Answer: P means "This statement is false"
then, P is a "function" of some statement,
if i write P( 3> 1932) this could be read as:
3>1932, this statement is false.
You could see that 3> 1932 is false, so P( 3>1932) is true.
Then you could se P(x) at something that is false if x is true, and true if x is false, so p is a negation.
Find all solutions of the equation algebraically. Check your solutions. (Enter your answers as a comma-separated list x^4-7x^2-144=0
Answer:
The solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]
Step-by-step explanation:
Consider the provided equation.
[tex]x^4-7x^2-144=0[/tex]
Substitute [tex]u=x^2\mathrm{\:and\:}u^2=x^4[/tex]
[tex]u^2-7u-144=0[/tex]
[tex]u^2-16u+9u-144=0[/tex]
[tex](u-16)(u+9)=0[/tex]
[tex]u=16,\:u=-9[/tex]
Substitute back [tex]\:u=x^2[/tex] and solve for x.
[tex]x^2=16\\x=\sqrt{16}\\ \quad x=4,\:x=-4[/tex]
Or
[tex]x^2=-9\\x=\sqrt{-9}\\ \quad x=3i,\:x=-3i[/tex]
Hence, the solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]
Check:
Substitute x=4 in provided equation.
[tex]4^4-7(4)^2-144=0[/tex]
[tex]256-112-144=0[/tex]
[tex]0=0[/tex]
Which is true.
Substitute x=-4 in provided equation.
[tex](-4)^4-7(-4)^2-144=0[/tex]
[tex]256-112-144=0[/tex]
[tex]0=0[/tex]
Which is true.
Substitute x=3i in provided equation.
[tex](3i)^4-7(3i)^2-144=0[/tex]
[tex]81+63-144=0[/tex]
[tex]0=0[/tex]
Which is true.
Substitute x=-3i in provided equation.
[tex](-3i)^4-7(-3i)^2-144=0[/tex]
[tex]81+63-144=0[/tex]
[tex]0=0[/tex]
Which is true.
A company makes greeting cards and their research shows that that price and demand are related linearly: p = mx + b. They know that for every additional card they wish to sell they need to drop the price by $0.05. They also know that in order to sell 360 cards they need to set the price at $4. Find the linear equation relating price to demand. p= 0 Preview
Answer:
The linear equation relating price to demand is [tex]p=-0.05x+360.2[/tex]
Step-by-step explanation:
A Linear Demand Function expresses demand p (the number of items demanded) as a function of the unit price x (the price per item).
From the information given we know two facts:
For every additional card sold, the price decrease $0.05To sell 360 cards they need to set the price at $4Let x be the price and p the number of cards sold.
The fact 1. is the slope of the function because is the change in demand per unit change in price.
We can use fact 2. to find the value of b in the equation
[tex]p=-0.05x+b\\b=p+0.05x[/tex]
[tex]b=360+0.05\cdot 4\\b=360+0.2\\b=360.2[/tex]
The linear equation relating price to demand is [tex]p=-0.05x+360.2[/tex]
In an arithmetic sequence, the nth term an is given by the formula An=a1+(n−1)d, where a1
is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by an=a1•rn−1.
Here r is the common ratio. Use these formulas to determine the indicated term in the given sequence.
The 30th term of 1, 4, 7, 10,...
Answer:
88
Step-by-step explanation:
We are given that in arithmetic sequence , the nth term [tex]a_n[/tex] is given by the formula
[tex]A_n=a_1+(n-1)d[/tex]
Where [tex]a_1=first term[/tex]
d=Common difference
In an geometric sequence, the nth term is given by
[tex]a_n=a_1r^{n-1}[/tex]
Where r= Common ratio
1,4,7,10,..
We have to find 30th term.
[tex]a_1=1,a_2=4,a_3=7,a_4=10[/tex]
[tex]d=a_2-a_1=4-1=3[/tex]
[tex]d=a_3-a_2=7-4=3[/tex]
[tex]d=a_4-a_3=10-7=3[/tex]
[tex]r_1=\frac{a_2}{a_1}=\frac{4}{1}=4[/tex]
[tex]r_2=\frac{a_3}{a_2}=\frac{7}{4}[/tex]
[tex]r_1\neq r_2[/tex]
Therefore, given sequence is an arithmetic sequence because the difference between consecutive terms is constant.
Substitute n=30 , d=3 a=1 in the given formula of arithmetic sequence
Then, we get
[tex]a_{30}=1+(30-1)(3)=1+29(3)=1+87=88[/tex]
Hence, the 30th term of sequence is 88.
In this card game, a player is dealt 10 cards. A player wins $1,000,000 if he is dealt 8 hearts. Use combinatorics to set up your problem a. Calculate the probability a player is dealt exactly 8 hearts b. Calculate the probability a player is dealt exactly 8 hearts if the first tweo cards he was dealt were diamonds. 2
Assume a standard deck of 52 cards with 4 suits of 13 cards each.
a. There are [tex]\dbinom{13}8\dbinom{39}2[/tex] ways of being dealt a hand consisting of 8 hearts and 2 non-hearts, so the probability of being dealt such a hand is
[tex]\dfrac{\dbinom{13}8\dbinom{39}2}{\dbinom{52}{10}}\approx0.0000602823[/tex]
b. This time, the non-hearts specifically belong to the suit of diamonds, for which there are [tex]\dbinom{13}2[/tex] ways of getting drawn, so the probability is
[tex]\dfrac{\dbinom{13}8\dbinom{13}2}{\dbinom{52}{10}}\approx0.0000063455[/tex]
a. The probability P of being dealt exactly 8 hearts is:
[tex]\[ P = \frac{k}{n} = \frac{\binom{13}{8} \times \binom{39}{2}}{\binom{52}{10}} \][/tex]
b. The probability P' of being dealt exactly 8 hearts given the first two cards were diamonds is:
[tex]\[ P' = \frac{k'}{n'} = \frac{\binom{8}{6} \times \binom{42}{2}}{\binom{50}{8}} \][/tex]
To solve this problem using combinatorics, we can calculate the probability by considering the total number of possible outcomes and the number of favorable outcomes.
Let's denote:
- n as the total number of ways to deal 10 cards from a standard deck (52 cards).
- k as the number of ways to deal exactly 8 hearts and 2 non-hearts from the remaining 44 cards in the deck.
a. To calculate the probability a player is dealt exactly 8 hearts:
Total number of ways to choose 8 hearts out of 13 hearts:
[tex]\[ \binom{13}{8} \][/tex]
Total number of ways to choose 2 non-hearts out of 39 non-hearts:
[tex]\[ \binom{39}{2} \][/tex]
Therefore, the number of favorable outcomes is:
[tex]\[ k = \binom{13}{8} \times \binom{39}{2} \][/tex]
The total number of ways to deal 10 cards from a deck of 52 cards is:
[tex]\[ n = \binom{52}{10} \][/tex]
So, the probability P of being dealt exactly 8 hearts is:
[tex]\[ P = \frac{k}{n} = \frac{\binom{13}{8} \times \binom{39}{2}}{\binom{52}{10}} \][/tex]
b. To calculate the probability a player is dealt exactly 8 hearts if the first two cards dealt were diamonds:
If the first two cards are diamonds, then there are 50 cards remaining, out of which 8 are hearts and 42 are non-hearts.
Total number of ways to choose 6 more hearts out of the remaining 8 hearts:
[tex]\[ \binom{8}{6} \][/tex]
Total number of ways to choose 2 non-hearts out of the remaining 42 non-hearts:
[tex]\[ \binom{42}{2} \][/tex]
Therefore, the number of favorable outcomes is:
[tex]\[ k' = \binom{8}{6} \times \binom{42}{2} \][/tex]
The total number of ways to deal 8 cards from the remaining 50 cards is:
[tex]\[ n' = \binom{50}{8} \][/tex]
So, the probability P' of being dealt exactly 8 hearts given the first two cards were diamonds is:
[tex]\[ P' = \frac{k'}{n'} = \frac{\binom{8}{6} \times \binom{42}{2}}{\binom{50}{8}} \][/tex]
Given a set Ω, let P(Ω) denote the the power set of Ω, that is P(Ω) is the collection of all subsets of Ω. Prove that Ω and P(Ω) do not have the same cardinality. Hint: Given a function Φ : Ω → P(Ω), consider the set X := {ω ∈ Ω : ω /∈ Φ(ω)}.
Step-by-step explanation:
As the hint says, for any function [tex]f:\Omega\to\mathcal{P}(\Omega)[/tex], we can think of the set [tex] X=\{ \omega\in\Omega : \omega \notin f(\omega)\}[/tex] (which is the set of all those elements of [tex]\Omega[/tex] which don't belong to their image). So [tex]X[/tex] is made of elements of [tex]\Omega[/tex], and so it belongs to [tex]\mathcal{P}(\Omega)[/tex].
Now, this set [tex]X[/tex] is NOT the image of any element in [tex] \Omega[/tex], since if there was some [tex]a\in\Omega[/tex] such that [tex]f(a)=X[/tex], then the following would happen:
If [tex]a\in X=f(a)[/tex], then by definition of the set [tex]X[/tex], [tex]a\notin f(a)[/tex], so we're getting that [tex]a\in f(a)[/tex] and also [tex] a\notin f(a)[/tex], which is a contradiction.
On the other hand, if [tex]a\notin f(a)[/tex], then by definition of the set [tex]X[/tex], we would get that [tex]a\in X=f(a)[/tex], so we're getting that [tex]a\in f(a)[/tex] and also [tex] a\notin f(a)[/tex], which is a contradiction again.
So in any case, the assumption that this set [tex]X[/tex] is the image of some element in [tex]\Omega[/tex] leads us to a contradiction, therefore this set [tex]X[/tex] is NOT the image of any element in [tex]\Omega[/tex], and so there cannot be a bijection from [tex]\Omega[/tex] to [tex]\mathcal{P}(\Omega)[/tex], and so the two sets cannot have the same cardinality.
List the digit that has the given place value for the number
59.3274659.32746.
Thousandths:
Ten-thousandths:
Tenths:
Answer:
Thousandths: 7
Ten-thousandths: 4
Tenths: 3
Step-by-step explanation:
The given number is 59.3274659
The place values of a numner is given in the followwing format:
Tens Ones.Tenths Hundredths Thousandths Tenthousandths and so on
We can fit our number
Tens Ones.Tenths Hundredths Thousandths Tenthousandths HT M
5 9 . 3 2 7 4
Thousandths: 7
Ten-thousandths: 4
Tenths: 3
Is .3 greater than .32
Answer:
No
.3 = .30
.32 > .30
Step-by-step explanation:
Consider the following data points.
P1(1, 3), P2(2, 4), P3(3, 5), P4(4, 7), P5(5, 8)
(a) Find the equation of the least-squares line for the data.
y(x) =
Answer:
the equation of the least-squares line for the data is: [tex]\hat Y=9.3+1.3x[/tex]
Step-by-step explanation:
In a simple linear regression model, such as, [tex]\hat Y=b_0+b_1x[/tex], the coefficients bo and b1 are estimated through the method of least squares by the use of the equations:
[tex]b_1=\frac{S{xy}}{S_x^2}\\\\b_0=\bar{y}+b_1 \bar{x}[/tex]
For the data provided you have to:
[tex]S_{xy}=\frac{\sum {(x_i-\bar x)(y_i-\bar y)}}{n-1}=3.25\\\\S_x^2=\frac{\sum {(x_i-\bar x)^2}}{n-1}=2.5\\\\\bar y=5.4[/tex], thus:
[tex]b_1=\frac{3.25}{2,5}=1.3\\\\b_0=5.4+1.3(3.0)=9.3[/tex]
the equation of the least-squares line for the data is:
[tex]Y=9.3+1.3x[/tex]
Find the next number in the pattern
-8, -2, 2 ,4 .....
Answer:
5
Step-by-step explanation:
We have to see the pattern, we start with -8, the second number is -2, the distance between -8 and -2 is 6 units. Now from -2 to 2 the distance is 4 units. From 2 to 4 distance is 2 units. Then we can conclude that with each step the distance is divided by 2, then the next number is 5 because the distance between 4 and 5 is 1 unit.
You wish to prove that three propositions p1, p2, and p3 are equivalent. will it suffice to show that p1 --> p2, p2 --> p3, and p3 --> p1? justify your answer
Answer:
It is sufficient to prove that [tex] p_1\implies p_2, p_2\implies p_3, p_3\implies p_1[/tex]
Step-by-step explanation:
The propositions [tex] p_1,p_2,p_3[/tex] being equivalent means they should always have the same truth value. If one of them is true, then all of them must be true. And if one of them is false, then all of them must be false.
Suppose we've proven that [tex] p_1\implies p_2, p_2\implies p_3, p_3\implies p_1[/tex] (call these first, second and third implications).
If [tex]p_1[/tex] was true, then by the first implication that we proved, it would follow that [tex]p_2[/tex] is also true. And then by the second implication that we prove it would follow then that [tex]p_3 [/tex] is also true. Therefore the three of them would be true. Notice the reasoning would have been the same if we had started assuming that the one that was true was either [tex]p_2~or~p_3[/tex]. So one of them being true makes all of them be true.
On the other hand, if [tex]p_1[/tex] was false, then by the third implication that we proved, it would follow that [tex]p_3[/tex] has to be false (otherwise [tex]p_1[/tex] would have to be true, which would be a contradiction). And then, since [tex]p_3[/tex] is false, by the second implication that we proved it would follow that [tex] p_2[/tex] is false (otherwise [tex] p_3[/tex] would have to be true, which would be a contradiction). Therefore the three of them would be false. Notice the reasoning would have been the same if we had started assuming that the one that was false was either [tex]p_2~or~p_3[/tex]. So one of them being false makes all of them be false.
So, the three propositions always have the same truth value, and so they're all equivalent.
Drag a statement or reason to each box to complete this proof.
If −6(x−4)=42, then x=−3
Answer:
Distributive property
Subtraction property of equality
Division property of equality
Step-by-step explanation:
Given
-6(x-4)=42
-6*x+-4*-6=42-------------------distributive property
-6x+24-24=42-24-----------------subtraction property of equality
-6x=18
-6x/-6=18/-6--------------------------division property of equality
x= -3
Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question. A jar contains 70 nickels and dimes worth $6.10. How many of each kind of coin are in the jar? nickels dimes
Answer:
There are 52 dimes and 18 nickles
Step-by-step explanation:
Lets call x= number of dimes and y= number of nickles
then we have the first equation
(1) x + y = 70
As a nickel is worth 0.05 US$ and a dime is worth 0.10 US$, we have the second equation
(2) 0.10x + 0.05y = 6.10
We then have a linear system of 2 equations and 2 unknowns
(1) x +y = 70
(2) 0.10x + 0.05y = 6.10
In order to solve the system by the elimination method, we have to multiply on of the equations by a suitable number to eliminate one unknown when adding the two equations.
There are several ways of doing this. We could, for example, multiply (1) by -0.05 and then add it to (2)
(1) -0.05x – 0.05y = (-0.05)70
(2) 0.10x + 0.05y = 6.10
That is to say,
(1) -0.05x – 0.05y = -3.5
(2) 0.10x + 0.05y = 6.10
Adding (1) and (2) we get
-0.05x = -2.6 => x = (-2.6)/(-0.05) = 52 => x = 52
So we have 52 dimes.
Substituting this value in equation 1, we obtain
y = 70 - x = 70 – 52 = 18
Then we have 18 nickels
A _______ conveys the notion of "or." We use the symbol logical or ∨ to represent a _______.
Answer:
v and or.
Step-by-step explanation:
A ∨conveys the notion of "or". We use the symbol logical ∨ to represent a disjunction.
The "or" in logical lenguage indicates that one thing can happen but another thing also can happen, that is what we call a disjunction. The symbol to represent it is ∨.
The "and" in logical lenguage indicates that a situation is true only if all the statements are true. In other words we see all the statements as needs, not as options, that is called a conjunction. The symbol to represent it is ^.
Write the equation in slope-intercept form of the line below
A. Y= 2
B. Y= -2
C. X= 2
D. X= -2
Answer:
y = 2
Step-by-step explanation:
Answer:
y=2
Step-by-step explanation:
Dr. Who owns a phone booth worth $7,500, which, for tax purposes, is assumed to depreciate linearly to zero over a 15 year period. a. Express the value of the doctor's phone booth as a function of time. b. What is the value of the phone booth after 6 years?
Answer:
a. y = 7500 - 500x
b. $ 4,500
Step-by-step explanation:
a. Given,
The original value of the phone = $ 75,00
Let the linear function that shows the value of booth after x years,
y = b - ax,
For x = 0 years, y = 7500
⇒ 7500 = b - a(0) ⇒ b = 7500
For x = 15, y = 0,
0 = b - 15x ⇒ 0 = 7500 - 15a ⇒ 15a = 7500 ⇒ a = 500
Hence, the function that shows the value of car after x years,
y = 7500 - 500x
b. if x = 6,
The value of the booth after 6 years would be,
y = 7500 - 500(6) = 7500 - 3000 = $ 4,500
The area of a flower bed is 24 square feet. If the other sides were whole number demensions, how many lengths and widths are possible for the flower bed.
Answer:
The possible combinations are (4 & 6) ,(8 &3), (2 &12), (24 &1)
Step-by-step explanation:
The area of a flower bed is 24 square feet.
Now, the factors of 24 are 2 x 2 x 2 x 3.
Hence, if the other sides were whole number dimensions, then the possible combinations will be (4 & 6) ,(8 &3), (2 &12), (24 &1)
The amount of red blood cells in a blood sample is equal to the total amount in the sample minus the amount of plasma. What is the total amount of blood drawn?
Answer:
Step-by-step explanation:
The question is:
The amount of red blood cells in a blood sample is equal to the total amount in the sample minus the amount of plasma. What is the total amount of blood drawn? Red blood cells = 45% Plasma = 5.5 ml
Solution:
We have given:
The amount of red blood cells in a blood sample is equal to the total amount in the sample minus the amount of plasma. We need to find how many ml is 1% of blood.
The equation we get is:
Red blood cells = total sample - amount of plasma
45% = 100% - 5.5 ml
Combine the percentages:
5.5 ml = 100%-45%
5.5 ml = 55%
0.1 ml = 1%
1% = 0.1 ml
Now you can find out red blood cells volume or total sample volume.
Red blood cells volume = 45% x 0.1ml/%
= 4.5ml
Total sample volume =100% x 0.1ml/%
= 10ml .
On a given day, 36 of the 445 students in a school were absent. What was the appproximate absentee rate that day?
Answer: The approximate absentee rate that day would be 8.09%.
Step-by-step explanation:
Since we have given that
Number of students who were absent = 36
Total number of students = 445
We need to find the approximate absentee rate that day :
Rate of absentee of that day would be
[tex]\dfrac{\text{Number of absentee}}{\text{Total number of students}}\times 100\\\\=\dfrac{36}{445}\times 100\\\\=8.09\%[/tex]
Hence, the approximate absentee rate that day would be 8.09%.
A pond contains 2760 L of pure water and an uknown amount of an undesirable chemical. Water contaninig 0.01 kg of this chemical per liter flows into the pond at a rate of 4 L/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond.
Let Q(t) be the amount of chemical (in kg) in the pond at time t hours.
(a) Write a differential equation for the amount of chemical in the pond? at any time time (enter Q for Q(t))
(b) How much chemical will be in the pond after a long time?
Let [tex]q[/tex] be the unknown amount of the chemical originally in the pond, so [tex]Q(0)=q[/tex].
a. The incoming water introduces the chemical at a rate of
[tex]Q'_{\rm in}=\left(0.1\dfrac{\rm kg}{\rm L}\right)\left(4\dfrac{\rm L}{\rm hr}\right)=\dfrac25\dfrac{\rm kg}{\rm hr}[/tex]
and the mixture flows out at a rate of
[tex]Q'_{\rm out}=\left(\dfrac Q{2760}\dfrac{\rm kg}{\rm L}\right)\left(4\dfrac{\rm L}{\rm hr}\right)=\dfrac Q{690}\dfrac{\rm kg}{\rm hr}[/tex]
so that the net rate of change (in kg/hr) of the chemical in the pond is given by the differential equation,
[tex]\boxed{Q'=\dfrac25-\dfrac Q{690}}[/tex]
b. The ODE is linear; multiplying both sides by [tex]e^{t/690}[/tex] gives
[tex]e^{t/690}Q'+\dfrac{e^{t/690}}{690}Q=\dfrac{2e^{t/690}}5[/tex]
Condense the left side into the derivative of a product:
[tex]\left(e^{t/690}Q\right)'=\dfrac{2e^{t/690}}5[/tex]
Integrate both sides to get
[tex]e^{t/690}Q=276e^{t/690}+C[/tex]
and solve for [tex]Q[/tex] to get
[tex]Q=276+Ce^{-t/690}[/tex]
The pond starts with [tex]q[/tex] kg of the chemical, so when [tex]t=0[/tex] we have
[tex]q=276+C\implies C=q-276[/tex]
so that the amount of chemical in the water at time [tex]t[/tex] is
[tex]Q(t)=276+(q-276)e^{-t/690}[/tex]
As [tex]t\to\infty[/tex], the exponential term will converge to 0, leaving a fixed amount of 276 kg of the chemical in the pond.
The differential equation for the amount of chemical in the pond is dQ/dt = 0.01 kg/L * 4 L/h. After a long time, the amount of chemical in the pond is Q = 0.04 kg/h * t.
Explanation:(a)
To write a differential equation for the amount of chemical in the pond at any time, we need to consider the rate of change of the chemical in the pond. The chemical flows into the pond at a rate of 0.01 kg/L and flows out at the same rate, so the rate of change of the chemical Q(t) in the pond is 0.01 kg/L multiplied by the rate of change of the volume of water in the pond, which is 4 L/h. Therefore, the differential equation for the amount of chemical in the pond is:
dQ/dt = 0.01 kg/L * 4 L/h
(b)
To determine how much chemical will be in the pond after a long time, we can solve the differential equation.
We can rewrite the differential equation as:
dQ = 0.01 kg/L * 4 L/h * dt
Integrating both sides:
∫dQ = ∫0.01 kg/L * 4 L/h dt
Q = 0.04 kg/h * t + C
Where C is a constant of integration. Given that the amount of chemical in the pond is initially 0 (since the pond starts with only pure water), we can substitute Q = 0 and solve for C:
0 = 0.04 kg/h * 0 + C
C = 0
Therefore, the amount of chemical in the pond after a long time is given by:
Q = 0.04 kg/h * t
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Find the augmented matrix for each of the following systems of linear equations. (a) x₂ - 2x2 = 0 (6) x + x₃ = 1 .3x₂ + 4x2 = -1 - x₂ + 2x2x₃ = 3. 2x, - x₂ = 3 0 (c) x₂ + x3 =1 (d) x₂ = 1 2x₂ - x3 + x₂=2 X₂=2 2x3 + x4 = 3
Answer:
The augmented matrix for each set of linear equations is:
a) [tex]x_1-2x_2=0\\3x_1+4x_2=-1\\2x_1-x_2=3[/tex]
Augmented matrix:
[tex]\left[\begin{array}{ccc}1&-2&0\\3&4&-1\\2&-1&3\end{array}\right][/tex]
b) [tex]x_1+x_3=1\\-x_1+2x_2-x_3=3[/tex]
Augmented matrix:
[tex]\left[\begin{array}{cccc}1&0&1&1\\-1&2&-1&3\end{array}\right][/tex]
c) [tex]x_1+x_3=1\\2x_2-x_3+x_5=2\\2x_3+x_4=3[/tex]
Augmented matrix:
[tex]\left[\begin{array}{cccccc}1&0&1&0&0&1\\0&2&-1&0&1&2\\0&0&2&1&0&3\end{array}\right][/tex]
d) [tex]x_1=1\\x_2=2[/tex]
Augmented matrix:
[tex]\left[\begin{array}{ccc}1&0&1\\0&1&2\end{array}\right][/tex]
Step-by-step explanation:
In order to find the augmented matrix, you have to take the numeric values of each variable and make a matrix with them. For example, in the linear system a) you can make a matrix out of the numeric values accompanying x_1 and x_2, this matrix will be:
[tex]\left[\begin{array}{cc}1&-2\\3&4\\2&-1\end{array}\right][/tex]
Then you have to make a vector with the constants in the linear equations, for the case of system a) the vector will be:
[tex]\left[\begin{array}{c}0&-1&3\end{array}\right][/tex]
To construct the augmented matrix, you append those matrices together and create a new one:
[tex]\left[\begin{array}{ccc}1&-2&0\\3&4&-1\\2&-1&3\end{array}\right][/tex]
Lysera enjoys exploring her land on horseback with its lush green valleys and ancient forests. She can cover a great deal of ground on her horse, Princess Grey Dawn, traveling at 9 km/h. Unfortunately Lysera has allergies. How far would Lysera and Princess Gray Dawn have moved while Lysera’s eyes were shut for 0.50 s during a hard sneeze? (answer in kilometers)
Answer:
0.00125 Km
Step-by-step explanation:
Data provided in the question:
Speed of the horse = 9 Km/h
Duration for which the Lysera's eyes were shut = 0.50 seconds
now,
1 hour = 3600 seconds
or
1 second = [tex]\frac{\textup{1}}{\textup{3600}}\ textup{hours}[/tex]
Thus,
0.50 seconds = [tex]\frac{\textup{0.50}}{\textup{3600}}\ textup{hours}[/tex]
Also,
Distance = speed × Time
on substituting the values, we get
Distance = 9 × [tex]\frac{\textup{0.50}}{\textup{3600}}[/tex]
or
Distance = 0.00125 Km
Find the area of the shaded region. All angles are right angles.
Area = Length time width.
The area of the shaded area is Length x width of the entire shape, minus the length x width of the unshaded area.
Full area: 27 x 10 = 270 square m.
Unshaded area: 17 x 5 = 85 square m.
Area of shaded region: 270 - 85 = 185 square m.
Use De Morgan’s laws to write negations to the following statements:
1. The train is late or my watch is fast.
2. Dogs bark and cats meow.
Answer: The negations are:
1. The train is not late and my watch is not fast.
2. Dogs don't bark or cats don't meow.
Step-by-step explanation:
Hi!
De Morgan's laws for two propositions P and Q are:
1. ¬(P ∨ Q) = (¬P) ∧ (¬Q)
2. ¬(P ∧ Q) = (¬P) ∨ (¬Q)
where the symbols are,
¬ = not
∨ = or
∧ = and
1. In this case the proposition is P ∨ Q, with
P = "the train is late"
Q = "my watch is fast"
Then by law 1: ¬(P ∨ Q) = "The train is not late and my watch is not fast"
2. In this case the proposition is P ∧ Q, with
P = "dogs bark"
Q = "cats meow"
Then by law 2: ¬(P ∧ Q) = "Dogs don't bark or cats don't meow"
Answer:
2
Step-by-step explanation:
Which choice could be the equation of a line perpendicular to the line represented by this equation? y = 5x− 2
A. y= -1/5x + 5
B. y = 5x + 2
C. y= 1/5x - 7
D. y = -5x + 5
Answer:
Option A. y= -1/5x + 5
Step-by-step explanation:
step 1
Find the slope of the given line
we have
y=5x-2
The slope m is
m=5
step 2
Find the slope of a line perpendicular to the given line
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is -1)
m1*m2=-1
we have
m1=5
(5)*m2=-1
m2=-1/5
therefore
The equation of a line perpendicular to the given line could be
y= -1/5x + 5
In the voting for City Council Precinct 5, only 1/2 of all eligible voters cast votes. Shelley received 3/10 of the votes and Morgan received 5/8 of the votes. What fraction of all eligible voters voted for Shelly? Morgan? Who received the most votes?
Answer: Fraction of all eligible voters voted for Shelly is [tex]\dfrac{3}{20}[/tex] .
Morgan received the most votes .
Step-by-step explanation:
Given : The fraction of all eligible voters cast votes =[tex]\dfrac{1}{2}[/tex]
The fraction of votes received by Shelly = [tex]\dfrac{3}{10}[/tex] (1)
Now, the fraction of all eligible voters voted for Shelly is given by :_
[tex]\dfrac{3}{10}\times\dfrac{1}{2}=\dfrac{3}{20}[/tex]
Thus, the fraction of all eligible voters voted for Shelly is [tex]\dfrac{3}{20}[/tex] .
The fraction of vote received by Morgan= [tex]\dfrac{5}{8}[/tex] (2)
To compare the fractions given in (1) and (2), we need to find least common multiple of 10 and 8 .
LCM (10, 8)=40
Now, make denominator 40 (to make equivalent fraction) in (1), (2) we get
Fraction of all eligible voters voted for Shelly = [tex]\dfrac{3\times4}{10\times4}=\dfrac{12}{40}[/tex]
Fraction of all eligible voters voted for Morgan =[tex]\dfrac{5\times5}{8\times5}=\dfrac{25}{40}[/tex]
Since, [tex]\dfrac{25}{40}>\dfrac{12}{40}[/tex] [By comparing numerators]
Therefore, Morgan received the most votes .
We define the relation =m (read "equal mod m") on Z x Z to be the set: {(p,q): m|(p-q)}. Please show work.
a.) Give two pairs which are in the relation =4 and two pairs that are not.
b.) Show the =m is an equivalence relation.
Step-by-step explanation:
a) Give two pairs which are in the relation [tex]\equiv \mod 4[/tex] and two pairs that are not.
As stated before, a pair [tex](x,y)\in \mathbb{Z}\times\mathbb{Z}[/tex] is equal mod m (written [tex]x\equiv y\mod m[/tex]) if [tex]m\mid (x-y)[/tex]. Then:
x=0 and y=4 is an example of a pair [tex]\equiv \mod 4[/tex]x=0 and y=1 is an example of a pair [tex]\not \equiv \mod 4[/tex]b) Show the [tex]\equiv \mod m[/tex] is an equivalence relation.
An equivalence relation is a binary relation that is reflexive, symmetric and transitive.
By definition [tex]\equiv \mod m[/tex] is a binary relation. Observe that:
Reflexive. We know that, for every m, [tex]m\mid 0[/tex]. Then, by definition, [tex]x\equiv x \mod m[/tex].Symmetry. It is clear that, given x,y and m such that [tex]m\mid (x-y)[/tex], then [tex]m\mid (y-x)[/tex]. Therefore [tex]x\equiv y \mod m \iff y\equiv x \mod m[/tex]Transitivity. Let x,y,z and m such that [tex]x\equiv y \mod m[/tex] and [tex]y\equiv z \mod m[/tex]. Then, [tex]m\mid (y-x)[/tex] and [tex]m\mid (z-y)[/tex]. Therefore:[tex]m\mid [(y-x)+(z-y)] \implies m\mid (z-x) \implies x\equiv z \mod m[/tex].
In conclusion, [tex]\equiv \mod m[/tex] defines an equivalence relation.
A _______ represents the idea of "if and only if." Its symbol is a double arrow, left right arrow↔.
Answer:
A bi-conditional statement represents the idea of "if and only if." Its symbol is a [tex]\leftrightarrow[/tex].
Step-by-step explanation:
We have been given an incomplete sentence. We are supposed to fill in the given blank.
A _______ represents the idea of "if and only if." Its symbol is a [tex]\leftrightarrow[/tex].
We know that "if and only if" stands for bi-conditional statement, which represents either both statements are true or both are false.
The symbol [tex]\leftrightarrow[/tex] represent a bi-conditional statement.
Therefore, our complete statement would be: A bi-conditional statement represents the idea of "if and only if." Its symbol is a [tex]\leftrightarrow[/tex].
A bi-conditional statement in mathematics signifies 'if and only if' situation. Represented by the symbol ↔, it means both conditions in the statement must be true concurrently.
Explanation:In mathematics, a bi-conditional statement represents the concept of 'if and only if'. Its symbol is ↔ which is a double arrow or a left right arrow. This statement essentially means that both the conditions present need to be true for the entire statement to be true. For example, in the statement 'x is even if and only if x is divisible by 2', both parts (x being even and x being divisible by 2) need to be true together.
Learn more about bi-conditional here:https://brainly.com/question/29210714
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Find a general solution of y" + 8y' + 16y=0.
Answer:
The general solution: [tex]C_{1}e^{-4x} + xC_{2}e^{-4x}[/tex]
Step-by-step explanation:
Differential equation: y'' + 8y' + 16y = 0
We have to find the general solution of the above differential equation.
The auxiliary equation for the above equation can be writtwn as:
m² + 8m +16 = 0
We solve the above equation for m.
(m+4)² = 0
[tex]m_{1}[/tex] = -4, [tex]m_{2}[/tex] = -4
Thus we have repeated roots for the auxiliary equation.
Thus, the general solution will be given by:
y = [tex]C_{1}e^{m_{1}x} + xC_{2}e^{m_{2}x}[/tex]
y = [tex]C_{1}e^{-4x} + xC_{2}e^{-4x}[/tex]
Final answer:
To find the general solution of y'' + 8y' + 16y = 0, determine the characteristic roots, and utilize them to form the solution expression with constants. The primary topic is solving second-order homogeneous differential equations.
Explanation:
To find a general solution of the differential equation y'' + 8y' + 16y = 0, we first look for the characteristic equation by assuming y = e^(rt). Substituting this into the equation gives us the characteristic equation r^2 + 8r + 16 = 0. Solving this quadratic equation gives us the roots r = -4, -4. Therefore, the general solution is y(x) = c1e^(-4x) + c2xe^(-4x), where c1 and c2 are constants.
An baseball player’s batting average decreases from 0.312 to 0.278. What is the percent decreased?
Answer:
[tex]10.897[/tex]
Step-by-step explanation:
An baseball player’s batting average decreases from 0.312 to 0.278 .
Let [tex]x_0[/tex] be the initial baseball player’s batting average and [tex]x_1[/tex] be the final baseball player’s batting average .
Initial value [tex]\left ( x_0 \right )[/tex] = 0.312
Final value [tex]\left ( x_1 \right )[/tex] = 0.278
So, change in value =Final value - Initial value = [tex]x_1-x_0[/tex] = [tex]0.278-0.312=-0.034[/tex]
Therefore , decrease in value = 0.034
We know that percent decreased = ( decrease in value × 100 ) ÷ Initial value
i.e. , [tex]\frac{0.034}{0.312}\times 100=\frac{3400}{312}=10.897[/tex]
Or we can say percentage change in baseball player’s batting average = [tex]-10.897 \%[/tex]
Show that the sum of any two odd numbers is even.
Hi here´s a way to solve it
Let m and n be odd integers. Then, we can express m as 2r + 1 and n as 2s + 1, where r and s are integers.
This means that any odd number can be written as the sum of some even integer and one.
Substituting, we have that m + n = (2r + 1) + 2s + 1 = 2r + 2s + 2.
As we defined r and s as integers, 2r + 2s + 2 is also an integer.
Now It is clear that 2r + 2s + 2 is an integer divisible by 2 becasue we have 2 in each of the integers.
Therefore, 2r + 2s + 2 = m + n is even.
So, the sum of two odd integers is even.