Let [tex]\vec h[/tex] and [tex]\vec\eta[/tex] be two vectors in [tex]H[/tex].
[tex]H[/tex] is a subspace of [tex]V[/tex] if (1) [tex]\vec h+\vec\eta\in H[/tex] and (2) for any scalar [tex]k[/tex], we have [tex]k\vec h\in H[/tex].
(1) True;
[tex]\mathrm{tr}(\vec h+\vec\eta)=\mathrm{tr}(\vec h)+\mathrm{tr}(\vec eta)=0[/tex]
so [tex]\vec h+\vec\eta\in H[/tex].
(2) Also true, since
[tex]\mathrm{tr}(k\vec h)=0k=k[/tex]
Therefore [tex]H[/tex] is a subspace of [tex]V[/tex].
Answer: Yes, H is a subspace of V
Step-by-step explanation:
We know that V is the space of all the 2x2 matrices with real entries.
H is the set of all 2x2 matrices with real entries that have trace equal to 0.
Obviusly the matrices that are in the space H also belong in the space V (because in H you have some selected matrices and in V you have all of them). The thing we need to prove is if H is an actual subspace.
Suppose we have two matrices that belong to H, A and B.
We must see that:
1) if A and B ∈ H, then (A + B)∈H
2) for a scalar number k, k*A ∈ H
lets write this as:
[tex]A = \left[\begin{array}{ccc}a1&a2\\a3&a4\\\end{array}\right] B = \left[\begin{array}{ccc}b1&b2\\b3&b4\\\end{array}\right][/tex]
where a1 + a4 = 0 = b1 + b4
then:
[tex]A + B = \left[\begin{array}{ccc}a1 + b1&a2 + b2\\a3 + b3&a4 + b4\\\end{array}\right][/tex]
the trace is:
a1 + b1 - (a4 + b4) = (a1 - a4) + (b1 - b4) = 0
then the trace is nule, and (A + B) ∈ H
and:
[tex]kA = \left[\begin{array}{ccc}k*a1&k*a2\\k*a3&k*a4\end{array}\right][/tex]
the trace is:
k*a1 - k*a4 = k(a1 - a4) = 0
so kA ∈ H
then H is a subspace of V
Answer all questions: 1) The electric field of an electromagnetic wave propagating in air is given by E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10 t -2z) (V/m). Find the associated magnetic field H(z,t)
The magnetic field H(z,t) of an electromagnetic wave is related to the electric field E(z,t) by a factor of the speed of light. Therefore, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is speed of light, approximately 3 x 10^8 m/s.
Explanation:The question is asking for the associated magnetic field H(z,t) of an Electromagnetic wave given the electric field E(z,t). A crucial fact to know for this question is that the electric and magnetic fields in an electromagnetic wave are perpendicular to each other and the direction of propagation. They also have a constant ratio of magnitudes in free space or air, which is the speed of light given by c = 1/√εOMO. Because of these relations, we know that we can find the magnetic field by simply dividing the given electric field by the speed of light in units that match the given Electric field.
So, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), then the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is the speed of light, approximately 3 x 10^8 m/s.
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To find the associated magnetic field H(z, t), you can use Faraday's law of electromagnetic induction. This law states that the rate of change of magnetic flux through a surface is equal to the induced electromotive force (EMF) along the boundary of the surface. By following a step-by-step process, you can find the magnetic field B(z, t) using the given electric field E(z, t).
Explanation:The associated magnetic field H(z, t) can be found by using Faraday's law of electromagnetic induction. Faraday's law states that the rate of change of magnetic flux through a surface is equal to the electromotive force (EMF) along the boundary of the surface. In this case, the magnetic field is changing due to the time-dependent electric field, so we can use Faraday's law to find the magnetic field.
Start by finding the magnetic flux through a surface with an area A in the z-direction.The magnetic field B is perpendicular to the surface, so the magnetic flux is given by Φ = B * A.By Faraday's law, the rate of change of magnetic flux is equal to the induced EMF around the boundary of the surface. In this case, the induced EMF is caused by the changing electric field.From the given electric field E(z, t), we can differentiate it with respect to time to find the rate of change, which gives us the induced EMF.Equating the rate of change of magnetic flux to the induced EMF, we can solve for the magnetic field B(z, t).By following these steps, you can find the associated magnetic field H(z, t) using Faraday's law of electromagnetic induction.
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d) neither one-to-one nor onto. 15. Determine whether each of these functions is a bijection from R to R. a) f(x)=2x+1 b) f(x)=x2+1 c) f(x) r3 d) f(x) (x2 +1)/(r2 +2) a function f(x)-ex from the set of real
Answer:
The only bijection is f(x)=2x+1.
I took r to be a constant.
Step-by-step explanation:
Bijections are both onto and one-to-one.
Onto means every element of the codomain gets hit. Here the codomain is the set of real numbers. So you want every y to get hit.
One-to-one means you don't want any y to get hit more than once.
f(x)=2x+1 is a linear function. It is diagonal line so every element of the codomain will get hit and hit only once so this function is onto and one-to-one.
f(x)=x^2+1 is a quadratic function. It is parabola so not every element of our codomain will get not get hit and of those that do get hit they get hit more than once. So this is neither onto or one-to-one.
f(x)=r^3 is a constant function. It is a horizontal line so not every y will get hit so it isn't onto. The same y is being hit multiple times so it isn't one-to-one.
f(x)=(x^2+1)/(r^2+2) is a quadratic. It is a parabola. Quadratic functions are not onto or one-to-one.
Find all values of x that are NOT in the domain of h.
If there is more than one value, separate them with commas
h(x) = x + 1 / x^2 + 2y + 1
Answer:
if x= -1 then its is NOT in the domain of h.
Step-by-step explanation:
Domain is the set of values for which the function is defined.
we are given the function
[tex]h(x) =\frac{x+1}{x^2 + 2x + 1}[/tex]
Solving the denominator by factorization
[tex]h(x) =\frac{x+1}{x^2 + x+x + 1}\\h(x) =\frac{x+1}{x(x+1)+1(x + 1)}\\h(x) =\frac{x+1}{(x+1)(x+1)}\\h(x) =\frac{1}{(x+1)}[/tex]
So, if x = -1 then its is NOT in the domain of h.
Write the equation that passes through the given point and has the slope indicated: (1, -3); with slope (-3/5)
Answer:
y = -3/5x - 12/5
Step-by-step explanation:
The equation I'm going to give is going to be in slope-intercept form. If you need it in point-slope, I can do so in an edit or the comments.
Slope-intercept form is: y = mx + b where m is the slope, b is the y-intercept.
So let's plug in our given slope:
y = -3/5x + b
Using this, we now plug in our x- and y-coordinates from the given point to solve for b.
-3 = -3/5(1) + b
-3 = -3/5 + b
Add 3/5 to both sides to isolate variable b.
-3 + 3/5 = b
-15/5 + 3/5 = b
-12/5 = b
Plug this new info back into the original equation and your answer is
y = -3/5x - 12/5
Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected (with replacement). Find the probability that for one such phone number, the last four digits include at least one 0.
Answer: 0.3439
Step-by-step explanation:
Total number of digits : 10
The number of digits except zero =9
The probability of selecting non-zero number :-
[tex]\dfrac{9}{10}=0.9[/tex]
Then, the probability that one such phone number , the last four digits do not include any zero:-
[tex]\text{P(no zero )}=(0.9)^4[/tex]
Then , the probability that for one such phone number, the last four digits include at least one 0:-
[tex]\text{P(at-least one zero )}=1-(0.9)^4=0.3439[/tex]
Hence, the probability that for one such phone number, the last four digits include at least one 0 is 0.3439 .
To find the probability of at least one 0 in a four-digit number, one can use the complement of the probability of no 0s occurring in any of the digits. The probability of at least one 0 is approximately 34.39%.
Explanation:The student is inquiring about the probability of a certain event, specifically related to randomly generating telephone numbers. To find the probability that at least one digit is a 0 in a randomly generated four-digit number, we can use the complementary probability approach. The probability of not getting a 0 in one digit is 9/10 since there are 9 other possible digits (1-9). Hence, the probability of not getting a 0 in any of the four digits is (9/10)^4. Subtracting this from 1 will give the probability of having at least one 0 in the four-digit sequence:
Probability of at least one 0 = 1 - (Probability of no 0 in any digit)
Probability of at least one 0 = 1 - (9/10)^4
After calculating, we have:
Probability of at least one 0 = 1 - 0.6561 = 0.3439
Therefore, the probability of having at least one 0 in a randomly chosen telephone number with four digits is approximately 34.39%.
Four marbles are to be selected at random with replacement from a jar that contains 10.0 red marbles, 9.0 blue marbles, 6.0 green marbles, and 7.0 yellow marbles. Find the probability of getting exactly 1.0 yellow marbles.
Answer:
7/32
Step-by-step explanation:
Is the given function phi(x) = x^2 - x^-1 an explicit solution to the linear equation d^2y/dx^2 - 2/x^2 y = 0? Circle your answer. (a) yes (b) no
Answer:
Yes
Step-by-step explanation:
We are given that a function [tex]\phi(x)=x^2-x^{-1}[/tex]
We have to find that given function is an explicit solution to the linear equation
[tex]\frac{d^2y}{dx^2}-\frac{2}{x^2}y=0[/tex]
If given function is an explicit solution of given linear equation then it satisfied the given differential equation
Differentiate w.r.t x
Then we get [tex]\phi'(x)=2x+x^{-2}[/tex]
Again differentiate w.r.t x
Then we get
[tex]\phi''(x)=2-\frac{2}{x^3}[/tex]
Substitute all values in the given differential equation
[tex]2-\frac{2}{x^3}-\frac{2}{x^2}(x^2-x^{-1})[/tex]
=[tex]2-\frac{2}{x^3}-2+\frac{2}{x^3}=0[/tex]
Hence, given function is an explicit solution of given differential equation.
Therefore, answer is yes.
Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.){1/2,1/4,1/6,1/8,1/10,...}a_n = ?
Answer:
[tex]a_{n}=\frac{1}{2n}[/tex] [Where a ≥ 1 ]
Step-by-step explanation:
The pattern of the given sequence is {[tex]\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10},......[/tex]
We have to find a formula for the general term [tex]a_{n}[/tex] of the given sequence.
We can rewrite the terms of the sequence as
[tex]\frac{1}{2}=\frac{1}{(2)(1)}[/tex]
[tex]\frac{1}{4}=\frac{1}{(2)(2)}[/tex]
[tex]\frac{1}{6}=\frac{1}{(2)(3)}[/tex]
[tex]\frac{1}{8}=\frac{1}{(2)(4)}[/tex]
[tex]\frac{1}{10}=\frac{1}{(2)(5)}[/tex]
Now we can write the term [tex]a_{n}[/tex] as
[tex]a_{n}=\frac{1}{2n}[/tex]
Where n = 1, 2, 3, 4, 5......
Calculate the mean, median, and mode for each of the following populations of numbers: (a) 17, 23, 19, 20, 25, 18, 22, 15, 21, 20 N (Population) Mean Median Mode (b) 505, 497, 501, 500, 507, 510, 501 N (Population) Mean Median Mode
Answer: i dont now
Step-by-step explanation:
u have to add them togther i guess
Find X.
A.124
B.138
C.282
D.69
To find x, find the difference between the outer angle and middle angle, then divide by two.
X = (210 - 72) / 2
x = 138 / 2
x = 69
The answer is D.
Answer
subtract 210 with 72 and then divide that by 2 and you get 138. so x=138.
Dana leaves Las Vegas for LA at 2 p.m. driving at 55 mph. At 4 p.m. Lance leaves LA for Las Vegas driving at 45 mph along the same route. If the cities are 260 miles, what time do they meet?
Answer: They meet after 1 hour 42 minutes.
Step-by-step explanation:
Since we have given that
Dana leaves Las Vegas for LA at 2 p.m. driving at 55 mph.
Let the time taken by Dana be 't'.
Distance traveled by Dana would be 55t.
At 4 p.m. Lance leaves LA for Las Vegas driving at 45 mph along the same route.
It means after 2 hours Lance leave for LA.
So, time taken by Lance be 't-2'.
Distance traveled by Lance would be 45(t-2)
Total distance = 260 miles
According to question, it becomes,
[tex]55t+45(t-2)=260\\\\55t+45t-90=260\\\\100t=260-90\\\\100t=170\\\\t=1.7\ hours=1\dfrac{7}{10}=1\ hour\ and\ \dfrac{7\times 60}{10}\ minutes=1\ hour\ 42\ minutes[/tex]
Hence, they meet after 1 hour 42 minutes.
1000 mL of D5W is ordered to be infused over 5 hours. The drop factor is 10 gtt/mL. How many gtt/min should be given to infuse the 1000 mL over 5 hours?
Answer:
flow rate of to infuse the 1000 mL over 5 hours is 33.33 gtt/min
Step-by-step explanation:
Given data
volume = 1000 mL
time = 5 hours
drop factor = 10 gtt/min
to find out
flow rate of to infuse the 1000 mL over 5 hours
Solution
we know flow rate formula i.e.
flow rate = ( volume × drop factor ) / time .................1
here time will be in min so time = 5 hours = 5 × 60 = 300 min
put volume, drop factor and time value in equation 1 and we get flow rate
flow rate = ( volume × drop factor ) / time
flow rate = ( 1000 × 10 ) / 300
flow rate = 33.33 gtt/min
flow rate of to infuse the 1000 mL over 5 hours is 33.33 gtt/min
what is the solution if the inequality shown below? a-1>11
Answer:
a > 12
Step-by-step explanation:
Isolate the variable a. Treat the > sign like the = sign. What you do to one side, you do to the other. Add 1 to both sides:
a - 1 (+1) > 11 (+1)
a > 11 + 1
a > 12
a > 12 is your answer.
~
Answer:
[tex]\huge \boxed{a>12}[/tex]
Step-by-step explanation:
Add by 1 from both sides.
[tex]\displaystyle a-1+1>11+1[/tex]
Simplify, to find the answer.
[tex]\displaystyle 11+1=12[/tex]
[tex]\displaystyle a>12[/tex], which is our answer.
Use a substitution method to solve both of the following DEs, stating the general solution clearly, and showing all work clearly. a. dy/dx - y = e^xy^2 (Solve explicitly for y.) b. dy/dx = x + y/x - y (You can leave the General Solution in implicit form.)
[tex]1.\rightarrow \frac{dy}{dx}-y=e^x y^2\\\\\rightarrow \frac{1}{y^2}\frac{dy}{dx}-\frac{1}{y}=e^x\\\\ \text{put},\frac{-1}{y}=z\\\\ \frac{dy}{y^2} =d z\\\\ \frac{dy}{dx} \times \frac{1}{y^2}=\frac{dz}{dx}\\\\\frac{dz}{dx} +z=e^x\\\\ \text{Integrating factor}=e^{\int {1} \, dx}\\\\=e^x \\\\ \text{Multiplying both sides by }e^x\\\\e^x(\frac{dz}{dx} +z)=e^{2x}\\\\ \text{Integrating both sides}\\\\z e^x=\frac{e^{2x}}{2}+C\\\\ \frac{-e^x}{y}=\frac{e^{2x}}{2}+C[/tex]
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[tex]\rightarrow \frac{dy}{dx}=x+\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}-x=\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}+y(1-\frac{1}{x})=x\\\\\text{Integrating factor}=e^{\int{1-\frac{1}{x}}\,dx}\\\\=e^{x-\log x}\\\\ \text{Multiplying both sides by} e^{x-\log x}\\\\e^{x-\log x}\times[\frac{dy}{dx}+y(1-\frac{1}{x})]=x \times e^{x-\log x}\\\\y\times e^{x-\log x} =\int x \times e^{x-\log x} \, dx}\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{e^{\log x}}\,dx\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{x} \, dx\\\\y\times e^{x-\log x}=e^x+K[/tex]
Find the least squares approximation of the the data (0, 1), (1, 2), (2, 1/2) (3, 3) using the quadratic function p(x) = a_0 + a_1 x + a_2 x^2. Plot p(x) along with the data to compare.
Answer:
The required function is [tex]p\left(x\right)=1.325-0.675x+0.375x^2[/tex].
Step-by-step explanation:
The given data points are (0, 1), (1, 2), (2, 1/2) and (3, 3).
Let the quadratic function is defined as
[tex]p(x)=a_0+a_1x+a_2x^2[/tex] .... (1)
Using graphing calculator, we get
[tex]a_0=1.325[/tex]
[tex]a_1=-0.675[/tex]
[tex]a_2=0.375[/tex]
Substitute [tex]a_0=1.325[/tex], [tex]a_1=-0.675[/tex] and [tex]a_2=0.375[/tex] in function (1), to find the quadratic function.
[tex]p\left(x\right)=1.325-0.675x+0.375x^2[/tex]
Therefore the required function is [tex]p\left(x\right)=1.325-0.675x+0.375x^2[/tex].
The graph of data points and quadratic function is shown below.
â(Future valueâ) Sarah Wiggum would like to make a singleâ lump-sum investment and have â$ 1.7 million at the time of her retirement in 34 years. She has found a mutual fund that expects to earn 8 percent annually. How much must Sarah investâ today? If Sarah earned an annual return of 16 16 âpercent, how much must she investâ today? a. If Sarah can earn 8 percent annually for the next 34 âyears, how much will she have to investâ today? â $ nothing â(Round to the nearestâ cent.)
Answer:
a) at 16%: $10,936.47
b) at 8%: $124,177.02
Step-by-step explanation:
At annual rate of return "r", the multiplier of Sarah's initial investment will be ...
k = (1+r)^34
For r = 0.16, k ≈ 155.433166, and Sarah's investment needs to be ...
$1.7·10^6/k ≈ $10,936.47
__
For r= 0.08, k ≈ 13.6901336, and Sarah's investment needs to be ...
$1.7·10^6/k ≈ $124,177.02
Find the? inverse, if it? exists, for the given matrix.
@MATX{{3;3;-1};{-12;-12;4};{2;6;0}}
Answer:
Step-by-step explanation:
The Inverse of the matrix doesn't exist because the determinant is equal to 0.
Answer:
The inverse of given matrix is not exist, since determinant is 0.
Step-by-step explanation:
The inverse of a square matrix [tex]A[/tex] is [tex]A^{-1}[/tex] such that
[tex]A A^{-1}=I[/tex] where I is the identity matrix.
Consider, [tex]A = \left[\begin{array}{ccc}3&3&-1\\-12&-12&4\\2&6&0\end{array}\right][/tex]
[tex]\mathrm{Matrix\:can\:only\:be\:inverted\:if\:it\:is\:non-singular,\:that\:is:}[/tex]
[tex]\det \begin{pmatrix}3&3&-1\\ -12&-12&4\\ 2&6&0\end{pmatrix}\ne 0[/tex]
[tex]\det \begin{pmatrix}3&3&-1\\ -12&-12&4\\ 2&6&0\end{pmatrix}[/tex]
[tex]\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \:[/tex]
[tex]\det \begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ g&h\end{pmatrix}[/tex]
[tex]=3\cdot \det \begin{pmatrix}-12&4\\ 6&0\end{pmatrix}-3\cdot \det \begin{pmatrix}-12&4\\ 2&0\end{pmatrix}-1\cdot \det \begin{pmatrix}-12&-12\\ 2&6\end{pmatrix}[/tex]
[tex]=3\left(-24\right)-3\left(-8\right)-1\cdot \left(-48\right)[/tex]
[tex]3\left(-24\right)-3\left(-8\right)-1\cdot \left(-48\right)=0[/tex]
Therefore, the inverse of given matrix is not exist, since determinant is 0.
M1Q7.) Construct a box plot from the data below
There are 16 numbers.
The median is 92.5 ( find the middle two values and divide by 2).
Minumum is 81
Maximum is 109
First quartile is 88.25 (Find median of the lower half of numbers).
Third quartile is 97.75 (Find median of the upper half of numbers.)
The interquartile range is 9.5 ( Difference between the first and third quartile).
Plotting that data in a box plot, the correct one looks like #1
Find the median.
92 and 93 are both middle numbers so add and divide by two.
92 + 93 = 185
185 / 2 = 92.5
Minimum (smallest number): 81
Maximum (largest number): 109
Find the median of the lower values behind the median.
88.25
Find the mean of the higher values ahead of the median.
97.75
Subtract to find the interquartile range.
97.75 - 88.25 = 9.5
The only option with these characteristics is Option A.
Best of Luck!
A new drug on the market is known to cure 20% of patients with breast cancer. If a group of 20 patients is randomly
selected, what is the probability of observing, at most, one patient who will be cured of breast cancer?
A• (20)/1 (0-20)^1 (0.80)^19
B. 1-(20)/1 (.20)'(0.230)
C. (20)/0( .80)^20+(20/1)(.20)^1(.80)^19
D • (20)/0 (.80)^20
E. 1-(20/0)(.80)^20
01-(20)10.2010
Answer:
It's actually C
Step-by-step explanation:
don't forget about the probability of 0 too
you have to add the two probability formulas
The probability of observing, at most, one patient who will be cured of breast Cancer will be P = (20 / q²⁰) + 20 / (0.20 x 0.80¹⁹). Then the correct option is C.
What is probability?Its basic premise is that something will almost certainly happen. The percentage of favorable events to the total number of occurrences.
A new drug on the market is known to cure 20% of patients with breast cancer.
p = 0.20
q = 1 – 0.20
q = 0.80
If a group of 20 patients is randomly selected.
The probability of observing, at most, one patient who will be cured of breast Cancer will be
P = (20 / q²⁰) + 20 / (0.20 x 0.80¹⁹)
Then the correct option is C.
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A medical laboratory tested 8 samples of human blood for acidity on the pH scale, with the results below. 7.1 7.5 7.6 7.4 7.3 7.3 7.3 7.5 a. Find the mean and standard deviation. b. What percentage of the data is within 2 standard deviations of the mean?
Answer:
Mean = 7.38
SD = 0.148
b. 95%
Step-by-step explanation:
Given data is:
7.1 7.5 7.6 7.4 7.3 7.3 7.3 7.5
Mean:
Mean = Sum/No. of values
= (7.1+7.5+7.6+7.4+7.3+7.3+7.3+7.5)/8
=59/8
=7.38
Standard Deviation:
x x-x' (x-x')^2
7.1 -0.28 0.0784
7.5 0.12 0.0144
7.6 0.22 0.0484
7.4 0.02 0.0004
7.3 -0.08 0.0064
7.3 -0.08 0.0064
7.3 -0.08 0.0064
7.5 0.12 0.0144
Total : 0.1752
Variance = Sum of squares/No of items
= 0.1752/8 = 0.0219
SD =√0.0219 = 0.148
b. What percentage of the data is within 2 standard deviations of the mean?
95% of data is within two standard deviations of mean in a standard normal distribution ..
The mean of the pH test results is 7.375, and the standard deviation is approximately 0.086.
Explanation:
To find the mean and standard deviation of the pH test results, we can use the following formulas:
Mean: Add up all the pH values and divide by the total number of samples (in this case, 8). So, (7.1 + 7.5 + 7.6 + 7.4 + 7.3 + 7.3 + 7.3 + 7.5) / 8 = 7.375.
Standard Deviation: Calculate the difference between each pH value and the mean, square each difference, calculate the mean of those squared differences, and then take the square root. Let's break it down into steps: Subtract the mean from each pH value: (7.1 - 7.375), (7.5 - 7.375), (7.6 - 7.375), (7.4 - 7.375), (7.3 - 7.375), (7.3 - 7.375), (7.3 - 7.375), (7.5 - 7.375). Square each difference: (0.0425)^2, (0.125)^2, (0.225)^2, (0.025)^2, (-0.075)^2, (-0.075)^2, (-0.075)^2, (0.125)^2. Calculate the mean of the squared differences: (0.0018 + 0.0156 + 0.0506 + 0.000625 + 0.005625 + 0.005625 + 0.005625 + 0.0156) / 8 = 0.0074. Take the square root of the mean: √0.0074 ≈ 0.086.
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If possible, find a matrix B such that AB = A2 + 2A.
Answer:
[tex]\large\boxed{B=A+2I}[/tex]
Step-by-step explanation:
It's possible if dimensions of a matrix A and matrix B are n × n
[tex]AB=A^2+2A\qquad\text{multiply both sides on the left by}\ A^{-1}\\\\A^{-1}AB=A^{-1}A^2+A^{-1}(2A)\qquad\text{we know}\ A^{-1}A=I\\\\IB=A^{-1}A\cdot A+2A^{-1}A\\\\IB=IA+2I\qquad\text{we know}\ IA=A\\\\B=A+2I[/tex]
Matrices is an array of numbers, usually 2 dimensional, but can be single dimensional too.
A matrix B such that [tex]AB = A^2 + 2A[/tex] is given as
[tex]B = A + 2I = \left[\begin{array}{cc}4&0\\0&4\end{array}\right][/tex]
(Assuming A is left invertible and [tex]A = \left[\begin{array}{cc}2&0\\0&2\end{array}\right][/tex])
When can we cancel out matrix multiplied on both sides of an equation?Suppose that there is an equation
[tex]AB = AC[/tex]
We cannot always say that [tex]B = C[/tex]
If we assume that A is left invertible, then only we can surely say that we have got [tex]B = C[/tex]
Similarly, for [tex]BA = CA[/tex] to imply [tex]B = C[/tex], we need A to be right invertible.
Assuming that we have A as a left invertible matrix, say
[tex]A = \left[\begin{array}{cc}2&0\\0&2\end{array}\right][/tex]
and [tex]L_A[/tex] be its left inverse, then [tex]L_A A = I[/tex] ([tex]I[/tex] is identity matrix)
Then,
[tex]AB = A^2 + 2A\\A(B) = A(A + 2I_2)\\\\\Multiplying L_{A}\text{ on left side of both terms,}\\\\L_{A} AB = L_{A}A(A + 2I_2)\\B = A + 2I_2\\\\B = \left[\begin{array}{cc}2&0\\0&2\end{array}\right] + \left[\begin{array}{cc}2&0\\0&2\end{array}\right] = \left[\begin{array}{cc}4&0\\0&4\end{array}\right] = 4I_2\\\\B = 4I_2[/tex]
Thus, i
A matrix B such that [tex]AB = A^2 + 2A[/tex] is given as
[tex]B = \left[\begin{array}{cc}4&0\\0&4\end{array}\right][/tex]
(Assuming A is left invertible and [tex]A = \left[\begin{array}{cc}2&0\\0&2\end{array}\right][/tex])
Learn more about invertible matrices here:
https://brainly.com/question/17027442
Show that the differential equation (on the left) is a solution of the function (on the right)
d^2u/dt^2 = a^2 * (d^2u/dx^2) u(x,t) = f(x-at) + g(x+at)
We have to show that
[tex]\frac{\partial ^{2}u}{\partial t^{2}}=a^{2}\frac{\partial ^{2}u}{\partial x^{2}}[/tex]
for [tex]\frac{\partial ^{2}u}{\partial t^{2}}[/tex] we have
[tex]\frac{\partial ^{2}u}{\partial t^{2}}=a^{2}\frac{\partial ^{2}u}{\partial x^{2}}[/tex]
[tex]\frac{\partial ^{2}u}{\partial t^{2}}=\frac{\partial ^{2}[f(x-at)+g(x+at)]}{\partial t^{2}}[/tex]
[tex]=\frac{\partial }{\partial t}[\frac{\partial[f(x-at)+g(x+at)] }{\partial t}][/tex]
[tex]\frac{\partial }{\partial t}[-a\cdot f'(x-at)+a\cdot g'(x+at)][/tex]
[tex]=a^{2}f''(x-at)+a^{2}g''(x+at)[/tex]
[tex]=a^{2}[f''(x-at)+g''(x+at)].............(i)[/tex]
similarly,
[tex]\frac{\partial ^{2}u}{\partial x^{2}}=\frac{\partial ^{2}[f(x-at)+g(x+at)]}{\partial x^{2}}[/tex]
[tex]=\frac{\partial }{\partial x}[\frac{\partial[f(x-at)+g(x+at)] }{\partial x}][/tex]
[tex]=\frac{\partial }{\partial x}[f'(x-at)+g'(x+at)][/tex]
[tex]=f''(x-at)+g''(x+at).......(ii)[/tex]
Comparing i and ii we get
[tex]a^{2}\frac{\partial ^{2}u}{\partial x^{2}}=\frac{\partial ^{2}u}{\partial t^{2}}[/tex]
Hence proved
A 20% TIP ON A MEAL THAT COSTS $29.17. CHOOSE THE CORRECT ESTIMATE BELOW. A.$ 58.00 B.$ 5.80 C.$ 0.58 D. $ 8.70
Answer:
$5.80 Option B.
Step-by-step explanation:
It is given that a 20% tip on a meal that costs $29.17.
The cost of the meal = $29.17
Tip on a meal = 20%
Therefore, 20% of $29.17
= [tex]\frac{20}{100}[/tex] × 29.17
= 0.20 × 29.17
= 5.834
= $5.80
The correct estimate would be $5.80 Option B.
Assume that when adults with smartphones are randomly selected, 51% use them in meetings or classes. If 11 adult smartphone users are randomly selected, find the probability that fewer than 5 of them use their smartphones in meetings or classes.
Answer:
The probability is 0.2356.
Step-by-step explanation:
Let X is the event of using the smartphone in meetings or classes,
Given,
The probability of using the smartphone in meetings or classes, p = 51 % = 0.51,
So, the probability of not using smartphone in meetings or classes, q = 1 - p = 1 - 0.51 = 0.49,
Thus, the probability that fewer than 5 of them use their smartphones in meetings or classes.
P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3)+P(X=4)
Since, binomial distribution formula is,
[tex]P(x) = ^nC_r p^x q^{n-x}[/tex]
Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Here, n = 11,
Hence, the probability that fewer than 5 of them use their smartphones in meetings or classes
[tex]=^{11}C_0 (0.5)^0 0.49^{11}+^{11}C_1 (0.5)^1 0.49^{10}+^{11}C_2 (0.5)^2 0.49^{9}+^{11}C_3 (0.5)^3 0.49^{8}+^{11}C_4 (0.5)^4 0.49^{7} [/tex]
[tex]=(0.5)^0 0.49^{11}+11(0.5)0.49^{10} + 55(0.5)^2 0.49^{9}+165 (0.5)^3 0.49^{8} +330(0.5)^4 0.49^{7} [/tex]
[tex]=0.235596671797[/tex]
[tex]\approx 0.2356[/tex]
A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 7% vinegar, and the second brand contains 12% vinegar. The chef wants to make 270 milliliters of a dressing that is 9% vinegar. How much of each brand should she use?
Answer: There is 162 ml of first brand and 108 ml of second brand.
Step-by-step explanation:
Since we have given that
Percentage of vinegar that the first brand contains = 7%
Percentage of vinegar that the second brand contains = 12%
Percentage of vinegar in mixture = 9%
Total amount of dressing = 270 ml
We will use "Mixture and Allegation":
First brand Second brand
7% 12%
9%
--------------------------------------------------------
12%-9% : 9%-7%
3% : 2%
So, ratio of first brand to second brand in a mixture is 3:2.
So, Amount of first brand she should use is given by
[tex]\dfrac{3}{5}\times 270\\\\=162\ ml[/tex]
Amount of second brand she should use is given by
[tex]\dfrac{2}{5}\times 270\\\\=108\ ml[/tex]
Hence, there is 162 ml of first brand and 108 ml of second brand.
Completion time (from start to finish) of a building remodeling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days. To be 99% sure that we will not be late in completing the project, we should request a completion time of _______ work-days.
Answer:
233 days.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 200, \sigma = 10[/tex]
To be 99% sure that we will not be late in completing the project, we should request a completion time of ...
This is the value of X when Z has a pvalue of 0.99. So X when Z = 2.325.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.325 = \frac{X - 200}{10}[/tex]
[tex]X - 200 = 10*2.325[/tex]
[tex]X = 232.5[/tex]
So the correct answer is 233 days.
The manufacturer of a certain engine treatment claims that if you add their product to your engine, it will be protected from excessive wear. An infomercial claims that a woman drove 33 hours without oil, thanks to the engine treatment. A magazine tested engines in which they added the treatment to the motor oil, ran the engines, drained the oil, and then determined the time until the engines seized. Determine the null and alternative hypotheses that the magazine will test.
Answer: [tex]H_0:\mu\geq33[/tex]
[tex]H_a:\mu<33[/tex]
Step-by-step explanation:
Let [tex]\mu[/tex] be the average number of hours a person drive without adding the product.
Given claim : An infomercial claims that a woman drove 33 hours without oil.
i.e. [tex]\mu\geq33[/tex]
It is known that the null hypothesis always contains equal sign and alternative hypothesis is just opposite of the null hypothesis.
Thus the null and alternative hypothesis for the given situation will be :-
[tex]H_0:\mu\geq33[/tex]
[tex]H_a:\mu<33[/tex]
Give the largest interval I over which the general solution is defined. PLEASE EXPLAIN HOW!!!
(x^2-1)dy/dx+2y=(x+1)^2
Divide both sides by [tex]x^2-1[/tex] to get a linear ODE,
[tex]\dfrac{\mathrm dy}{\mathrm dx}+\dfrac2{x^2-1}y=\dfrac{x+1}{x-1}[/tex]
In order for this operation to be valid in the first place, we require that [tex]x\neq\pm1[/tex] (since that would make [tex]\dfrac1{x^2-1}[/tex] undefined, which we don't want to happen). Then we are forcing any solution to the ODE to exist on any of the three intervals, [tex](-\infty,-1)[/tex], [tex](-1, 1)[/tex], or [tex](1,\infty)[/tex], and either the first or third of these can be chosen as the largest interval.
In case you also need to solve the ODE: Multiply both sides by [tex]\dfrac{1-x}{1+x}[/tex], so that
[tex]\dfrac{1-x}{1+x}\dfrac{\mathrm dy}{\mathrm dx}-\dfrac2{(1+x)^2}y=-1[/tex]
Then the left side can be condensed as the derivative of a product, since
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}\right]=-\dfrac2{(1+x)^2}[/tex]
and we have
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}y\right]=-1[/tex]
Integrate both sides:
[tex]\displaystyle\int\frac{\mathrm d}{\mathrm dx}\left[\frac{1-x}{1+x}y\right]\,\mathrm dx=-\int\mathrm dx[/tex]
[tex]\dfrac{1-x}{1+x}y=-x+C[/tex]
[tex]\implies\boxed{y=\dfrac{(-x+C)(1+x)}{1-x}}[/tex]
The largest interval over which the general solution is defined for the given differential equation is [-1, ∞).
Here's how:
Rewrite the differential equation in proper form.Analyze the coefficients to determine the interval of definition.In this case, the interval is determined by the denominator of the coefficient of dy/dx.. CAR WASH Shea and Tucker are washing their father's car. Shea can wash it by herself in 20 minutes. Tucker can wash it by himself in 30 minutes. 19 How long does it take them to wash the car if they work together?
Answer:
12 minutes
Step-by-step explanation:
First we figure out how much of a car each person can wash in 1 minute.
Shea can wash the car by herself in 20 minutes
Therefore, she can wash a car in 1 minute = [tex]\frac{1}{20}[/tex]
Tucker can wash a car in 30 minutes.
Tucker can wash the car in 1 minute = [tex]\frac{1}{30}[/tex]
Thus, in one minute together they wash a car
[tex]\frac{1}{20}[/tex] + [tex]\frac{1}{30}[/tex] = [tex]\frac{50}{600}[/tex] = [tex]\frac{1}{12}[/tex]
In one minute together they can wash = [tex]\frac{1}{12}[/tex] of a car
Time needed to wash entire car together = 12 minutes.
It takes them 12 minutes to wash the car.
Final answer:
Shea and Tucker can wash their father's car in 12 minutes if they work together.
Explanation:
To find out how long it takes Shea and Tucker to wash the car together, we can use the concept of work rates.
Shea can wash the car by herself in 20 minutes, so her work rate is 1/20 of the car per minute.
Tucker can wash the car by himself in 30 minutes, so his work rate is 1/30 of the car per minute.
When two people work together, their work rates are additive. So, if Shea and Tucker work together, their combined work rate is 1/20 + 1/30 = 1/12 of the car per minute.
Since the work rate is the reciprocal of the time taken, the time it takes them to wash the car together is 12 minutes. Therefore, it takes Shea and Tucker 12 minutes to wash their father's car if they work together.
A civil service exam yields scores which are normally distributed with a mean of 81 and a standard deviation of 5.5. If the civil service wishes to set a cut-off score on the exam so that 15% of the test takers fail the exam, what should the cut-off score be? Remember to round your z-value to 2 decimal places.
Answer:
The cutoff score should be 75.8 marks
Step-by-step explanation:
The cut-off should be set as the value corresponding to an area 15% in the normal distribution diagram.
Using the standard distribution tables we have value of standard normal deviate (Z) corresponding to area of 15% = -1.04
Thus we have
[tex]-1.04=\frac{X-\bar{X}}{\sigma }\\\\X=-1.04\times 5.5+81\\X=75.8[/tex]