Solve the given differential equation by undetermined coefficients.


y''' − 3y'' + 3y' − y = ex − x + 21

Answer:

The value of the 3 is 30,000,000.

Step-by-step explanation:

From the digit at the right, you go multiplying each element by 10 powered to a counter that starts at zero and increases at every digit. So:

Our counter is i

i = 0;

v(7) is the value of the 7

[tex]v(7) = 7*10^{0} = 7[/tex]

i = 1;

v(5) is the value of the 5

[tex]v(5) = 5*10^{1} = 50[/tex]

i = 2;

v(1) is the value of the 1

[tex]v(1) = 1*10^{2} = 100[/tex]

i = 3;

v(5) is the value of the 5

[tex]v(5) = 5*10^{3} = 5,000[/tex]

i = 4;

v(9) is the value of the 9

[tex]v(9) = 9*10^{4} = 90,000[/tex]

i = 5;

v(6) is the value of the 6

[tex]v(6) = 6*10^{5} = 600,000[/tex]

i = 6;

v(8) is the value of the 8

[tex]v(8) = 8*10^{6} = 8,000,000[/tex]

i = 7;

v(3) is the value of the 3

[tex]v(3) = 3*10^{7} = 30,000,000[/tex]

The value of the 3 is 30,000,000.

The digit 3 in the number 138,695,157 represents a value of 300 million, indicating its substantial contribution to the overall magnitude of the figure in the context of place value and powers of 10.

The value of the digit 3 in the number 138,695,157 is 3 hundred million. In this number, each place value represents a power of 10, with the rightmost digit being ones, the next one being tens, the next hundreds, and so on. The digit 3 in the hundred million's place means that it represents 3 multiplied by 100,000,000.

In other words, the digit 3 in this context signifies 300 million. This is because when you see a digit in a number, its place value determines its weight in terms of powers of 10. So, the digit 3 in the hundred million's place is equivalent to 3 x 100,000,000, which is indeed 300 million.

So, in the number 138,695,157, the digit 3 holds the value of 300 million, contributing significantly to the overall magnitude of the figure.

Learn more about place value here:

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Answer:

The 105th term of given sequence is [tex]\frac{105}{2}[/tex].

Step-by-step explanation:

The given sequence is

[tex]\frac{1}{2},1,\frac{3}{2},2[/tex]

It can be rewritten as

[tex]0.5,1,1.5,2[/tex]

Here the first term is 0.5.

It is an arithmetic​ sequence because it has common difference.

[tex]d=a_2-a_1=1-0.5=0.5[/tex]

[tex]d=a_3-a_2=1.5-1=0.5[/tex]

[tex]d=a_4-a_3=2-1.5=0.5[/tex]

The nth term of an AP is

[tex]a_n=a_1+(n-1)d[/tex]

where, [tex]a_1[/tex] is first term and d is common difference.

Substitute [tex]a_1=0.5[/tex] and [tex]d=0.5[/tex] in the above formula.

[tex]a_n=0.5+(n-1)0.5[/tex]

[tex]a_n=0.5+0.5n-0.5[/tex]

[tex]a_n=0.5n[/tex]

We need to find the 105th term of given sequence.

Substitute n=105 in the above equation.

[tex]a_n=0.5(105)[/tex]

[tex]a_n=52.5[/tex]

[tex]a_n=\frac{105}{2}[/tex]

Therefore the 105th term of given sequence is [tex]\frac{105}{2}[/tex].

Answer:

96%

Step-by-step explanation:

Conditional probability is defined as:  

P(A|B) = P(A∩B) / P(B)  

Or, in English:  

Probability that A occurs, given that B has occurred = Probability that both A and B occur / Probability that B occurs

We want to find the probability that a woman uses an ATM to get cash, given that she uses an ATM to check her balance.

P(withdraws cash | checks account)

Using the definition of condition probability, this equals:

P = P(withdraws cash AND checks account) / P(checks account)

We know that P(checks account) = 0.28.

But we don't know what P(withdraws cash AND checks account) is.  To find that, we need to use the definition of P(A∪B):

P(A∪B) = P(A) + P(B) − P(A∩B)

This says that the probability of A or B occurring (or both) is the probability of A occurring plus the probability of B occurring minus the probability of both A and B occurring.

P(withdraws cash OR checks account) = P(withdraws cash) + P(checks account) − P(withdraws cash AND checks account)

0.95 = 0.94 + 0.28 − P(withdraws cash AND checks account)

P(withdraws cash AND checks account) = 0.27

Therefore:

P = 0.27 / 0.28

P ≈ 0.96

Final answer:

The probability that a woman who checks her account balance at an ATM also withdraws cash is approximately 96.43%.

Explanation:

To solve the problem, we can apply the probability rule for conditional probability. We are provided with the following probabilities:

Using this information, we're interested in finding the probability that a user who checks their account balance also withdraws cash, represented as P(Cash|Balance).

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

Where A and B are two events, and P(A|B) is the conditional probability of A given B.

Using the inclusion-exclusion principle, we can express P(Cash ∩ Balance) as:

P(Cash ∩ Balance) = P(Cash) + P(Balance) - P(Cash ∪ Balance)

Substitute the given probabilities:

P(Cash ∩ Balance) = 0.94 + 0.28 - 0.95 = 0.27

The probability that a woman who checks her balance also gets cash (P(Cash|Balance)) is:

P(Cash|Balance) = P(Cash ∩ Balance) / P(Balance)

P(Cash|Balance) = 0.27 / 0.28 ≈ 0.9643

Therefore, the probability is approximately 96.43%.

Answer:

[tex]\frac{\pi}{43082}\text{ radians per second}[/tex]

Step-by-step explanation:

Given,

Time taken in one rotation of earth = 23 hours, 56 minutes and 4 seconds.

Since, 1 minute = 60 seconds and 1 hour = 3600 seconds,

⇒ Time taken in one rotation of earth = (23 × 3600 + 56 × 60 + 4) seconds

= 86164  seconds,

Now,  the number of radians in one rotation = 2π,

That is, 86164 seconds = 2π radians

[tex]\implies 1\text{ second }=\frac{2\pi}{86164}=\frac{\pi}{43082}\text{ radians}[/tex]

Hence, the number of radians in one second is [tex]\frac{\pi}{43082}[/tex]

The Earth completes a 2π radian rotation about its axis in 23 hours, 56 minutes, and 4 seconds. After converting this time to 86,164 seconds, the number of radians the Earth rotates in one second can be calculated by dividing 2π by 86,164, giving a result of approximately 0.00007292115 radians.

The Earth completes one full rotation about its axis in 23 hours, 56 minutes and 4 seconds. This rotation can be converted into radians, using the principle that one complete rotation is equivalent to 2π radians. So first, convert the rotation time into seconds: (23 x 60 x 60) + (56 x 60) + 4 = 86,164 seconds. Therefore, the Earth rotates through 2π radians in this time.

Now, we want to find out how many radians the Earth rotates in one second. To calculate this, divide 2π (which represent a full rotation in radians), by the total number of seconds in one rotation: 2π/86,164. This will give you approximately 0.00007292115 radians, which is the angular velocity or the number of radians the Earth rotates in one second.

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Step-by-step explanation:

Proof for i)

We will prove by mathematical induction that, for every natural [tex]n\geq 4[/tex], the number of diagonals of a convex polygon with n vertices is [tex]\frac{n(n-3)}{2}[/tex].

In this proof we will use the expression d(n) to denote the number of diagonals of a convex polygon with n vertices

Base case:

First, observe that:, for n=4, the number of diagonals is

[tex]2=\frac{n(n-3)}{2}[/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, given a convex polygon with n vertices, wich we will denote by P(n), if we add a new vertix (transforming P(n) into a convex polygon with n+1 vertices, wich we will denote by P(n+1)) we have that:

Because of these observation we know that, for every [tex]n\geq 4[/tex],

[tex]d(n+1)=d(n)+1+(n-2)=d(n)+n-1[/tex]

Therefore:

[tex]d(n+1)=d(n)+n-1=\frac{n(n-3)}{2}+n-1=\frac{n^2-3n+2n-2}{2}=\frac{n^2-n-2}{2}=\frac{(n+1)(n-2)}{2}[/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Proof for ii)

Observe that:

Then, the statement is not true for n=1,2,3.

We will prove by mathematical induction that, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex].

Base case:

For n=4, [tex]2n=8<24=n![/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex], [tex]2n<n![/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that,

[tex]n!+2\leq (n+1)! \iff n!+2\leq n!(n+1) \iff 1+\frac{2}{n!}\leq n+1 \iff 2\leq n*n![/tex]

wich is true as we are assuming [tex]n\geq 4[/tex]. Therefore:

[tex]2(n+1)=2n+2<n!+2\leq (n+1)![/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex]

Answer: [tex](0.0628,\ 0.1186)[/tex]

Step-by-step explanation:

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Critical value : [tex]z_{\alpha/2}=\pm1.96[/tex]

Sample size : n= 408

Proportion of new websites registered on the Internet were anonymous :

[tex]\hat{p}=\dfrac{37}{408}\approx0.0907[/tex]

The formula to find the confidence interval for population proportion is given by :-

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

i.e. [tex]0.0907\pm (1.96)\sqrt{\dfrac{0.0907(1-0.0907)}{408}}[/tex]

[tex]=0.0907\pm0.0278665515649\\\\\approx 0.0907\pm0.0279\\\\=(0.0907-0.0279,\ 0.0907+0.0279)\\\\=(0.0628,\ 0.1186)[/tex]

Hence,  the 95 percent confidence interval for the proportion of all new websites that were anonymous = [tex](0.0628,\ 0.1186)[/tex]

Final answer:

To find the percent of the original price you end up paying after a 25% discount and an additional 40% discount, first calculate the discounted prices and then determine the final price. In this case, you end up paying 45% of the original price.

Explanation:

To find the percent of the original price you end up paying, you need to calculate the final price after both discounts. Let's say the original price of the item is $100. First, apply the 25% discount by multiplying the original price by 0.75 (1 - 0.25 = 0.75). This gives you a price of $75. Next, apply the additional 40% discount by multiplying the discounted price by 0.60 (1 - 0.40 = 0.60). This gives you a final price of $45. Therefore, you end up paying 45% of the original price.

Answer:

Cost function C(x) == FC + VC*Q

Revenue function R(x) = Px * Q

Profit function P(x) =(Px * Q)-(FC + VC*Q)

P(12000) = -38000 Loss

P(23000) = 28000 profit

Step-by-step explanation:

Total Cost is Fixed cost plus Variable cost multiplied by the produce quantity.  

(a)Cost function

C(x) = FC + vc*Q

Where  

FC=Fixed cost

VC=Variable cost

Q=produce quantity

(b)

Revenue function

R(x) = Px * Q

Where  

Px= Sales Price

Q=produce quantity

(c) Profit function

Profit = Revenue- Total cost

P(x) =(Px * Q)-(FC + vc*Q)

(d) We have to replace in the profit function

at 12,000 units

P(12000) =($20 * 12,000)-($110,000 + $14*12,000)

P(12000) = -38000

at 23,000 units

P(x) =($20 * 23,000)-($110,000 + $14*23,000)

P(23000) = 28000

Answer:

The statement is true

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

We will prove our base case, when n=1, to be true.

base case:

[tex]\sum^{1}_{i=1}\frac{1}{(2-1)(2+1)} =\frac{1}{3}=\frac{n}{2n+1}[/tex]

Inductive hypothesis:

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Now, we will assume the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.

Inductive step:

[tex]\sum^{n+1}_{i=1}\frac{1}{(2i-1)(2i+1)} =\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)}+\frac{1}{(2(n+1)-1)(2(n+1)+1)}=\frac{n}{2n+1}+\frac{1}{(2n+1)(2n+3)}=\frac{n(2n+3)+1}{(2n+1)(2n+3)}=\frac{2n^2+3n+1}{(2n+1)(2n+3)}=\frac{(2n+1)(n+1)}{(2n+1)(2n+3)}=\frac{n+1}{2n+3}=\frac{n+1}{2(n+1)+1}[/tex]

With this we have proved our statement to be true for n+1.

In conlusion, for every natural [tex]n[/tex].

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Answer:   99 minutes

Step-by-step explanation:

Given: While completing a race, Edward spent 54 minutes walking.

The ratio of time walking to jogging was 6:5 i.e. [tex]\dfrac{6}{5}[/tex]     (1)

Let x be the time taken ( in minutes ) by him for jogging.

then, the ratio of time walking to jogging will be [tex]\dfrac{54}{x}[/tex]  (2)

From (1) and (2), we have

[tex]\dfrac{6}{5}=\dfrac{54}{x}\\\\\Rightarrow\ 6x=54\times5\\\\\Rightarrow\ x=\dfrac{54\times5}{6}=45[/tex]  

So, the number of minutes he took for jogging = 45 minutes

Now, the total time he spent on completing the race= 54+45=99 minutes

Answer:

0.2 kW-hr

Step-by-step explanation:

First, we are going to transform 100W in kW. We can use a rule of three in which we know that 1000W is equivalent to 1 kW, then how many kW are equivalent to 100W. This is:

1000W ------------- 1 kW

100W -------------- X

Where X is the the number of kW that are equivalent to 100W. Solving for X, we get:

[tex]X=\frac{100W * 1kW}{1000W} =0.1kW[/tex]

Then, for calculate the number of kW-hr we need to multiplicate the number of kW with the number of hours, This is:

0.1 kW * 2 hours = 0.2 kW-hr

Finally, when one 100W light bulbs is used for 2 hours, it used 0.2 kW-hr

Answer:

(A) [tex]y=ke^{2t}[/tex] with [tex]k\in\mathbb{R}[/tex].

(B) [tex]y=ke^{2t}/2-1/2[/tex] with [tex]k\in\mathbb{R}[/tex]

(C) [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex]

(D) [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex],

Step-by-step explanation

(A) We can see this as separation of variables or just a linear ODE of first grade, then [tex]0=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y \Rightarrow  \frac{1}{2y}dy=dt \ \Rightarrow \int \frac{1}{2y}dy=\int dt \Rightarrow \ln |y|^{1/2}=t+C \Rightarrow |y|^{1/2}=e^{\ln |y|^{1/2}}=e^{t+C}=e^{C}e^t} \Rightarrow y=ke^{2t}[/tex]. With this answer we see that the set of solutions of the ODE form a vector space over, where vectors are of the form [tex]e^{2t}[/tex] with [tex]t[/tex] real.

(B) Proceeding and the previous item, we obtain [tex]1=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y+1 \Rightarrow  \frac{1}{2y+1}dy=dt \ \Rightarrow \int \frac{1}{2y+1}dy=\int dt \Rightarrow 1/2\ln |2y+1|=t+C \Rightarrow |2y+1|^{1/2}=e^{\ln |2y+1|^{1/2}}=e^{t+C}=e^{C}e^t \Rightarrow y=ke^{2t}/2-1/2[/tex]. Which is not a vector space with the usual operations (this is because [tex]-1/2[/tex]), in other words, if you sum two solutions you don't obtain a solution.

(C) This is a linear ODE of second grade, then if we set [tex]y=e^{mt} \Rightarrow y''=m^2e^{mt}[/tex] and we obtain the characteristic equation [tex]0=y''-4y=m^2e^{mt}-4e^{mt}=(m^2-4)e^{mt}\Rightarrow m^{2}-4=0\Rightarrow m=\pm 2[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in the first items the set of solutions form a vector space.

(D) Using C, let be [tex]y=me^{3t} [/tex] we obtain that it must satisfies [tex]3^2m-4m=1\Rightarrow m=1/5[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in (B) the set of solutions does not form a vector space (same reason! as in (B)).  

By considering the given information we have ,

[tex]H_0: \mu = 10\\\\ H_a: \mu\neq10[/tex]

Since, the alternative hypothesis is two tailed so the test is a two-tailed test.

Given : Population mean : [tex]\mu=10[/tex]

Standard deviation: [tex]\sigma= 3[/tex]

Sample size : n=25 , whihc is less than 30 so the sample is small and we use t-test.

Sample mean : [tex]\overline{x}=11.3[/tex]

Significance level : [tex]\alpha= 0.5[/tex]

Formula to find t-test statistic is given by :-

[tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

i.e. [tex]t=\dfrac{11.3-10}{\dfrac{3}{\sqrt{25}}}\approx2.17[/tex]

By using the standard normal distribution table,

The p-value corresponds 2.17 (two-tailed)=0.0300068

Since , the p-value is less than the significance level, so we reject the null hypothesis.

Hence, we conclude that there are enough evidence to to support the claim that movie rental rates in Yuma, Arizona, were different from those of the country as a whole .

A hypothesis test is conducted to see if the average movie rental rate in Yuma, Arizona, is statistically different from the national average. This problem is resolved in several steps including stating hypotheses, formulating an analysis plan, analyzing sample data, and interpreting the results. The rental rate is then compared to a critical value determined by the significance level (α = .05).

The subject here is a hypothesis testing problem related to the mean rental rate of DVDs in Yuma, Arizona. Blockbuster found that the average nation-wide movie rental rate was 10 movies per year with a standard deviation of 3. In Yuma, a sample of 25 people resulted in a mean rental rate of 11.3 movies per year. The company wanted to check whether this difference was significant or not. So, they used an alpha level of .05 to conduct this hypothesis test.

Here are the steps of the hypothesis test:

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Answer: 433

Step-by-step explanation:

The given sequence : 19​,42​,65​,88​,...

Here we can see that the difference in each of the two consecutive terms is 23.  [88-65=23, 65-42=23, 42-19=23]

i.e. it has a common difference of 23.

Therefore, it is an arithmetic sequence .

In an arithmetic​ sequence, the nth term an is given by the formula[tex]A_n=a_1+(n-1)d[/tex] , where [tex]a_1[/tex] is the first term and d is the common difference.​

For the given sequence , [tex]a_1=19[/tex]  and [tex]d=23[/tex]

Then,  to find the 19th term of  the sequence, we put n= 19 in the above formula:-

[tex]A_{19}=19+(19-1)(23)=19+(18)(23)=19+414+433[/tex]

Hence, the 19th term of  the sequence = 433

Final answer:

To find the 19th term of the arithmetic sequence 19, 42, 65, 88, ..., the common difference (23) is determined from the sequence and applied in the arithmetic sequence formula. Substituting the values into the formula, the 19th term is calculated to be 433.

Explanation:

To find the 19th term, we must first determine the common difference, d, of the sequence. Observing the given sequence, we see that the difference between consecutive terms is 42 - 19 = 23. Therefore, the common difference is 23.

Next, we apply the formula for the nth term of an arithmetic sequence which is An = a1 + (n-1)d. Here, a1 is the first term, n is the term number, and d is the common difference.

Substituting the values for the 19th term, we have: A19 = 19 + (19-1) × 23 = 19 + 18 × 23 = 19 + 414 = 433. Therefore, the 19th term of the sequence is 433.

Answer:

(x,y,z)=(2,0,-4)

Step-by-step explanation:

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\2&-3&2&-4\\4&-1&3&-4\end{array}\right][/tex]

Using the elementary operations

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\0&-5&5&-20\\0&-5&9&-36\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&4&-16\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

The answer is determined:

x=2

y=0

z=-4

You can check they are correct, by entering in the original formulas.

Answer:

2 500 000 rad.

Step-by-step explanation:

Mega is the metric prefix for [tex]10^{6}[/tex], therefore you just need to multiply by 1 000 000 to find the value in rads.

Answer:

θ = -33.69°

Step-by-step explanation:

For Φ>0 and Φ<0  (in general Φ≠nπ  where n is an integer), sin(Φ) ≠ 0

Dividing both equations:

[tex]\frac{sin(\phi) sin(\theta)}{sin(\phi)cos(\theta)} = tan(\theta) = 0.2/(-0.3)=-2/3\\[/tex]

Therefore:

arctan(θ) = -2/3

  θ = -33.69°

The answer does not depend on the sign of Φ, in fact we just need that the sine does not become zero, which occurs when Φ is equal to an integer times π (radians) or 180 (degrees)

Have a nice day!

To find theta (θ) given that sin phi (φ) sin theta (θ) = 0.2 and sin phi (φ) cos theta (θ) = -0.3 with sin phi (φ) being positive or negative, one must first eliminate sin phi (φ) by manipulating the given equations, then solve for theta (θ) using trigonometric identities and inverse functions based on the signs of sin and cos.

We have two equations involving sin φ and θ (theta): sin φ sin θ = 0.2 and sin φ cos θ = -0.3. Also, it is given that sin φ > 0 or sin φ < 0. To find θ, first, we need to derive an equation involving only θ by eliminating sin φ. We can do this by squaring and adding both equations.

∑: (sin φ sin θ)^2 + (sin φ cos θ)^2 = 0.2^2 + (-0.3)^2 = 0.04 + 0.09 = 0.13

Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, we can rewrite ∑ as sin^2 φ = 0.13. To solve for θ, we can take either of the initial equations, say sin φ sin θ = 0.2, and substitute sin^2 φ from ∑ giving sin θ = (0.2 / √0.13) or cos θ = (-0.3 / √0.13). Both positive and negative values of sin φ lead to the calculation for different θ values. The actual values of θ are determined by using the arc functions (arcsin, arccos) for both positive and negative scenarios of sin φ, taking into account the range of θ based on the signs of sin and cos.

Answer:

Step-by-step explanation:

Given are the observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added:

2767 2924 3042 2844 2895

No of items = 5

If written in ascending order the order would be

2767   2844   2895   2924   3042

Hence median is the middle value in the ordered row = 2895

Mean = sum/5

=[tex]\frac{14472}{5} =2894.4[/tex]

Answer:

P(A)=1/7

P(B)=1/2

P(C)=1/42

P(A|C)=1/10

P(B|C)=1/10

P(A|B)=1/7

A and B are independent

A and C aren't independent

B and C aren't independent

Step-by-step explanation:

A="b falls in the middle"

- b can fall in seven possible places, but only one is the middle. So, P(A)=1/7

B="c falls to the right of b"

X=i means "b falls in the i-th position"

Y=j means "c falls in the j-th position"

if b falls in the first place, c can fall in the 2nd, 3rd, 4th, 5th, 6th or 7th place.

if b falls in the 2nd place, c can fall in the 3rd, 4th, 5th, 6th or 7th place

 ...

If b falls in the 6th place, c can fall in the 7th place

then:

[tex][tex]P(B)=\displaystyle\sum_{i=1}^{6}( P(X=i)\displaystyle\sum_{j=i+1}^{7} P(Y=j))=\displaystyle\sum_{i=1}^{6}( \frac{1}{7}\displaystyle\sum_{j=i+1}^{7} \frac{1}{6})=\frac{1}{42}\displaystyle\sum_{i=1}^{6}(\displaystyle\sum_{j=i+1}^{7}1)=\frac{6+5+4+3+2+1}{42}=\frac{1}{2}[/tex][/tex]

- if d falls in the 1st place, e falls in the 2nd and f in the 3rd place

- if d falls in the 2nd place, e falls in the 3rd and f in the 4th place

- if d falls in the 3rd place, e falls in the 4th and f in the 5th place

- if d falls in the 4th place, e falls in the 5th and f in the 6th place

- if d falls in the 5th place, e falls in the 6th and f in the 7th place

X=i means "d falls in the i-th position"

Y=j means "e falls in the j-th position"

Z=k means "f falls in the k-th position"

[tex]P(C)=\displaystyle\sum_{i=1}^{5}( P(X=i)P(Y=i+1)P(Z=i+2))=\displaystyle\sum_{i=1}^{5}(\frac{1}{7}\times\frac{1}{6}\times\frac{1}{5})=\frac{1}{210}\displaystyle\sum_{i=1}^{5}(1)=\frac{1}{42}[/tex]

P(A|C)=P(A∩C)/P(C)=?

A∩C:

- d falls in the 1st place, e in the 2nd, f in the 3rd and b in the 4th place

- b falls in the 4th place, d in the 5th place, e in the 6th, f in the 7th place

P(A∩C)=2*(1/7*1/6*1/5*1/4)=1/420

P(A|C)=(1/420)/(1/42)=1/10

P(B|C)=P(B∩C)/P(C)=?

X=i means "d falls in the i-th position"

Y=j means "e falls in the j-th position"

Z=k means "f falls in the k-th position"

V=k means "b falls in the k-th position"

W=k means "c falls in the k-th position"

[tex]P(B\cap C)=\displaystyle\sum_{i=1}^{3} P(X=i)P(Y=i+1)P(Z=i+2)\displaystyle\sum_{j=i+3}^{6}P(V=j)P(W=j+1)[/tex]

[tex]P(B\cap C)=\displaystyle\sum_{i=1}^{3} \frac{1}{7}\times\frac{1}{6}\times\frac{1}{5}(\displaystyle\sum_{j=i+3}^{6}\frac{1}{4}\times\frac{1}{3})=\frac{1}{2520}\displaystyle\sum_{i=1}^{3} \displaystyle\sum_{j=i+3}^{6}1=\frac{1}{420}[/tex]

P(B|C)=(1/420)/(1/42)=1/10

P(A|B)=P(B∩A)/P(B)=?

B∩A:

- b falls in the 4th place and c in the 5th

- b falls in the 4th place and c in the 6th

- b falls in the 4th place and c in the 7th

P(B∩A)=3*(1/7*1/6)=1/14

P(A|B)=(1/14)(1/2)=1/7

If one event is independent of another, P(X∩Y)=P(X)P(Y)

So:

P(A∩B)=1/14=(1/7)*(1/2)=P(A)P(B), A and B are independent

P(A∩C)=1/420≠(1/7)*(1/42)=1/294=P(A)P(C), A and C aren't independent

P(B∩C)=1/420≠(1/2)*(1/42)=1/84=P(A)P(C), B and C aren't independent

Answer:

a) The probability that an order requests at least one optional feature is 34%+26% = 60%.

b) The probability that an  order does not request more than one optional feature is 40% + 34% = 74%.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

In our problem, the probabilities are:

-40% that no optional features are requested.

-34% that one optional feature is requested

-26% that more than one optional feature is requested.

(a) What is the probability that an order requests at least one optional feature?

There is a 34% probability that one optional feature is requested and a 26% probability that more than one optional feature is requested.

So the probability that an order requests at least one optional feature is 34%+26% = 60%.

(b) What is the probability that an order does not request more than one optional feature?

There is a 40% probability that no optional features are requested and a 34% probability that one optional feature is requested.

So the probability that an  order does not request more than one optional feature is 40% + 34% = 74%.

Final answer:

The probability an order requests at least one optional feature is 0.60, and the probability an order does not request more than one optional feature is 0.74.

Explanation:

The question involves calculating probabilities based on provided proportions of orders with optional features.

Part (a): Probability of at least one optional feature

The proportion of orders with no optional features is 0.40. Therefore, the probability that an order requests at least one optional feature is 1 - 0.40 = 0.60. So, the probability is 0.60 when rounded to two decimal places.

Part (b): Probability of not more than one optional feature

We are given that orders with one optional feature make up 0.34 and those with no optional features constitute 0.40. Adding these together gives us a probability of 0.34 + 0.40 = 0.74 for orders not requesting more than one optional feature. Thus, this probability is 0.74, rounded to two decimal places.

Answer:

n = 60

Step-by-step explanation:

GIVEN DATA:

Total sum of consecutive number is 1800

sum of n number is given as

[tex] sum = \frac{ n(n+1)}{2}[/tex]

where n is positive number and belong to natural number i.e 1,2,3,4,...

from the data given we have[tex]1800 = \frac{n(n+1)}{2}[/tex]

solving for n we get

[/tex]n^2 + n -3600 = 0[/tex]

n = 59.5, -60.5

therefore n = 60

Answer:757

Step-by-step explanation:

Answer:

19/30

Explanation:

1. Exchange them to a common factor which happens to be 30 for both of them

2. Multiply by that factor on both the top and bottom to get the number equivalent to a fraction of that category

3. Subtract

4. Simplify, however in this case simplification isn't doable.

Answer: Our required probability is 0.406.

Step-by-step explanation:

Since we have given that

Probability of selecting an adult over 40 years of age with cancer = 0.05

Probability of a doctor correctly diagnosing a person with cancer as having the disease = 0.78

Probability of incorrectly diagnosing a person without cancer as having the disease = 0.06

Let A be the given event i.e. adult over 40 years of age with cancer. P(A) = 0.05.

So, P(A')=1-0.05 = 0.95

Let C be the event that having cancer.

P(C|A)=0.78

P(C|A')=0.06

So, using the Bayes theorem, we get that

[tex]P(A|C)=\dfrac{P(A).P(C|A)}{P(A).P(C|A)+P(A')P(C|A')}\\\\P(A|C)=\dfrac{0.78\times 0.05}{0.78\times 0.05+0.06\times 0.95}\\\\P(A|C)=0.406[/tex]

Hence, our required probability is 0.406.

Answer:

total mass of 6 samples = 538.896 g

in terms of X = 5.283 g

Step-by-step explanation:

Given:

Number of crude oil samples = 6

Density of each sample = 0.8240 kg/L

Volume filled by each sample = 1.09 × 10⁻⁴ m³

now,

1 m³ = 1000 L

thus,

1.09 × 10⁻⁴ m³ = 1.09 × 10⁻⁴ m³ × 1000 = 0.109 L

also,

Mass = Density × Volume

or

Mass of each sample = 0.8240 × 0.109 = 0.089816 kg

Thus,

total mass of 6 samples = Mass of each sample × 6

or

total mass of 6 samples = 0.089816 kg × 6 = 0.538896 kg

or

total mass of 6 samples = 538.896 g

or

in X = [tex]\frac{\textup{total mass of 6 samples}}{\textup{102}}[/tex]

= 5.283 g

Answer:

The two readings that are cutoff values are T=1.84 deg C and T=-1.84 deg C.

Step-by-step explanation:

Thermometers rejected by measurements above normal represent 3.3% of the total, which indicates, by normal probability distribution data, that accepted thermometers are 96.7% likely to measure less than the maximum allowable temperature.  

This value (P(X>x)=0.967) corresponds to z = 1.8388. Since the mean and standard deviation values are the same as the standard normal probability distribution (mean = 0, sd = 0), the z value is equivalent to the measured value (temperature).

Given the symmetry of the probability distribution, we can affirm that the thermometers rejected by measurements below the permissible measured a temperature lower than -1.8388.

Final answer:

The cutoff values separating the rejected thermometers are -1.88°C for the low end and 1.88°C for the high end, based on a normal distribution with a mean of 0°C and a standard deviation of 1.00°C.

Explanation:

To find the two readings that are cutoff values separating the rejected thermometers, we look at the normal distribution curve with a mean of 0°C and a standard deviation of 1.00°C. Since 3% of thermometers are rejected for high readings and another 3% for being too low, these correspond to the tail ends of the distribution.

Using the Z-score table, find the Z-score that has 3% in the tail. For the lower end, we seek the Z-score where the left tail (the area to the left of the Z-score) is 0.03, and for the higher end, we look for a Z-score where the right tail is 0.03. These Z-scores are approximately -1.88 for the low end and +1.88 for the high end (using the 97th percentile since we want the upper 3%).

To find the actual thermometer readings:

For the low cutoff: cutoff low = mean + (Z-score * standard deviation) = 0 + (-1.88 * 1) = -1.88°C.

For the high cutoff: cutoff high = mean + (Z-score * standard deviation) = 0 + (1.88 * 1) = 1.88°C.

Therefore, thermometers reading lower than -1.88°C or higher than 1.88°C will be rejected.

Answer:

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Step-by-step explanation:

To find : Convert [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}[/tex] into [tex]\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Solution :

We convert units one by one,

[tex]1\text{ m}^2=10.7639\text{ ft}^2[/tex]

[tex]1\text{ sec}=\frac{1}{3600}\text{ hour}[/tex]

[tex]1\text{ cal}=0.003968\text{ BTU}[/tex]

Converting temperature unit,

[tex]^\circ C\times \frac{9}{5}+32=^\circ F[/tex]

[tex]1^\circ C\times \frac{9}{5}+32=33.8^\circ F[/tex]

So, [tex]1^\circ C=33.8^\circ F[/tex]

Substitute all the values in the unit conversion,

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{10.7639\times \frac{1}{3600}\times 33.8}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{0.101061}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Therefore, The conversion of unit is [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Answer:

The price of calculator before 30 years = $30+$122.36 = $145.36

Step-by-step explanation:

We have given price current price of calculator = $23

It is given that current price of calculator is after decrease of 532 %

We have to find the price of calculator before 30 years

The price of calculator will be more than 532 % from the current price

So 532% of 23 [tex]=\frac{23\times 532}{100}=$122.36[/tex]

So the price of calculator before 30 years = $30+$122.36 = $145.36

Step-by-step explanation:

When we need to subtract a number from another number, in that case, we can take the complement of the first number to add it to the second number, the result will be the same. It is because when we take the complement of 9 of that number, it will represent the negative of that number. Hence, by adding the negative of a number we will get the same result as we get after subtraction.

For example:

Subtract 213 from 843

843 - 213 = 630

complement of 9 of 213= 999-213

                                       =786

Now, add 786 and 843

786+843=1629

We got the result in 4 digits so by adding the left-most digit to the right-sided three-digit number of the result, we will get

629+1 = 630

To calculate the time Izzy takes to run to school on rainy days, we use the distance and speed to find that she runs 1.7 miles in approximately 13.6 minutes. We are unable to calculate the distance through the woods or the time saved without further information.

Calculation of Time and Distance

To calculate the time Izzy takes to run to school on rainy days, we use the following formula:

Time (in hours) = Distance (in miles) / Speed (in miles per hour)

Izzy runs a total distance of 1.2 miles on West St and then 0.5 miles on North Ave, summing up to 1.7 miles. Given Izzy's speed is 7.5 miles per hour, the time taken to run to school on rainy days can be calculated as:

Time = (1.2 + 0.5) miles / 7.5 mph = 1.7 / 7.5

To find the time in minutes, multiply the time in hours by 60:

Time in minutes = (1.7 / 7.5)  imes 60
= 13.6 minutes (approximately)

As the dashed path through the woods on dry days is not described in the question, we cannot calculate the exact distance Izzy is traveling through the woods. Without this information, we also cannot calculate the time saved by cutting through the woods.