Answer:
As consequence of the Taylor theorem with integral remainder we have that
[tex]f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \int^a_x f^{(n+1)}(t)\frac{(x-t)^n}{n!}dt[/tex]
If we ask that [tex]f[/tex] has continuous [tex](n+1)[/tex]th derivative we can apply the mean value theorem for integrals. Then, there exists [tex]c[/tex] between [tex]a[/tex] and [tex]x[/tex] such that
[tex] \int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}dt = \frac{f^{(n+1)}(c)}{n!} \int^a_x (x-t)^n d t = \frac{f^{(n+1)}(c)}{n!} \frac{(x-t)^{n+1}}{n+1}\Big|_a^x[/tex]
Hence,
[tex] \int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}d t = \frac{f^{(n+1)}(c)}{n!} \frac{(x-t)^{(n+1)}}{n+1} = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} .[/tex]
Thus,
[tex] \int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}d t = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} [/tex]
and the Taylor theorem with Lagrange remainder is
[tex] f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}[/tex].
Step-by-step explanation:
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.
Compare the numbers 26^39 and 39^26. Which one is bigger? Are they the same? No calculators.
Answer:
[tex]26^{39}[/tex] is greater.
Step-by-step explanation:
Given numbers,
[tex]26^{39}\text{ and }39^{26}[/tex]
∵ HCF ( 26, 39 ) = 13,
That is, we need to make both numbers with the exponent 13.
[tex]26^{39}=((13\times 2)^3)^{13}=(13^3\times 2^3)^{13}=(13^3\times 8)^{13}=(13^2\times 104)^{13}[/tex]
[tex](\because (a)^{mn}=(a^m)^n\text{ and }(ab)^m=a^m.a^n)[/tex]
[tex]39^{26}=((13\times 3)^2)^{13}=(13^2\times 3^2)^{13}=(13^2\times 9)^{13}[/tex]
Since,
[tex]13^2\times 104>13^2\times 9[/tex]
[tex]\implies (13^2\times 104)^{13} > (13^2\times 9)^{13}[/tex]
[tex]\implies 26^{39} > 39^{26}[/tex]
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
Proof by contradiction!
Prove that the following statement is true: the sum of two odd numbers is an even number
Answer:
The proposition: The sum of two odd numbers is an even number is true.
Step-by-step explanation:
A proof by contradiction is a proof technique that is based on this principle:
To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false.
Facts that we need:
Any even number has the form 2nAny odd number has the form 2n + 1Proposition. The sum of two odd numbers is an even number
Proof. Suppose this proposition is false in this case we assume that the sum of two odd numbers is not even. (That would mean that there are two odd numbers out there in the world somewhere that'll give us an odd number when we add them.)
Let a, b be odd numbers. Then there exist numbers m, n, such that a = 2m + 1, b = 2n + 1 .Thus a + b = (2m + 1) + (2n + 1) = 2(m + n + 1) which is even. This contradicts the assumption that the sum of two odd numbers is not even.
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.
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"
A patient is to receive 2 tablets po ACHS for 30 days. How many tablets should you dispense? DO NOT include units
Answer: We should dispense 60 tablets.
Step-by-step explanation:
Given : A patient is to receive 2 tablets po ACHS for 30 days.
i.e. Dose for each day = 2 tablets
Number of days = 30
If a patient takes 2 tablets each day , then the number of tablets he require for 30 days will be :-
[tex]2\times30=60[/tex] [Multiply 2 and 30]
Therefore, the number of tablets we should dispense = 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.
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
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]
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]
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.
Can 5/8 be estimated to 1/2
Answer:
Step-by-step explanation:
Estimated because there is no number that can go into 5 and 8 evenly
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 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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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]
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.
Use a definition, postulate, or theorem to find the value of x In the figure described
SV is an angle bisector of ZRST. IfmZRSV = (2x +9)° and mZRST = (6x - 26)°, find x.
Select each definition, postulate, or theorem you will use.
A
Angle Addition Postulate
definition of midpoint
C
definition of angle bisector
D
Linear Palr Theorem
The solution is x =
For the following linear system, put the augmented coefficient matrix into reduced row-echelon form.
x1 − 2x2 + 3x3 + 2x4 + x5 = 10
2x1 − 4x2 + 8x3 + 3x4 + 10x5 = 7
3x1 − 6x2 + 10x3 + 6x4 + 5x5 = 27
Answer:
This is the reduced row-echelon form
[tex]\left[\begin{array}{cccccc}1&-2&0&0&3&5\\0&0&1&0&2&-3\\0&0&0&1&-4&7\end{array}\right][/tex]
from the augmented matrix
[tex]\left[\begin{array}{cccccc}1&-2&3&2&1&10\\2&-4&8&3&10&7\\3&-6&10&6&5&27\end{array}\right][/tex]
Step-by-step explanation:
To transform an augmented matrix to the reduced row-echelon form we need to follow this steps:
1. Write the system of equations as an augmented matrix.
The augmented matrix for a system of equations is a matrix of numbers in which each row represents the constants from one equation (both the coefficients and the constant on the other side of the equal sign) and each column represents all the coefficients for a single variable.
[tex]\left[\begin{array}{cccccc}1&-2&3&2&1&10\\2&-4&8&3&10&7\\3&-6&10&6&5&27\end{array}\right][/tex]
2. Make zeros in column 1 except the entry at row 1, column 1 (this is the pivot entry). Subtract row 1 multiplied by 2 from row 2 [tex]\left(R_2=R_2-\left(2\right)R_1\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&3&2&1&10\\0&0&2&-1&8&-13\\3&-6&10&6&5&27\end{array}\right][/tex]
3. Subtract row 1 multiplied by 3 from row 3 [tex]\left(R_3=R_3-\left(3\right)R_1\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&3&2&1&10\\0&0&2&-1&8&-13\\0&0&1&0&2&-3\end{array}\right][/tex]
4. Since element at row 2 and column 2 (pivot element) equals 0, we need to swap rows. Find the first non-zero element in the column 2 under the pivot entry. As can be seen, there are no such entries. So, move to the next column. Make zeros in column 3 except the entry at row 2, column 3 (pivot entry). Divide row 2 by 2 [tex]\left(R_2=\frac{R_2}{2}\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&3&2&1&10\\0&0&1&-1/2&4&-13/2\\0&0&1&0&2&-3\end{array}\right][/tex]
5. Subtract row 2 multiplied by 3 from row 1 [tex]\left(R_1=R_1-\left(3\right)R_2\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&0&7/2&-11&59/2\\0&0&1&-1/2&4&-13/2\\0&0&1&0&2&-3\end{array}\right][/tex]
6. Subtract row 2 from row 3 [tex]\left(R_3=R_3-R_2\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&0&7/2&-11&59/2\\0&0&1&-1/2&4&-13/2\\0&0&0&1/2&-2&7/2\end{array}\right][/tex]
7. Make zeros in column 4 except the entry at row 3, column 4 (pivot entry). Subtract row 3 multiplied by 7 from row 1 [tex]\left(R_1=R_1-\left(7\right)R_3\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&0&0&3&5\\0&0&1&-1/2&4&-13/2\\0&0&0&1/2&-2&7/2\end{array}\right][/tex]
8. Add row 3 to row 2 [tex]\left(R_2=R_2+R_3\righ)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&0&0&3&5\\0&0&1&0&2&-3\\0&0&0&1/2&-2&7/2\end{array}\right][/tex]
9. Multiply row 3 by 2 [tex]\left(R_3=\left(2\right)R_3\right)[/tex]
[tex]\left[\begin{array}{cccccc}1&-2&0&0&3&5\\0&0&1&0&2&-3\\0&0&0&1&-4&7\end{array}\right][/tex]
using a directrixof y = -3 and a focus of (2, 1), what quadratic funtion is created
Answer:
The function is a parabola described by: [tex]y = 1 +\frac{(x-2)^{2}}{8}[/tex]
Step-by-step explanation:
Since a directrix and a focus are given, then we know that they are talking about a parabola.
We know that the distance from any point in a parabola (x, y), to the focus is exactly the same as the distance to the directrix. Therefore, in order to find the required equation, first we will compute the distance from an arbitrary point on the parabola (x, y) to the focus (2, 1):
[tex]d_{pf} = \sqrt{(x-2)^2+(y-1)^2}[/tex]
Next, we will find the distance from an arbitrary point on the parabola (x, y) to the directrix (y= -3). Since we have an horizontal directrix, the distance is easily calculated as:
[tex]d_{pd}=y-(-3)=y+3[/tex]
Finally, we equate the distances in order to find the parabola equation:
[tex]\sqrt{(x-2)^2+(y-1)^2}=y+3[/tex]
[tex](x-2)^2+(y-1)^2=(y+3)^2[/tex]
[tex](x-2)^{2} +y^{2}-2y+1=y^{2}+6y+9[/tex]
[tex](x-2)^{2}-8=8y[/tex]
[tex]y = \frac{(x-2)^{2}}{8}-1[/tex]
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.
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]
Write the base number for each expression: Please I need with steps
1. 5^12
2. 1.2^2
3. (1/3)^4
Answer:
1) Base - 5
2) Base - 1.2
3) Base - [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
To find : Write the base number for each expression ?
Solution :
Base number is defined as the number written in exponent form [tex]a^n[/tex]
which tells you to multiply a by itself, so a is the base of power n.
1) [tex]5^{12}[/tex]
In number [tex]5^{12}[/tex] the base is 5 as the power is 12.
2) [tex]1.2^{2}[/tex]
In number [tex]1.2^{2}[/tex] the base is 1.2 as the power is 2.
3) [tex](\frac{1}{3})^{2}[/tex]
In number [tex](\frac{1}{3})^{2}[/tex] the base is [tex]\frac{1}{3}[/tex] as the power is 4.
The base number for 5^12 is 5, for 1.2^2 is 1.095, and for (1/3)^4 is 0.577.
Explanation:To find the base number for 5^12, we need to determine what number raised to the 12th power gives us 5. Since 5 is a prime number, it cannot be expressed as a perfect power of another number. Therefore, the base number for 5^12 is 5.To find the base number for 1.2^2, we need to determine what number raised to the 2nd power gives us 1.2. We can rewrite 1.2 as 6/5. Taking the square root of 6/5 gives us approximately 1.095. Therefore, the base number for 1.2^2 is 1.095.To find the base number for (1/3)^4, we need to determine what number raised to the 4th power gives us 1/3. We can rewrite 1/3 as 3^(-1). Taking the 4th root of 3^(-1) gives us approximately 0.577. Therefore, the base number for (1/3)^4 is 0.577.Learn more about Exponents here:https://brainly.com/question/5497425
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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]
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.
Step 3: write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1.
Answer: I don't know if you wanted to write sin(4x) or [tex]sin^{4} (x)[/tex] , but here we go:
ok, sin(4x) = sin(2x + 2x), and we know that:
sin (a + b) = sin(a)*cos(b) + sin(b)*cos(a)
then sin (2x + 2x) = sin(2x)*cos(2x) + cos(2x)*sin(2x) = 2cos(2x)*sin(2x)
So 2*cos(2x)*sin(2x) is equivalent of sin(4x)
If you writed [tex]sin^{4} (x)[/tex] then:
[tex]sin^{4} (x) = sin^{2} (x)*sin^{2} (x)[/tex]
and using that: [tex]sins^{2} (x) = \frac{1-cos(2x)}{2}[/tex]
we have: [tex]sin^{2} (x)*sin^{2} (x) = \frac{(1-cos(2x))*(1-cos(2x))}{4} = \frac{1-2cos(2x) + cos^{2}(2x) }{4}[/tex]
and using that: [tex]cos^{2} (x) = \frac{1 + cos(2x)}{2}[/tex]
[tex]\frac{1-2cos(2x) + cos^{2}(2x) }{4} = \frac{1-2cos(2x) + \frac{1+cos(4x)}{2} }{4}[/tex]
You can keep simplifying it, but there is your representation of [tex]sin^{4} (x)[/tex] that does not contain powers of trigonometric functions greater than 1.
To write an equivalent expression for sin(4x) without powers of trigonometric functions greater than 1, use the trigonometric identity sin(2x) = 2sin(x)cos(x). Apply this identity twice to get 4sin(x)cos(x)(cos^2(x) - sin^2(x)).
Explanation:To write an equivalent expression for sin(4x) that does not contain powers of trigonometric functions greater than 1, we can use the trigonometric identity sin(2x) = 2sin(x)cos(x). By applying this identity twice, we get:
sin(4x) = sin(2(2x)) = 2sin(2x)cos(2x)Using the identity sin(2x) = 2sin(x)cos(x) again, we have 2sin(2x)cos(2x) = 2(2sin(x)cos(x))(cos^2(x) - sin^2(x))Simplifying further, we obtain 2(2sin(x)cos(x))(cos^2(x) - sin^2(x)) = 4sin(x)cos(x)(cos^2(x) - sin^2(x))Therefore, an equivalent expression for sin(4x) that does not contain powers of trigonometric functions greater than 1 is 4sin(x)cos(x)(cos^2(x) - sin^2(x)).
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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.
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.
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.
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