

So, now the number of possible options for each line of the sonnet would be 8. And according to the  fundamental principle of counting, the number of sonnets with neither the line from 1st nor the last page would be = [tex]8^{14}[/tex]

This represents the number of favorable outcomes, as we want the randomly selected sonnet to be such that none of its lines came from either the first or the last sonnet in the book.

So,

Number of favorable outcomes = [tex]8^{14}[/tex]

Total possible outcomes of the event = [tex]10^{14}[/tex]

As the probability is defined as the ratio of favorable outcomes to the total outcomes, we can write:

The probability that none of its lines came from either the first or the last sonnet in the book = [tex]\frac{8^{14}}{10^{14}} = 0.044[/tex]

You can use the product rule from combinatorics to calculate the number of ways sonnets can be created.

The answers are:

a) The number of sonnets that can be created are [tex]10^{14}[/tex]

b) The probability needed is 0.04398

If a work A can be done in p ways, and another work B can be done in q ways, then both A and B can be done in  [tex]p \times q[/tex] ways.

Remember that this count doesn't differentiate between order of doing A first or B first then doing other work after the first work.

Thus, doing A then B is considered same as doing B then A

Now, each of the 14 lines of the sonnet we're going to create can be chosen from 10 of the sonnets available in book, thus, by using rule of product for these 14 events, each possible to be done in 10 ways, the total number of ways comes to be

[tex]10 \times 10 \times ... \times 10\: \: (14\rm \: times) = 10^{14}[/tex]sonnets.

Thus,

a) The number of sonnets that can be created are [tex]10^{14}[/tex]

Now, the number of ways we can select sonnet's lines such that it doesn't contain its lines from first or last sonnet can be calculated just as previous case but now there are only 8 options available for each line (as 2 of them are restricted from using).

Thus,

[tex]8 \times 8 \times ... \times 8\: \: (14\rm \: times) = 8^{14}[/tex]sonnets possible who doesn't contain any lines from first and last sonnet.

Let E be the event such that

E = Event of selecting sonnets which doesn't contain any line from first or last sonnet

Then,

[tex]P(E) = \dfrac{\text{Count of favorable cases}}{\text{Count of total cases}} = \dfrac{8^{14}}{10^{14}} = (0.8)^{14} \approx 0.04398[/tex]

Thus,

the probability that none of its lines came from either the first or the last sonnet in the book is 0.04398 approx.

Thus,

The answers are:

a) The number of sonnets that can be created are [tex]10^{14}[/tex]

b) The probability needed is 0.04398

Learn more about rule of product of combinatorics here:

https://brainly.com/question/2763785

Answer:

Lots of connections!

Step-by-step explanation:

I attached some images to make it more clear.

Visual and verbal discussion:

A good starting point is the circle. One can think of a circle as the set of points that are equidistant to a certain point. In that sense, one can define a circle using the distance. At the same time, given a point [tex](x_{0},y_{0})[/tex] in the plane, we can connect the point with the origin of the coordinates system forming a rectangle triangle! (See 2nd image)

Algebraic discussion:

1. The distance Formula:

Given two points in the plane [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] we can find the distance between both points with the distance formula:

[tex]d = \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}[/tex]

2. Equation of a circle centered at a point [tex](x_{0},y_{0})[/tex]

[tex]r = \sqrt{(x-x{_0}^2) + (y-y_{0})^2}[/tex]

3. The Pythagorean Theorem:

Given a triangle of sides [tex]a;b[/tex] and hypotenuse [tex]h[/tex] we have that

[tex]h^2 = a^2 + b^2[/tex]

Only by writing this equations we can already see the similarities between all three.

In fact, the most amazing thing about all three is that they are equivalents. That is, we can obtain every single one of them as an immediate result of another.

For instance, as I said before, one can think of a circle as the set of points equidistant to a certain origin or center point. So we could use the distance formula for each point on the circumference and we would obtain always the same value [tex]r[/tex] hence obtaining the equation of a circle.

Also as we discussed, any point on the plane form a rectangle triangle with its coordinates and calculating the distance of said point to the origin of the coordinate system would give us no other thing than the hypotenuse of said triangle!