Answer:
[tex]I=\frac{1000}{exp^{0,806725*t-0.6906755}+1}[/tex]
Step-by-step explanation:
The rate of infection is jointly proportional to the number of infected troopers and the number of non-infected ones. It can be expressed as follows:
[tex]\frac{dI}{dt}=a*I*(1000-I)[/tex]
Rearranging and integrating
[tex]\frac{dI}{dt}=a*I*(1000-I)\\\\\frac{dI}{I*(1000-I)}=a*dt\\\\\int\frac{dI}{I*(1000-I)}=\int a*dt\\\\-\frac{ln(1000/I-1)}{1000}+C=a*t[/tex]
At the initial breakout (t=0) there was one trooper infected (I=1)
[tex]-\frac{ln(1000/1-1)}{1000}+C=0\\\\-0,006906755+C=0\\\\C=0,006906755[/tex]
In two days (t=2) there were 5 troopers infected
[tex]-\frac{ln(1000/5-1)}{1000}+0,006906755=a*2\\\\-0,005293305+0,006906755=2*a\\a = 0,00161345 / 2 = 0,000806725[/tex]
Rearranging, we can model the number of infected troops (I) as
[tex]-\frac{ln(1000/I-1)}{1000}+0,006906755=0,000806725*t\\\\-\frac{ln(1000/I-1)}{1000}=0,000806725*t-0,006906755\\-ln(1000/I-1)=0,806725*t-0.6906755\\\\\frac{1000}{I}-1=exp^{0,806725*t-0.6906755} \\\\\frac{1000}{I}=exp^{0,806725*t-0.6906755}+1\\\\I=\frac{1000}{exp^{0,806725*t-0.6906755}+1}[/tex]
How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?
Answer:
The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.
Step-by-step explanation:
To find : How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?
Solution :
A 150-W light bulb is on for 20-hours.
The heat produced by bulb is given by,
[tex]H=150\times 20[/tex]
[tex]H=3000\ W-hr[/tex]
We know that,
[tex]1\ \text{W-hr}=3.4\ \text{BTU}[/tex]
Converting W-hr into BTU,
[tex]3000\ \text{W-hr}=3000\times 3.4\ \text{BTU}[/tex]
[tex]3000\ \text{W-hr}=10200\ \text{BTU}[/tex]
Therefore, The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.
Combinatorics problem. A web site was hit 300 times over a period of 15 days Show that over some period of 3 consecutive days, it was hit at least 60 times.
Answer:
There is a period of 3 consecutive days in which the website was hits at least 60 times.
Step-by-step explanation:
A web site was hit 300 times over a period of 15 days.
To solve this question we will use the Pigeonhole Principle.
Here, n = 300 and k = 5
We will find [tex]\frac{n}{k}[/tex] to get that there is a hole with at least [tex]\frac{300}{5}=60[/tex] pigeons.
Hence, there is a period of 3 consecutive days in which the website was hits at least 60 times.
Let A be the set represented by the bitstring 01011011100, let B be the set represented by the bitstring 10110111010. Find the bitstrings representing Ac, AUB, AnB, and A-E.
Answer:
Ac = 10100100011
[tex]A \cup B = 11111111110[/tex]
[tex]A \cap B = 00010011000[/tex]
[tex]A - B = 100100111010[/tex]
Step-by-step explanation:
All these operations are bitwise operations.
Ac is the complement of a. So where we have a bit 0, the complement is 1. Where we have a bit 1, the complement is 0. So
A = 01011011100
Ac = 10100100011
The second operation is the union between A and B. This is bitwise(bit 1 of A with bit 1 of B, bit 2 of A with bit 2 of B,...). The union operation is only 0 when it is between two zeros. So:
A = 01011011100
B = 10110111010
[tex]A \cup B = 11111111110[/tex]
The third operation is the intersection between A and B. Again, bitwise. The intersection is only 1 when it is between two bits that are 1. So
A = 01011110100
B = 10110111010
[tex]A \cap B = 00010011000[/tex]
The last operation is the bitwise subtraction between A and B. We start from the least significant bit(the last one). And we have to take care of the borrow operator also, similarly to a decimal subtraction.
We can only borrow from a previous bit 1, and this bit is set to 0
0-0 is 0 with no borrow
1-0 is 1 with no borrow
1 with borrow - 0 is 1 with borrow
1 with borrow - 1 is 1 with no borrow
0-1 is 1 with no borrow
0 with borrow -1 is 0 with no borrow
1-1 is 0 with no borrow
If we arrive at the first bit(the most significant) with a borrow, we must add a 1 at the front of the answer. So
A = 01011110100
B = 10110111010
[tex]A - B = 100100111010[/tex]
Prove that for all x and y in Z, x +3y is a multiple of 7 iff 3x +2y is a multiple of 7. Might be helpful to calculate 2(3x +2y)+(x +3y) and/or 4(x +3y)+(3x +2y).
Proof :
First, it is important to have in mind that a number [tex] m \in \mathbb{Z} [/tex] is a multiple of [tex]n\in\mathbb{Z} [/tex] iff there exists [tex]k\in\mathbb{Z}[/tex] such that [tex] m = n \cdot k[/tex].
Also, you have to prove a logical equivalence. To this end, it is possible to prove two logical implications.
Step-by-step explanation:
1.) Let x, y be integers such that x + 3y is a multiple of 7. You have to prove that 3x +2y is a multiple of 7.
In effect, by hypothesis there exists k [tex]\in\mathbb{Z}[/tex] such that x + 3y = 7 k . So, you get
[tex]\begin{equation*} 4(x+3y) + (3x + 2y) = 7x + 14y = 7 (x + 2y) \ \mbox{(direct computations and factoring)}\end{equation*} [/tex].
Therefore, 4(x +3y) + (3x +2y) is a multiple of 7. Then,
[tex](3x + 2y) = 7 (x + 2y) - 4(x + 3y) = 7 (x+2y) - 4 \cdot 7 k = 7 (x + 2y -4k) \ \mbox{(factoring)}[/tex].
Given that x,y,k are integers, then x + 2y - 4k is an integer and hence, 3x + 2y is a multiple of 7.
To finish, it remains to prove its reciprocal statement.
2.) Let x, y be integers such that 3x + 2y is a multiple of 7. You have to prove that x +3y is a multiple of 7. Reasoining as before , there exists q [tex]\in\mathbb{Z}[/tex] such that 3x + 2y = 7 \cdot q. Thus,
[tex]$$ \begin{equation*} 2(3x+2y) + (x + 3y) = 7x + 7y = 7 (x + y) \ \mbox{direct computations and factoring} \\\end{equation*} $$[/tex] Thus, [tex] 2(3x +2y) + (x +3y)[/tex] is a multiple of 7.
On the other hand, using the hypothesis [tex] $$ \begin{equation*} (x + 3y) = 7 (x + y) - 2(3x + 2y) = 7 (x+y) - 2 \cdot 7 q = 7 (x + y -2q) \ \mbox{(factoring)} \end{equation*} $$ [/tex] .
Finally, thanks that [tex]x,y,q [/tex] are integer numbers, then [tex] x + y - 2q[/tex] is a integer number and therefore, [tex] 3x + 2y [/tex] is a multiple of 7.
4. The salaries of a school cook, custodian, and bus driver are in the ratio 2:4:3. If their
combined monthly salaries for November total $8,280, what is the monthly salary for each
person?
Answer:
cook: $1840custodian: $3680bus driver: $2760Step-by-step explanation:
There are a total of 2+4+3 = 9 "ratio units", so each one is worth ...
$8,280/9 = $920
Multiplying the ratio by $920, we get ...
cook : custodian : bus driver = $920 × (2 : 4 : 3) = $1840 : $3680 : $2760
The monthly salaries for the school cook, custodian, and bus driver are $1,840, $3,680, and $2,760, respectively.
Explanation:To find the monthly salary for each person, we need to first determine the common ratio between their salaries. The given salaries are in the ratio 2:4:3. We can assign a constant to the ratio, such as 2x:4x:3x.
Next, we can set up an equation using the given information that the combined monthly salaries total $8,280:
2x + 4x + 3x = 8,280
9x = 8,280
To solve for x, we can divide both sides of the equation by 9:
x = 920
Now, we can find the monthly salary for each person by substituting x back into the ratio:
School cook's monthly salary: 2x = 2(920) = $1,840
Custodian's monthly salary: 4x = 4(920) = $3,680
Bus driver's monthly salary: 3x = 3(920) = $2,760
If f(x)=7/x^2, then what is the area enclosed by the graph of the function, the horizontal axis, and vertical lines at x=3 and x=4
Answer:
[tex]Area=\frac{7}{12}[/tex]
Step-by-step explanation:
[tex]Area=\int\limits^a_b {f(x)} \, dx =\int\limits^4_3 {\frac{7}{x^{2}} } \, dx =-7*\frac{1}{x}=-7(1/4-1/3)=\frac{7}{12}[/tex]
7. During a recent storm clean up there is a mound of m tons of sand at the beach. Throughout the day, 1200 tons of sand is added to the mound. Two dump trucks come in and take 800 tons of sand each from the mound. At the end of the day, the mound has 2,400 tons of sand. A. Write the equation that describes the situation. B. Solve for m.
Answer:
There was 2800 tons of sand in the mound initially.
Step-by-step explanation:
Let there be m tons of sand on the beach initially.
Throughout the day, 1200 tons of sand is added to the mound. So, total sand becomes = [tex]m+1200[/tex]
Two dump trucks come in and take 800 tons of sand each from the mound.
Means they took [tex]800\times2=1600[/tex] tons of sand
At the end of the day, the mound has 2,400 tons of sand.
This can be modeled as:
[tex]m+1200-1600=2400[/tex]
Solving for m:
[tex]m-400=2400[/tex]
[tex]=> m=2400+400[/tex]
m = 2800
Hence, there was 2800 tons of sand in the mound initially.
The initial amount of sand at the beach was 4000 tons.
Explanation:To solve this problem, we can set up an equation based on the given information. Let's assume that the initial amount of sand at the beach is m tons. Throughout the day, 1200 tons of sand is added, so the total amount of sand becomes m + 1200 tons. Two dump trucks take 800 tons of sand each, so the amount of sand remaining is (m + 1200) - (2 * 800) tons. At the end of the day, the mound has 2400 tons of sand, so we can set up the equation (m + 1200) - (2 * 800) = 2400.
Simplifying the equation, we have m - 1600 = 2400, which can be further simplified to m = 4000.
Therefore, the initial amount of sand at the beach was 4000 tons.
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PLZ HELP I BEG DUE IN 30 MIN!!! 30 POINTS!!!!
The total distance from A to B is 5 ( -3 to 2 = 5).
Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.
Answer:
A to B is 5 ( -3 to 2 = 5).//!!//Then you are gonna be Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.
a number has seven digits. all the digits are 6 except the hundred-thousands' digit, which is 2, and the thousands' digit, which is 4. what is the number?
Answer:
264,666.
Step-by-step explanation:
We have been given that a number has seven digits.
All digits are 6 except the hundred-thousands' digit.
Hundred thousands: 100,000.
The hundred thousand's digit is 2, so its value would be 200,000.
We have been given that thousands's digit is 4, so its value would be:
4 thousands: 4,000
The number with given hundred thousand's and thousands's digit would be 204,000.
Since all digits are 6 except the hundred-thousands' digit, therefore, our number would be 264,666.
A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 85 and standard deviation σ = 24. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (a) x is more than 60
(b) x is less than 110
(c) x is between 60 and 110
(d) x is greater than 140 (borderline diabetes starts at 140)
Answer:
(a) 0.8512 (b) 0.8512 (c) 0.7024 (d) 0.0110
Step-by-step explanation:
The blood glucose follows a normal distribution N(μ=85;σ=24).
For every value of X, we can calculate the z-score (equivalent for a N(0;1)) and compute the probability.
(a) P(x>60)
z = (x-μ)/σ = (60-85)/24 = -1.0417
P(x>60) = P(z>-1.0417) = 0.8512
(b) P(x<110)
z = (x-μ)/σ = (110-85)/24 = 1.0417
P(x<110) = P(z<1.0417) = 0.8512
(c) P(60<x<110) = P(x<110)-P(x<60)
P(60<x<110) = P(z<1.0417) - P(z<-1.0417)
P(60<x<110) = 0.8512 - (1-0.8512) = 0.8512 - 0.1488 = 0.7024
(d) P(x>140)
z = (x-μ)/σ = (140-85)/24 = 2.2917
P(x>140) = P(z>2.2917) = 0.0110
Final answer:
Explanation of probabilities for different blood glucose levels using mean and standard deviation.
Explanation:
Probability calculations for blood glucose levels:
(a) x is more than 60: Calculate the z-score using the formula z = (x - μ) / σ. With x = 60, μ = 85, and σ = 24, find the probability using a standard normal distribution table.
(b) x is less than 110: Use the z-score formula with x = 110, μ = 85, and σ = 24 to determine the probability.
(c) x is between 60 and 110: Find the individual probabilities for x = 60 and x = 110, then subtract the two values to get the probability in this range.
(d) x is greater than 140: Similar to the previous steps, find the z-score for x = 140 and calculate the probability.
Let P={ 1,2,3,5,7,9}, Q= { 1,2,3,4,5}, and R= {2,3,5,7,11}
Find P n R?
Answer:
[tex]P \cap R=\{2,3,5,7\}[/tex]
Step-by-step explanation:
Given : Let P={ 1,2,3,5,7,9}, Q= { 1,2,3,4,5}, and R= {2,3,5,7,11}
To find : The value of [tex]P \cap R[/tex] ?
Solution :
The intersection of two sets is defined as the set of their common elements or the elements appear in both the sets.
The sets are P={ 1,2,3,5,7,9} and R= {2,3,5,7,11}
Common elements in P and R are {2,3,5,7}
So, [tex]P \cap R=\{2,3,5,7\}[/tex]
Therefore, The value of [tex]P \cap R=\{2,3,5,7\}[/tex]
The average number of points you scored in the first 9 basketball games of the season was 23. What do you need in the 10th game to get your average to n? OA. n-23 OB, 10n-207 O C. 9n-23 D. 207-9n
Answer:
B. 10n-207
Step-by-step explanation:
Given,
The average number of points of 9 games = 23,
∵ Sum of observations = their average × number of observations
So, the total points for 9 games = 9 × 23 = 207,
Now, if the average of 10 games is n,
Then the total points for 10 games = 10n,
Hence, the number of points earned in 10th game = the total points for 10 games - the total points for 9 games
= 10n - 207
i.e. OPTION B is correct.
The correct option is (B) 10n-207. To raise the average points per game to 'n' after the 10th basketball game, one would need to score '10n - 207' points in the 10th game, based on the previous average of 23 points in the first 9 games.
To solve the problem, we need to use the formula for calculating an average. The average score after 10 games would be the total points scored divided by 10. If the first 9 games had an average of 23 points, the total points from these games is 9 × 23 = 207.
To get an average of n after the 10th game, the total points should be 10× n. Therefore, the points needed in the 10th game would be the difference between 10 × n and the sum score from the first 9 games (207). That gives us the equation 10n - 207 for the number of points needed in the 10th game.
A tube feeding formula contains 6 grams of protein per each 80 ml of the formula. If the patient needs 120 grams of protein per day, how much tube-feeding formula should he get every day?
Answer:
1600 ml.
Step-by-step explanation:
Let x represent amount of tube-feeding formula.
We have been given that a tube feeding formula contains 6 grams of protein per each 80 ml of the formula.
To solve our given problem, we will use proportions as:
[tex]\frac{x}{\text{120 gram}}=\frac{\text{80 ml}}{\text{ 6 grams}}[/tex]
[tex]\frac{x}{\text{120 gram}}*\text{120 gram}=\frac{\text{80 ml}}{\text{ 6 grams}}*\text{120 gram}[/tex]
[tex]x=\text{80 ml}*20[/tex]
[tex]x=\text{1600 ml}[/tex]
Therefore, the patient should get 1600 ml of tube feeding formula every day.
A bird feeder has a diameter of 3 inches and is composed of a cylinder and a cone. A diagram of the feeder is shown below.
What is the volume of this bird feeder, to the nearest tenth of cubic inch?
The total volume of the bird feeder is 47.1 cubic inches.
The total volume of the bird feeder.
Given:
Diameter (D) of the bird feeder = 3 inchesHeight of the cone (hcone) = 2 inchesHeight of the cylinder (hcylinder) = 6 inchesCalculate the Radius (r):So, the total volume of the bird feeder is 47.1 cubic inches.
Solve the system of linear equations using the Gauss-Jordan elimination method.
2x + 2y + z = 18
x + z = 7
4y − 3z = 20
Answer:
Values for each variable are:
x = 19
y = -4
z = -12
Step-by-step explanation:
As we can remember the Gauss-Jordan elimination method consists of creating a matrix with all the equations of the system. Remember that, if a variable does not appear in one of the equations, we give a value of 0 to its coefficient . Each equation will constitute a line of the matrix. So, the matrix will look like this:
2 2 1 18
1 0 1 7
0 4 -3 20
For the Gauss-Jordan elimination we can multiply lines, add or subtract one line to another or we can rearrange the order at any given time. The goal is to get only 1s in the matrix diagonal, to determine the value of each variable.
Since we already have a line with a 1, we'll take that line as our starting point, and we'll rearrange it as our 1st line. By multiplying the 1st line for 2 and then subtracting the result to the second line:
1 0 1 7
0 2 -1 4
0 4 -3 20
Now, we multiply the second line by 2 and subtract the result to the third line
1 0 1 7
0 2 -1 4
0 0 -1 12
In order to get the value of Z all we have to do is multiply the third line by (-1).
1 0 1 7
0 2 -1 4
0 0 1 -12
Now, we add the third line to the second line.
1 0 1 7
0 2 0 -8
0 0 1 -12
Then, multiply the second line by a fraction 1/2, to get the value for Y
1 0 1 7
0 1 0 -4
0 0 1 -12
Finally, we subtract the third line to the 1st line to get the value for X
1 0 0 19
0 1 0 -4
0 0 1 -12
All we got left is to prove our answer is correct by replacing the variables in the system with the values found:
First equation
2(19) + 2(-4) + (-12) = 18
38 - 8 - 12 = 18
38 - 20 = 18
Second equation
19 + (-12) = 7
19 -12 = 7
Third equation
4(-4) - 3(-12) = 20
-16 + 36 = 20
In the following ordinary annuity, the interest is compounded with each payment, and the payment is made at the end of the compounding period. Find the amount of time needed for the sinking fund to reach the given accumulated amount. (Round your answer to two decimal places.) $255 monthly at 5.8% to accumulate $25,000.
Answer:
Ans. the amount of time needed for the sinking fund to reach $25,000 if invested $255/month at 5.8% compounded monthly (Effective monthly=0.4833%) is 80.45 months.
Step-by-step explanation:
Hi, first we need to transform that 5.8% compounded monthly into an effective monthly rate, that is as follows.
[tex]r(EffectiveMonthly)=\frac{r(Comp.Monthly)}{12} =\frac{0.058}{12} =0.00483[/tex]
That means that our effective monthly rate is 0.483%
Now, we need to solve for "n" the following formula.
[tex]FutureValue=\frac{A((1+r)^{n}-1) }{r}[/tex]
Let´s start solving
[tex]25,000=\frac{255((1.00483)^{n}-1) }{0.00483}[/tex]
[tex]\frac{25,000*0.00483}{255} =1.00483^{n} -1\\[/tex]
[tex]0.47352941=1.00483^{n} -1[/tex]
[tex]1+0.47352941=1.00483^{n[/tex]
[tex]1.47352941=1.00483^{n}[/tex]
[tex]Ln(1.47352941)=n*Ln(1.00483)[/tex]
[tex]\frac{Ln(1.47352941)}{Ln(1.00483)} =n=80.45[/tex]
This means that it will take 80.45 months to reach $25,000 with an annuity of $255 at a rate of 5.8% compounded monthly (0.4833% effective monthly).
Best of luck.
There are N passengers in a plane with N assigned seats (N is a positive integer), but after boarding, the passengers take the seats randomly. Assuming all seating arrangements are equally likely, what is the probability that no passenger is in their assigned seat? Compute the probability when N → [infinity]
The probability that no passenger is in their assigned seat is 0.6321.
Given data:
There are N passengers in a plane with N assigned seats (N is a positive integer), but after boarding, the passengers take the seats randomly.
The probability that no passenger is in their assigned seat is often referred to as the "surprising" or "paradoxical" result.
The problem is analyzed for a specific case with N = 3 passengers.
Passenger 1 sits in Seat 1: In this case, the remaining two passengers have a 1/2 chance of sitting in their assigned seats.
Passenger 1 sits in Seat 2: Again, the remaining two passengers have a 1/2 chance of sitting in their assigned seats.
Passenger 1 sits in Seat 3: In this case, the remaining two passengers will definitely sit in their assigned seats.
Now, calculate the probability for each scenario and find the overall probability that no passenger is in their assigned seat:
Scenario 1: Probability = 1/3 * 1/2 = 1/6
Scenario 2: Probability = 1/3 * 1/2 = 1/6
Scenario 3: Probability = 1/3
Overall Probability = Probability of Scenario 1 + Probability of Scenario 2 + Probability of Scenario 3
= 1/6 + 1/6 + 1/3
= 1/2
Thus, for N = 3, the probability that no passenger is in their assigned seat is 1/2.
Now, consider the case when N → ∞ (approaching infinity).
In this scenario, the probability can be calculated as follows:
Probability = 1 - 1/e
Hence, as N approaches infinity, the probability that no passenger is in their assigned seat approaches 1 - 1/e, or approximately 0.6321.
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Final answer:
The derangement problem questions the probability that no passenger sits in their assigned seat, which asymptotically approaches 1/e (approximately 0.3679) as the number of seats and passengers approaches infinity.
Explanation:
The student's question is about calculating the probability that no passenger is seated in their assigned seat on a plane where passengers sit randomly, especially when the number of passengers (and seats) approaches infinity. This is known as the derangement problem or a problem of calculating permutations where no element appears in its original position. A specific case of permutations without fixed points is termed as a derangement, and the probability of a derangement occurring in a permutation of N elements approaches 1/e where e is Euler's number, approximately equal to 2.71828. Through an approximation known as the asymptotic probability, when N → [infinity], this probability approaches 1/e, which is approximately 0.3679 or 36.79%.
What is always true of the diagonals of a parallelogram?
Answer:
The diagonals of parallelogram bisect each other.
Step-by-step explanation:
Consider the provided statement.
The parallelogram has few properties.
1: The opposite side of parallelogram are parallel by the definition of parallelogram.
2: The opposite sides and angles of a parallelogram are congruent.
3: The consecutive angles of a parallelogram are supplementary.
4: The diagonals of parallelogram bisect each other.
From the above properties of a parallelogram we can say that:
The diagonals of parallelogram bisect each other.
The diagonals of a parallelogram always bisect each other, dividing each other into two equal parts. However, unless the parallelogram is a rectangle or square, the diagonals themselves are not necessarily of equal length.
Explanation:In a parallelogram, the diagonals bisect each other, meaning they intersect and divide each other into two equal parts. So, if you have a parallelogram ABCD, its diagonals AC and BD will intersect at a point E, making the two parts of each diagonal (AE and EC, BE and ED) equal in length. In other words, AE equals EC and BE equals ED. However, unless the parallelogram is a special case, like a rectangle or a square, the diagonals themselves are not of equal length. Hence, the diagonals of a parallelogram are always bisecting each other but are not necessarily equal in length.
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Give the first three non-zero terms of the Taylor series for f(x) = tan(x) about x 0· Use this to approximate tan(1) and give an upper bound on the error in this approximation
Answer:
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]
Approximate error = 0.4426
Step-by-step explanation:
f(x)=tanx, a=0
Maclaurin series formula used is given below
[tex]f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)x^{n}}{n!}=f(0)+f'(0)x+\dfrac{f''(0)}{2!}x^{2}+\dfrac{f'''(0)}{3!}x^{3}+....[/tex]
f(x)=tanx
f(0)=tan0=0
[tex]f'(x)=sec^{2}x\\f'(0)=sec^{2}0=1\\f''(x)=2sec^{2}xtanx\\f''(0)=2sec^{2}0tan0=0\\f'''(x)=-4sec^{2}x+6sec^{4}x\\f'''(0)=-4sec^{2}0+6sec^{4}0=-4+6=2\\[/tex]
[tex]f''''(x)=-8(2sec^{2}xtan^{2}x+sec^{4}x)+24(4sec^{4}xtan^{2}x)+sec^{6})\\f''''(0)=-8(0+1)+24(0+1)=-8+24=16\\[/tex]
[tex]f(x)=0+x+0+\dfrac{2x^{3}}{3!}+\dfrac{16x^{4}}{4!}\\[/tex]
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}\\[/tex]
Hence, the Taylor series for f(x)=tanx is given by
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]
Maclaurin series upper bound error formula used is given as
R_n(x)=|f(x)-T_n(x)|
R_3(x)=|tanx-T_3(x)|
[tex]R_3(x)=|tanx-x-\dfrac{x^{3}}{3}-\dfrac{2x^{4}}{3}|[/tex]
Plugging this value x=1
[tex]R_3(x)=|tan(1)-1-\dfrac{1}{3}-\dfrac{2}{3}|\\[/tex]
R_3(x)=|1.5574-1-0.333-0.666|
R_3(x)=|-0.4426|=0.4426
Hence, upper bound on the error approximation
tan(1)=0.4426
The following data are direct solar intensity measurements (watts/m2 ) on different days at a location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768, 870, 918, 940, 946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939, 955, 960, 498, 653, 730, and 753. Calculate the sample mean and sample standard deviation. Prepare a dot diagram of these data. Indicate where the sample mean falls on this diagram. Give a practical interpretation of the sample mean.
Answer:
Mean: 810.51
Standard deviation: 128.32
Step-by-step explanation:
First, we calculate the sample mean. We have 35 data samples, so we compute it as [tex]\displaystyle \frac{562 + .... + 753}{35}[/tex].
In general, given a set of n samples [tex]\{ X_1,...,X_n\}[/tex], we calculate the sample mean as
[tex]\displaystyle \bar{X} = \sum_{i=1}^n \frac{X_i}{n}[/tex].
For the standard deviation [tex]\sigma[/tex], we first begin by calculating it's square. It can be obtained from the formula
[tex]\displaystyle \sigma ^2 = \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{n-1}[/tex]
By taking square root after computing the right hand side, we attain the desired value.
The attached image shows the dot diagram of this sample. The bottom pink vertical line shows the mean, and the two horizontal pink lines have a lenght of [tex]\sigma[/tex].
The standard deviation means "The average expected distance a new sample will be from the mean". That's why usually, data samples which are denser around the mean have smaller standard deviations (as opposed to distributions who have a lot of values far away from the mean, which will make the standard deviation grow bigger).
Three less than the quotient of a number and two has a result of thirteen. Which equation and solution correctly represents this sentence. N/2+3=13;n=5 , n/2-3=13;n=8 , n/2+3=13;=20 , n/2-3=13;n=32
Answer: n/2-3=13;n=32
Step-by-step explanation: the equation and solution correctly represents this sentence is n/2-3=13;n=32,
first we have to keep in mind the beginning of the exercise which is (three less) then we know that the 3 has a sign of subtraction(-3), when we meet this, we can know that this exercise has only two possible good answer.
Only the equations which have a (-3) inside will be good,the rest of equation has a 3 with a sum sign
N/2+3=13;n=5 ,
n/2+3=13;=20
then we can omit the past equations, and this let us this equations as possible.
n/2-3=13;n=32
n/2-3=13;n=8
after this we only need resolve the equation to get a correct result, only the equation with a correct result will be the correct answer,
then we proceed to clear n from each equation.
for this equation the result must be equal to 8 if we clear n.
n/2-3=13;n=8
n/2-3=13
we pass the three to the other side with sum sign
n/2=13 + 3
we resolve the sum
n/2=16
after we pass the 2 multiplying to the other side
n = 16 × 2
we resolve the product of the multiplication
n = 32
but this answer said the result must be equal to 8
n=8 ≠ n = 32
as this result is different, we can conclude that this is a bad answer,
the we get only one possibility
this equation
n/2-3=13;n=32
for this equation the result must be equal to 32 if we clear n.
n/2-3=13;n=8
n/2-3=13
we pass the three to the other side with sum sign
n/2=13 + 3
we resolve the sum
n/2=16
after we pass the 2 multiplying to the other side
n = 16 × 2
we resolve the product of the multiplication
n = 32
Answer:
C
Um I'm late but its not any of the others so....
120 cm = _______ inches Round UP to nearest 100th
Answer:
120cm = 46.8 inches
Step-by-step explanation:
This 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.
Each cm has 0.39 inches. How many inches are there in 120 cm?
1cm - 0.39inches
120cm - x inches
[tex]x = 120*0.39[/tex]
[tex]x = 46.8[/tex] inches
120cm = 46.8 inches
Use mathematical induction to prove that for each integer n ≥ 4, 5^n ≥ 2^2n+1 + 100. (it is 5 to power of n and 2 to the power of 2n+1)
Step-by-step explanation:
The statement to be proved using mathematical induction is:
"For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]We will begin the proof showing that the base case is satisfied (n=4).
[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex].
Then, 1 is true for n=4.
Now we will assume that the statement holds for some arbitrary natural number [tex]n\geq 4[/tex] and prove that then, the statement holds for n+1. Observe that
[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]
With this the inductive step has been proven and then, our statement is true,
For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]
Find the most general antiderivative of the function. (Check yo f(x) = 3^x + 7 sinh(x) F(x) = Need Help? Watch It Talk to a Tutor
Answer:
F(x)=[tex]\frac{3^x}{ln(3)}[/tex]+7cosh(x)+C
Step-by-step explanation:
The function is f(x)=3ˣ+7sinh(x), so we can define it as f(x)=g(x)+h(x) where g(x)=3ˣ and h(x)=7sinh(x).
Now we have to find the most general antiderivative of the function this means that we have to calculate [tex]\int\ {f(x)} \, dx[/tex] wich is the same as [tex]\int\ {(g+h)(x)} \, dx[/tex]
The sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. Then,
[tex]\int\ {(g+h)(x)} \, dx[/tex] = [tex]\int\ {g(x)} \, dx + \int\ {h(x)} \, dx[/tex]
1- [tex]\int\ {g(x)} \, dx =[/tex][tex]\int\ {3^x} \, dx = \frac{3^x}{ln(3)}+C[/tex] this is because of the rule for integration of exponencial functions, this rule is:
[tex]\int\ {a^x} \, dx =\frac{a^x}{ln(x)}[/tex], in this case a=3
2-[tex]\int\ {h(x)} \, dx =[/tex][tex]\int\ {7sinh(x)} \, dx =7\int\ {sinh(x)} \, dx =7cosh(x)+C[/tex] , number seven is a constant (it doesn´t depend of "x") so it "gets out" of the integral.
The result then is:
F(x)= [tex]\int\ {(h+g)(x)} \, dx=\int\ {h(x)} \, dx +\int\ {g(x)} \, dx[/tex]
[tex]\int\ {3^x} \, dx +\int\ {7sinh(x)} \, dx = \frac{3^x}{ln(3)} +7cosh(x) + C[/tex]
The letter C is added because the integrations is undefined.
How many milliequivalents of sodium chloride are contained with 3 L of normal saline?
Answer:
466mEq
Step-by-step explanation:
First, we need to know the concentration of NaCl in a normal saline solution, this is by definition 0.9%, meaning we have 0.9g of NaCl per 100ml of solution, we want to know how much NaCl we have in 3L (3000ml):
[tex]3000ml*\frac{0.9g}{100ml}=27g=27000mg[/tex]
So, we have 27000mg in 3L of normal saline solution.
Now, acording to our milliequivalent (mEq) equation ([tex]mEq=\frac{mg}{pE}[/tex]) where pE is de molecular mass of NaCl divided by their charges, in this case 1:
[tex]pE= \frac{23+35}{1}=\frac{58}{1} = 58[/tex]
Finally we substitute in the mEq formula:
[tex]mEq=\frac{mg}{pE}=\frac{27000}{58}=466mEq[/tex]
I hope you find this information useful! Good luck!
Give a power series representation for the function f(x) x^3/(1 + 9x^2)
Recall that for [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n\ge0}x^n[/tex]
Replace [tex]x[/tex] with [tex]-9x^2[/tex] and we get
[tex]\displaystyle\frac1{1-(-9x^2)}=\sum_{n\ge0}(-9x^2)^n=\sum_{n\ge0}(-9)^nx^{2n}[/tex]
Lastly, multiply this by [tex]x^3[/tex], so that
[tex]\boxed{f(x)=\displaystyle\sum_{n\ge0}(-9)^nx^{2n+3}}[/tex]
Let v1, v2, w be three linearly independent vectors in R 3 . That is, they do not all lie on the same plane. For each of the following (infinite) set of vectors, carefully sketch it in R 3 , and determine whether or not it is a vector space (i.e., a subspace of R 3 ). Explain your reasoning.
Answer:
Where are the sets of vectors?
Step-by-step explanation:
Which set of vectors?
Let V be the set of pairs (x; y) of real numbers and let the eld F be the
real number set. Dene the addition and scalar multiplication as follows:
(x1; y1) + (x2; y2) = (x1 + x2; 0)
c(x; y) = (cx; 0):
Is V , with these operations, a vector space? Explain.
To prove that V is a vector space we must prove that the sum define on it satisfy conmutativiy, asociativity and existence of the neutral element and inverses. Also, the scalar multiplication define on V must satisfy distributivity propertie with respect to the sum and viceversa, and an asosiativity too in the sense that [tex]x(y\cdot v)= (xy)\cdot v[/tex] for [tex]x,y\in \mathbb{R}[/tex] and [tex]v\in V[/tex]. One can prove with this that the neutral element for the sum is unique. But with your operations you have two neutral elements for [tex](1;2)[/tex]
[tex](1;2)+(-1;3)=(0;0)[/tex]
and
[tex](1;5)+(-1;11)=(0;0)[/tex]
So, you dont have a vector space.
Final answer:
The set V, with its defined addition and scalar multiplication operations, does not fulfill essential vector space properties such as the existence of an additive identity, presence of additive inverses, and correct scalar multiplication effects on components. Therefore, V is not a vector space.
Explanation:
To determine if a set V, defined with specific addition and scalar multiplication operations, is a vector space, it must satisfy several properties commonly defined in linear algebra.
For V to be considered a vector space, the addition operation must be associative and commutative, there must be an additive identity (zero vector), each vector must have an additive inverse, scalar multiplication must be associative, there must be a multiplicative identity (1), and both operations must distribute over vector addition and scalar addition.
The defined operations on V are (x₁; y₁) + (x₂; y₂) = (x₁ + x₂; 0) for vector addition and c(x; y) = (cx; 0) for scalar multiplication. These operations fail to satisfy several vector space properties, including:
The existence of an additive identity that affects both components, since (x₁; y₁) + (0; 0) should equal (x₁; y₁), but according to the given addition rule, it equals (x₁; 0).The presence of additive inverses, as there is no pair (x₂; y₂) such that (x₁; y₁) + (x₂; y₂) equals the zero vector (0; 0).The scalar multiplication does not adequately affect the y-component; it should leave it y unchanged, i.e., c(x; y) = (cx; cy), but the given rule yields (cx; 0).Due to these inadequacies, V does not meet the criteria for a vector space under the provided operations.
Enrollment at ELAC decreased by 5%, or 600 people, the year. How many people were enrolled last year?
Answer:
12,000.
Step-by-step explanation:
Let x be the number of people that were enrolled last year.
We have been given that enrollment at ELAC decreased by 5%, or 600 people, the year. We are asked to find the number of people that were enrolled last year.
We can set as equation such that 5% of x equals 600.
[tex]\frac{5}{100}\cdot x=600[/tex]
[tex]0.05x=600[/tex]
[tex]\frac{0.05x}{0.05}=\frac{600}{0.05}[/tex]
[tex]x=12,000[/tex]
Therefore, 12,000 people were enrolled last year.
Which of the following probability distributions is visualized by a histogram? a. continuous c. geometric b. discrete d. uniform
Answer:
d. uniform
Step-by-step explanation:
If the data is uniformly distributed i.e. it follows Uniform Probability Distributions then it can be visualized by a histogram. Then the shape of the histogram will be bell-shaped which means as the value of x is increases the value of y also increases for small values of y and it will decrease for a large value of y.
But, other types of probability distributions can't be visualized by a histogram.
Hence option (d) is correct.