Answer with Step-by-step explanation:
The given differential equation is variable separable in nature and hence will be solved accordingly as follows:
[tex]\frac{dy}{dx}=\frac{y}{x}\\\\=\frac{dy}{y}=\frac{dx}{x}\\\\\int \frac{dy}{y}=\int \frac{dx}{x}\\\\ln(y)=ln(x)+ln(c)\\\\ln(y)=ln(cx)\\\\(\because ln(ab)=ln(a)+ln(b))\\\\\therefore y=cx[/tex]
where 'c' is constant of integration whose value shall be obtained using the given condition [tex]y(1)=-2[/tex]
Thus we have
[tex]-2=c\times 1\\\\\therefore c=-2\\[/tex]
Thus solution becomes
[tex]y=-2x[/tex]
the amount of carbon 14 still present is a sample after t years
is given by the function C(t)=
Coe-.00012t
where co is the initial anong . estimate the age of a sample of
wood discoverd by a arecheologist if the carbon level in the sample
is only 20% of it orginal carbon 14 level.
Answer:
The age of this sample is 13,417 years.
Step-by-step explanation:
The amount of carbon 14 present in a sample after t years is given by the following equation:
[tex]C(t) = C_{0}e^{-0.00012t}[/tex]
Estimate the age of a sample of wood discoverd by a arecheologist if the carbon level in the sampleis only 20% of it orginal carbon 14 level.
The problem asks us to find the value of t when
[tex]C(t) = 0.2C_{0}[/tex]
So:
[tex]C(t) = C_{0}e^{-0.00012t}[/tex]
[tex]0.2C_{0} = C_{0}e^{-0.00012t}[/tex]
[tex]e^{-0.00012t} = \frac{0.2C_{0}}{C_{0}}[/tex]
[tex]e^{-0.00012t} = 0.2[/tex]
[tex]ln e^{-0.00012t} = ln 0.2[/tex]
[tex]-0.00012t = -1.61[/tex]
[tex]0.00012t = 1.61[/tex]
[tex]t = \frac{1.61}{0.00012}[/tex]
[tex]t = 13,416.7[/tex]
The age of this sample is 13,417 years.
You wish to ship six crude oil samples from your drill site to your laboratory. Each sample has a density of 0.8240 kg/L and fills a 1.090e-4 m3container. How much mass, X g , of crude oil will you be shipping?
(HINT: |X| is near an order of magnitude of 102 g ).
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
On rainy days, Izzy goes from his house to the school by running 1.2 miles on West St, then makes a 90º turn and runs 0.5 miles on North Ave.
a. If Izzy runs 7.5 miles per hour, approximately how much time will it take her to run to school on rainy days?
b. On dry days, Izzy runs on the dashed path through the woods. How far is she traveling?
c. If Izzy runs 7.5 miles per hour, how much time will she save by cutting through the woods?
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.
Explanation: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.
In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06, what is the probability that an adult over 40 years of age is diagnosed as having cancer?
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.
Prove that (from i=1 to n) sum([1/((2i-1)(2i+1))] = n/(2n+1). If true use induction, else give smallest value of n that it is false for.
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]
Solve the system of linear equations using the Gauss-Jordan elimination method.
(x,y,z)=__________________
2x + 2y − 3z = 16
2x − 3y + 2z = −4
4x − y + 3z =
−4
Answer:
(x,y,z)=(2,0,-4)
Step-by-step explanation:
First we create the extended matrix from the equations[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
Substract to the 2nd line the first one, and the 3rd one twice the first:[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\0&-5&5&-20\\0&-5&9&-36\end{array}\right][/tex]
Divide the first line by 2, the 2nd one by -5 and substract to the 3rd the 2nd:[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&4&-16\\\end{array}\right][/tex]
Divide the 3rd by 4:[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&1&-4\\\end{array}\right][/tex]
Add the 3rd to the 2nd:[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]
Substract the 2nd to the 1st[tex]\left[\begin{array}{ccc|c}1&0&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]
Add the 3rd multiplied by 3/2:[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.
A medical device is sterilized by gamma radiation at 2.5 megarads (Mrad). Express the equivalent quantity in rads.
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.
If sin phi sin theta = 0.2 and sin phi cos theta = -0.3 and sin phi > 0 what is theta ? Repeat for sin phi < 0.
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.
Explanation: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.
How many sets of two or more consecutive positive integers can be added to obtain a sum of 1800?
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
fraction subtract 4/5-1/6
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.
A manufacturer has a monthly fixed cost of $110,000 and a production cost of $14 for each unit produced. The product sells for $20 per unit.
(a) What is the cost function?
C(x) =
(b) What is the revenue function?
R(x) =
(c) What is the profit function?
P(x) =
(d) Compute the profit or loss corresponding to a production level of 12,000 and 23,000 units. (Input a negative value to indicate a loss.)
at 12,000 units $ ______
at 23,000 units $______
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
Way back in the olden days, Blockbuster tallied all their US movie rental data and found that on average, individuals rent 10 movies a year with a standard deviation of 3. Treat these as population statistics. They wanted to see if movie rental rates in Yuma, Arizona, were different from those of the country as a whole (why Yuma? Who knows ). A random sample of 25 blockbuster members in Yuma yielded a mean rental rate of 11.3 movies per year. Use alpha = .05
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).
Explanation: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:
State the hypotheses. The null hypothesis H0 would be that the mean rental rate in Yuma is the same as the average across the US (μ = 10). The alternative hypothesis Ha would be that the mean rental rate in Yuma is not equal to the average across the US (μ ≠ 10).Formulate an analysis plan. For this analysis, the significance level is defined as alpha (α) = .05. As per the conditions, the population standard deviation (σ) is known and equals 3.Analyze sample data. Using the sample data and the information provided, we can calculate the test statistic (z).Interpret the results. If the test statistic is beyond the critical value, we reject the null hypothesis. Otherwise, we do not have enough evidence to reject it.Learn more about Hypothesis Testing here:https://brainly.com/question/34171008
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For the function, f(x) = 8x + 5x, find the following. (a) f(5) D NUD K (b) f(-2) orea (c) f(4.2) muca (d) f(-4.2)
Answer:
(a) 65
(b) -26
(c) 54.6
(d) -54.6
Step-by-step explanation:
(a) [tex]f(5)=8(5)+5(5)=40+25=65[/tex]
(b) [tex]f(-2)=8(-2)+5(-2)=-16-10=-26[/tex]
(c) [tex]f(4.2)=8(4.2)+5(4.2)=33.6+21=54.6[/tex]
(d) [tex]f(-4.2)=8(-4.2)+5(-4.2)=-33.6-21=-54.6[/tex]
In a recent year, 17.7% of household watched the finals of a popular reality series. There are 110.2 million households in the United States. How many households watched the finals?
This is Find the percent of a number.
For this case we propose a rule of three:
110.2 million -------------> 100%
x --------------------------------------> 17.7%
Where the variable "x" represents the number of households (in millions) that watched the finals of a popular reality series.
[tex]x = \frac {17.7 * 110.2} {100}\\x = \frac {1950.54} {100}\\x = 19.5054[/tex]
Thus, a total of 19.5054 million homes watched the finals of a popular reality series.
Answer:
19.5054 million homes watched the finals of a popular reality series.
Among users of automated teller machines (ATMs), 94% use ATMs to withdraw cash and 28% use them to check their account balance. Suppose that 95% use ATMs to either withdraw cash or check their account balance (or both). Given a woman who uses an ATM to check her account balance, what the probability that she also uses an ATM to get cash?
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:
The probability that ATM users withdraw cash (P(Cash)) is 94%, or 0.94.The probability that ATM users check their account balance (P(Balance)) is 28%, or 0.28.The probability that ATM users either withdraw cash or check their account balance (or both) (P(Cash ∪ Balance)) is 95%, or 0.95.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%.
In an arithmetic sequence, the nth term an is given by the formula An=a1+(n−1)d, where a1is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by an=a1•rn−1.
Use these formulas to determine the indicated term in the given sequence.
The 19th term of 19,42,65,88,...
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.
convert 1 cal/(m^2 * sec * °C) into BTU/(ft^2 * hr * °F)
Its easy enough to convert the energy, time, and area units, but how am I suppose to convert the temp units?
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]
In a sample of 408 new websites registered on the Internet, 37 were anonymous (i.e., they shielded their name and contact information). (a) Construct a 95 percent confidence interval for the proportion of all new websites that were anonymous. (Round your answers to 4 decimal places.)
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]
Find the general solution to each of the following ODEs. Then, decide whether or not the set of solutions form a vector space. Explain your reasoning. Compare your answers to the previous problem. Recall that the general solution has the form y(t) = yh(t) + yp(t).
(A) y' - 2y = 0
(B) y' - 2y = 1
(C) y" - 4y = 0
(D) y" - 4y = e^(3t)
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)).
For the statement "if Fury is the director of SHIELD then Hill and Coulson are SHIELD agents" (a) Write the contrapositive (b) Write the converse (c) Write the inverse (d) Write the negation
Step-by-step explanation:
Consider the provided information.
If Fury is the director of SHIELD then Hill and Coulson are SHIELD agents"
For the condition statement [tex]p \rightarrow q[/tex] or equivalent "If p then q"
The rule for Converse is: Interchange the two statements. [tex]q \rightarrow p[/tex]The rule for Inverse is: Negative both statements. [tex]\sim p \rightarrow \sim q[/tex]The rule for Contrapositive is: Negative both statements and interchange them. [tex]\sim q \rightarrow \sim p[/tex]The rule for Negation is: If p then q" the negation will be: p and not q. [tex]p \rightarrow q=\sim p\vee q=p\vee \sim q[/tex]Part (A) Write the contrapositive.
Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
Contrapositive: If Hill and Coulson are not SHIELD agents, then Fury is not the director of SHIELD.
Part (b) Write the converse.
Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
Converse: If Hill and Coulson are SHIELD agents, then Fury is the director of SHIELD.
Part (c) Write the inverse.
Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
Inverse: If Fury is not the director of SHIELD then Hill and Coulson are not SHIELD agents
Part (D) Write the negation.
Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
Negation: Fury is the director of SHIELD and Hill and Coulson are not SHIELD agents"
Step-by-step explanation:
Consider the provided information.
If Fury is the director of SHIELD then Hill and Coulson are SHIELD agents." For the condition statement or equivalent "If p then q"
The rule for Converse is: Interchange the two statements.
The rule for Inverse is: Negative both statements.
The rule for Contrapositive is: Negative both statements and interchange them.
The rule for Negation is: If p then q" the negation will be: p and not q.
- Part (A) Write the contrapositive.
.Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
Contrapositive: If Hill and Coulson are not SHIELD agents, then Fury is not the director of SHIELD.
- Part (b) Write the converse.
.Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
.Converse: If Hill and Coulson are SHIELD agents, then Fury is the director of SHIELD.
- Part (c) Write the inverse.
.Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
.Inverse: If Fury is not the director of SHIELD then Hill and Coulson are not SHIELD agents
- Part (D) Write the negation.
.Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.
.Negation: Fury is the director of SHIELD and Hill and Coulson are not SHIELD agents"
You deposit the same $10,000 into a bank account at 4% annual interest. How long will it take for the $10,000 to compound to $30,000?
N= I/Y= PV= PMT= FV= P/Y=
Answer:
time = 28 years
Step-by-step explanation:
Given,
principal amount = $10,000
rate = 4%
total amount = $30,000
According to compound interest formula
[tex]A\ =\ P(1+r)^t[/tex]
where, A = total amount
P = principal amount
r = rate
t = time in years
so, from the question we can write,
[tex]30000\ =\ 10000(1+0.04)^t[/tex]
[tex]=>\ \dfrac{30000}{10000}\ =\ (1+0.04)^t[/tex]
[tex]=>\ 3\ =\ (1.04)^t[/tex]
by taking log on both sides, we will get
=> log3 = t.log(1.04)
[tex]=>\ t\ =\ \dfrac{log3}{log1.04}[/tex]
=> t = 28.01
So, the time taken to get the amount from 10000 to 30000 is 28 years.
Calculate: (Round two decimals places for the final answer):
1880 milliliter (mL)=_____ pints (pts) ?
Answer:
1880 milliliter (mL) = 3.97 pints (pts)
Step-by-step explanation:
This problem can be solved as a rule of three problem.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
1 milliliter (mL) is equal to 0.002 pints. How many pints are 1880 milliliter (mL)? We have the following rule of three
1 mL - 0.002 pints
1880 mL - x pints
x = 1880*0.002
x = 3.97 pints
There are 3.97 pints in 1880 milliliters.
Calculating conditional probabilities - random permutations. About The letters (a, b, c, d, e, f, g) are put in a random order. Each permutation is equally likely. Define the following events: A: The letter b falls in the middle (with three before it and three after it) B: The letter c appears to the right of b, although c is not necessarily immediately to the right of b. For example, "agbdcef" would be an outcome in this event. C: The letters "def occur together in that order (e.g. "gdefbca") Calculate the probability of each individual event. That is, calculate p(A), P(B), and p(c). What is p(AIC)? (c) What is p(BIC)? What is p(AIB)? (e) Which pairs of events among A, B, and C are independent? Feedback?
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
How many kW-hr are used when one 100W light bulbs is used for 2 hours?
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
the earth rotates about its axis once every 23 hours, 56 minutes and 4 seconds. Approximate the number of radians the earth rotates in one second.
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.
Explanation: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.
Learn more about Angular Velocity of Earth here:https://brainly.com/question/32821466
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Why does changing a subrtraction problem to an addition with the complement of 9 work
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
The accompanying observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added: 2767 2924 3042 2844 2895 (a) What are the values of the sample mean x and sample median x tilde?
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]
While completing a race, Edward spent 54 minutes walking. If his ratio of time walking to jogging was 6:5, how many minutes did he spend completing the race?
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
A store has clearance items that have been marked down by 25%. They are having a sale, advertising an additional 40% off clearance items. What percent of the original price do you end up paying? Give your answer accurate to at least one decimal place.
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.
Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4, (ii) 2n < n! for for all n > k > 0, discover the value of k before doing induction.
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:
Every diagonal in P(n) will still be a diagonal in P(n+1). One (and only one) side of P(n) will be a diagonal in P(n+1).There would be an extra n-2 diagonals (those that connect with the new added vertix).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:
For n=1 [tex]2n=2>1=n![/tex]For n=2 [tex]2n=4>2=n![/tex]For n=3 [tex]2n=6=n![/tex]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]