Answer:
Part 1.
In 12 hours 30 units of Humulin R insulin in 300 mL is to be infused.
So, per hour = [tex]\frac{30}{12}= 2.5[/tex] units are to be infused.
Part 2.
Now 30 units Humulin R insulin in 300 mL of normal saline (NS).
So, 2.5 units will be in : [tex]\frac{300\times2.5}{30}= 25[/tex] mL
Hence. 2.5 units Humulin R insulin in 25 mL of normal saline (NS).
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
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.
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Hello, I asked help for this problem several times to no answer. I dont understand it and can use any help. Please, and thank you. please see the picture attached!
Check the picture below.
let's recall that a straight-line has 180°, and that sum of all interior angles in a triangle is also 180°.
(b) Suppose the 4 × 6 coefficient matrix for a system of linear equations has 4 pivot columns. Say as much as you can about the solutions to the corresponding system of equations, with explanation.
Answer:
The system is consistent with infinitely many solutions.
Step-by-step explanation:
Consider the provided information.
It is given that the 4 × 6 coefficient matrix for a system of linear equations has 4 pivot columns.
4 pivot columns means that there is a pivot in every row but not in every column as we know that the number of columns are greater than 4.
In a coefficient matrix the columns without pivot elements correspond to free variables.
If a system of linear equations has no solution then it is known as inconsistent otherwise its called consistent.
Infinite solution: The system of equation is consistent but at least one of the variables is free.
As each row has pivot that means that means system of linear equations has solution. Also at least one of the variable is free that means it has infinitely many solutions.
Thus, the system is consistent with infinitely many solutions as there is a pivot in every row but not every column.
Final answer:
A 4 × 6 coefficient matrix with 4 pivot columns indicates an underdetermined system of linear equations, leading to an infinite number of solutions within a two-dimensional solution space.
Explanation:
When a 4 × 6 coefficient matrix for a system of linear equations has 4 pivot columns, it indicates the presence of 4 independent equations for solving the variables. However, since there are 6 columns in total and 4 of them are pivot columns, it implies that there are 6 variables in the system. The presence of 4 pivot columns means that 4 of these variables can be solved in terms of the remaining 2 variables, assuming the system has consistent equations. Therefore, the system does not have a unique solution; instead, it has an infinite number of solutions that form a two-dimensional solution space, because the system is underdetermined (more variables than independent equations).
The concept of pivot columns is crucial in linear algebra for understanding the solvability of linear equation systems. A pivot column in a matrix corresponds to an independent equation in the system, which directly affects the nature of the solutions. When dealing with a system of linear equations, it is essential to determine the number of pivot columns to understand the dimensions of the solution space and whether the system is over-determined, underdetermined, or exactly determined.
The equation for a parabola has the form y = ax^2 + bz + c, where a, b, and c are constants and aメ0. Find an equation for the parabola that passes through the points (-1,14), (2,-7), and (5, 8) Answer: y-
Answer:
a = 2, b = -9, c = 3
Step-by-step explanation:
Replacing x, y values of the points in the equation y = a*x^2 + b*x +c give the following:
(-1,14)
14 = a*(-1)^2 + b*(-1) + c
(2,-7)
-7 = a*2^2 + b*2 + c
(5, 8)
8 = a*5^2 + b*5 + c
Rearranging:
a - b + c = 14
4*a + 2*b + c = -7
25*a + 5*b + c = 8
This is a linear system of equations with 3 equations and 3 unknows. In matrix notation the system is A*x = b whith:
A =
1 -1 1
4 2 1
25 5 1
x =
a
b
c
b =
14
-7
8
Solving A*x = b gives x = Inv(A)*b, where Inv(A) is the inverse matrix of A. From calculation software (I used Excel) you get:
inv(A) =
0.055555556 -0.111111111 0.055555556
-0.388888889 0.444444444 -0.055555556
0.555555556 0.555555556 -0.111111111
inv(A)*b
2
-9
3
So, a = 2, b = -9, c = 3
Adam wishes to have $16,000 available in 18 yrs to purchase a new car for his son. To accomplish this goal, how much should adam invest now in a CD that pays 1.24% interest compounded monthly?
Answer:
To accomplish this goal, how much should adam invest now $12800.7525 in a CD that pays 1.24% interest compounded monthly.
Step-by-step explanation:
Amount = 16000
Time = 18 years
Interest = 1.24% interest compounded monthly
So, Formula : [tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Formula : [tex]16000=P(1+\frac{1.24}{100 \times 12})^{12 \times 18}[/tex]
[tex]16000=P(1.24992651224)[/tex]
[tex]\frac{16000}{1.24992651224}=P[/tex]
[tex]12800.7525=P[/tex]
Hence To accomplish this goal, how much should adam invest now $12800.7525 in a CD that pays 1.24% interest compounded monthly.
A standard deck of cards has 52 cards in four suits: spades, hearts, diamonds, and clubs. Spades and clubs are black; hearts and diamonds are red. Each suit has 13 cards: 2 through 10, Jack, Queen, King, Ace. The "face cards" are the Jack, Queen, and King. Robby Billity shuffles a standard deck and draws one card. Then Robby replaces the card, shuffles again, and draws another card. He repeats this procedure for a total of 1000 draws. About how many times would we expect Robby to draw these cards?
Answer:
231
Step-by-step explanation:
Let's compute the probability of this event and then multiply it by 100.
As each deck has 4 suits and each suit has 3 faces, there are 12 faces in a deck.
As a deck has 52 cards, the probability of drawing a face is
12/52 = 0.23076
This means that if you choose a card 1000 times from a deck of poker cards, you probably will select 231 times a face (rounding to the nearest integer).
Robby Billity is expected to draw approximately 500 red cards and 231 face cards out of 1000 draws, based on the probability of each type of card in a standard deck.
To determine how many times we would expect Robby Billity to draw a specific type of card over 1000 draws, we first need to calculate the probability of drawing each type of card from a standard deck. Since there are 52 cards in the deck and Robby replaces the card each time before drawing again, each draw is independent with the same probability.
Calculating Probabilities:
Red Cards: There are 26 red cards (hearts and diamonds) out of 52. The probability of drawing a red card is 26/52, which simplifies to 1/2 or 0.5.
Face Cards: Each suit has 3 face cards (Jack, Queen, King), totaling 4 x 3 = 12 face cards. Therefore, the probability of drawing a face card is 12/52, which simplifies to 3/13 or approximately 0.231.
Expected Number of Draws:
Red Cards: The expected number of red card draws in 1000 trials is calculated as 0.5 x 1000 = 500.Face Cards: The expected number of face card draws in 1000 trials is calculated as 0.231 x 1000 = 231.Therefore, Robby Billity is expected to draw approximately 500 red cards and 231 face cards out of 1000 draws.
In a certain region of the country it is known from
pastexperience that theprobability of selecting an adult over 40
yearsof age with cancer is 0.05. If the probability of a
doctorcorrectly diagnosing a person with cancer as having the
disease is0.78 and the probability of incorrectly diagnosing a
person withoutcancer as having the disease is .06, what is the
probability that aperson is diagnosed as having cancer?
Answer:
There is a 9.6% probability that a person is diagnosed as having cancer.
Step-by-step explanation:
In this problem, we have these following probabilities:
A 5% probability that an adult over 40 has cancer.
This also means that:
There is a 95% probability that an adult over 40 does not have cancer. (Since either the adult has cancer or does not have cancer, and the sum of the probabilities is 100%).
A 78% probability of a person that has cancer being diagnosed,
A 6% probability of a person that does not have cancer being diagnosed.
What is the probability that a person is diagnosed as having cancer?
[tex]P = P_{1} + P_{2}[/tex]
[tex]P_{1}[/tex] is the probability of those who have cancer being diagnosed. So it is 78% of 5%. So
[tex]P_{1} = 0.05*0.78 = 0.039[/tex]
[tex]P_{2}[/tex] is the probability of those who do not have cancer being diagnosed. So it is 6% of 95%. So
[tex]P_{1} = 0.06*0.95 = 0.057[/tex]
So
[tex]P = P_{1} + P_{2} = 0.039 + 0.057 = 0.096[/tex]
There is a 9.6% probability that a person is diagnosed as having cancer.
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
Determine whether the data shows a linear relationship. If so, write an equation of a line of it.
Answer:
The data do not show a linear relationshipStep-by-step explanation:
A plot of the points connected by straight lines makes it pretty clear they do not all fall on the same line. There is no linear relationship here.
Sales of a certain MP3 players are approximated by the relationship S(x) = 4740x + 31,000(0 x 5) where S(x) denotes the number of MP3 players sold in year x (x = 0 corresponds to the year 2000). Find the number of MP3 players expected to be sold in 2002.
Answer:
40480 MP3 players expected to be sold in 2002.
Step-by-step explanation:
Sales of a certain MP3 players are approximated by the relationship;
[tex]S(x)=4740x+ 31000[/tex] (0≤x≤5)
(x = 0 corresponds to the year 2000)
That means 2002 corresponds to x = 2
Now substituting x = 2 in the expression.
[tex]S(x)=4740(2)+ 31000[/tex]
= [tex]9480+31000[/tex]
= 40480.
Hence, 40480 MP3 players expected to be sold in 2002.
What is amount of interest I earned on $4,000 deposited in a savings account with 4% interest compounded annually after 4 years?
Give answer in US dollars and cents, rounded to the nearest cent. Do NOT enter "$" sign
Answer:
$ 679.43
Step-by-step explanation:
Since, the amount formula in compound interest is,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where,
P = initial amount,
r = annual rate of interest,
n = number of compounding periods in a year,
t = number of years,
Here, P = $ 4,000, r = 4% = 0.04, t = 4 years, n = 1,
So, the amount after 4 years would be,
[tex]=4000(1+0.04)^4\approx \$4679.43[/tex]
Hence, the amount of interest,
[tex]I=A-P=4679.43-4000 = \$ 679.43[/tex]
Estimate how many books can be shelved in a college library with 3500 m2 of floor space. Assume 8 shelves high, having books on both sides, with corridors 1.5 m wide. Assume books are approximately 25 cm deep and 5 cm wide, on average.
Approximately 347,200 books can be shelved in the college library with 3500 m² of floor space.
Given:
Floor space of the library = 3500 m²
Height of each shelf = 1.5 mWidth of each shelf = 0.5 m (assuming each book is 5 cm wide and there are two rows of books on each shelf)Depth of each shelf = 0.25 m (assuming each book is 25 cm deep)Width of each corridor = 1.5 mTo estimate how many books can be shelved in a college library with 3500 m² of floor space, we can use the following assumptions:
8 shelves high on both sidesCorridors 1.5 m wideBooks are approximately 25 cm deep and 5 cm wide, on average.To calculate the number of books that can be shelved, we need to find the volume of the shelving space and divide it by the volume of each book.
To find the volume of the shelving space, we need to subtract the volume of the corridors from the total volume of the library. The total volume of the library is:
[tex]V_{library} = (63\ yards) \times (32 \ yards) \times (6\ yards)[/tex]
Converting yards to meters, we get:
[tex]V_{library} = (63 \times 0.9144 \ meters) \times (32 \times 0.9144 \ meters) \times (6 \times 0.9144\ meters)[/tex]
Simplifying the equation, we get:
[tex]V_{library} \approx 1407\ m^3[/tex]
The volume of the corridors can be calculated as follows:
[tex]V_{corridors} = (8\ shelves) \times (0.5\ m + 1.5\ m) \times (1.5\ m) \times (63\ m + 32\ m)[/tex]
Simplifying the equation, we get:
[tex]V_{corridors} = 756\ m^3[/tex]
Therefore, the volume of the shelving space is:
[tex]V_{shelving} = V_{library} - V_{corridors} \\V_{shelving} \approx 651 \ m^3[/tex]
To find the volume of each book, we can multiply the depth, width, and height of each book:
[tex]V_{book} = (0.25\ m) \times (0.05\ m) \times (0.15\ m)[/tex]
Simplifying the equation, we get:
[tex]V_{book} = 0.001875 \ m^3[/tex]
Finally, we can divide the volume of the shelving space by the volume of each book to find the number of books that can be shelved:
[tex]Number\ of\ books = \frac{V_{shelving}} {V_{book}} \\Number\ of\ books \approx 347,200[/tex]
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The number of books can be shelved in a college library with [tex]3500 \ m^2[/tex] of floor space is 700,000 books.
Assume that the room is square-shaped.
The size of an average book is [tex]0.05\times0.25 \ m[/tex]. Thus, the thickness of the shelf is 0.5 m.
Area of the room, [tex]A=3500 \ m^2[/tex]
Width of the corridor space, [tex]W_c=1.5 \ m[/tex]
As the room is square-shaped, its width is as follows:
[tex]W_{room}=\sqrt{3500}[/tex]
[tex]= 59.16 m[/tex]
The area of a square with side a is [tex]a^2[/tex].
The total number of rows (r) is
[tex]r=\frac{W_{room}}{W_c+shelfsize}[/tex]
[tex]= \frac{59.16}{1.5+0.5}[/tex]
[tex]= 29.28 m[/tex]
The total number of shelves
[tex]= r\times 8(height of the shelves)\times2(facing both sides)[/tex]
[tex]= 473.28[/tex]
As the room is square-shaped, the length of each shelf is [tex]59.16 \ m[/tex].
The number of books on each shelf (n) is as follows:
[tex]n=\frac{Length \ of \ each \ row}{Thickness \ of \ each \ book}[/tex]
= [tex]\frac{59.16}{0.04}[/tex]
[tex]= 1479[/tex]
The total number of books is the sum of books on all shelves. Thus, the total number of books (N) is as follows:
[tex]N=n\times (Total \ number \ of \ shelves)[/tex]
[tex]= 1479\times473.28[/tex]
[tex]= 699981.12[/tex]
[tex]\approx 700000[/tex]
Thus, there are approximately 700,000 books.
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A theater has 10 seats in the first row and 30 seats in the 6th row. How many seats are in the 11th row?
(It's ok if you give me the equation, that's all I need please and thank you)
Find the difference per row:
10 seats in the first row
30 seats in the sixth row:
30 -10 = 20 seats difference.
6-1 = 5 rows difference.
20 seats / 5 rows = 4 seats per row.
This means for every additional row, there are 4 more seats per row.
The equation would be:
Sn = S +(n-1)*d
Where d is the difference = 4
S = number of seats from starting row = 10
n = the number of rows wanted
S(11) = 10 + (11-1)*4
S(11) = 10 + 10*4
S(11) = 10 + 40
S(11) = 50
Check:
Row 6 = 30 seats
Row 7 = 30 + 4 = 34 seats
Row 8 = 34 + 4 = 38 seats
Row 9 = 38 + 4 = 42 seats
Row 10 = 42 + 4 = 46 seats
Row 11 = 46 + 4 = 50 seats.
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%.
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.
Four teams A,B,C, and D compete in a tournament, and exactly one of them will win the tournament. Teams A and B have the same chance of winning the tournament. Team C is twice as likely to win the tournament as team D. The probability that either team A or team C wins the tournament is 0.6. Find the probabilities of each team winning the tournament.
Answer:
A= 0,2
B= 0,2
C= 0,4
D=0,2
Step-by-step explanation:
We know that only one team can win, so the sum of each probability of wining is one
P(A)+P(B)+P(C)+P(D)=1
then we Know that the probability of Team A and B are the same, so
P(A)=P(B)
And that the the probability that either team A or team C wins the tournament is 0.6, so P(A)+Pc)= 0,6, then P(C)= 0.6-P(A)
Also, we know that team C is twice as likely to win the tournament as team D, so P(C)= 2 P(D) so P(D) = P(C)/2= (0.6-P(A))/2
Now if we use the first formula:
P(A)+P(B)+P(C)+P(D)=1
P(A)+P(A)+0.6-P(A)+(0.6-P(A))/2=1
0,5 P(A)+0.9=1
0,5 P(A)= 0,1
P(A)= 0,2
P(B)= 0,2
P(C)=0,4
P(D)=0,2
Teams A and B each have a 0.2 probability of winning, team C has a 0.4 probability, and team D has a 0.2 probability of winning the tournament. The given conditions were used to calculate these probabilities step-by-step.
To find the probabilities of each team (A, B, C, and D) winning the tournament, let's denote the probability for each team as follows: P(A), P(B), P(C), and P(D). According to the problem, we are given these conditions:
Teams A and B have the same chance of winning: P(A) = P(B)Team C is twice as likely to win the tournament as team D: P(C) = 2P(D)The probability that either team A or team C wins is 0.6: P(A) + P(C) = 0.6Let's express everything in terms of P(D):
P(A) = P(B) (Let it be x)P(C) = 2P(D)From the total probability, we know that:
P(A) + P(B) + P(C) + P(D) = 1Substituting the given conditions:
x + x + 2P(D) + P(D) = 12x + 3P(D) = 1From condition 3:
x + 2P(D) = 0.6We now have two equations:
2x + 3P(D) = 1x + 2P(D) = 0.6First, solve for x in terms of P(D) from equation 2:
x = 0.6 - 2P(D)Substitute this into equation 1:
2(0.6 - 2P(D)) + 3P(D) = 11.2 - 4P(D) + 3P(D) = 11.2 - P(D) = 1P(D) = 0.2Now substitute P(D) back to find x:
x = 0.6 - 2(0.2) = 0.6 - 0.4 = 0.2Therefore, P(A) = P(B) = 0.2.
Using P(C) = 2P(D):P(C) = 2(0.2) = 0.4Summarizing the probabilities:
P(A) = 0.2P(B) = 0.2P(C) = 0.4P(D) = 0.2Draw a graph of order 4 and size equal to zero. What is this graph called?
Answer:
The draw in the file is a realization of a graph of order 4 and size zero.
In the book of Douglas West, Introduction to Graph Theory the name of this graph is 'Trivial graph'
Step-by-step explanation:
Remember that the order of a graph is the number of vertices and the size of the graph is the number of edges of the graph.
Vanillin is the substance whose aroma the human nose detectsin
the smallest amount. The threshold limit is 2.0
x10-11grams per liter of air. If the current priceof
50.0g of vanillin is $112, determine the cost to suppy
enoughvanillin so that the aroma could be detectable in a large
aircrafthangar of volume 5.0 x 107 m3.
Answer:
$2.24
Step-by-step explanation:
Given:
Threshold limit = 2.0 × 10⁻¹¹ grams per liter of air
Current price of 50.0 g vanillin = $112
Volume of aircraft hanger = 5.0 × 10⁷ m³
Now,
1 m³ = 1000 L
thus,
5.0 × 10⁷ m³ = 5.0 × 10⁷ × 1000 = 5 × 10¹⁰ L
therefore,
The mass of vanillin required = 2 × 10⁻¹¹ × 5 × 10¹⁰ = 1 g
Now,
50 grams of vanillin costs = $112
thus,
1 gram of vanillin will cost = [tex]\frac{\textup{112}}{\textup{50}}[/tex] = $2.24
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 ^.
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]
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.
Suppose that for a function f,f(2) is not defined. Also suppose that limx→2−f(x)=7 and limx→2+f(x)=7. Which, if any, of the following statements is false? a) limx→2f(x)=7 b) f has jump discontinuity at x = 2 c) If we re-define f so that f(2) = 7 then the new function will be continuous at x = 2 d) f has removable discontinuity at x = 2 e) All of the above statements are true.
All of the statements are true.
If the limit of a function f(x) at x = a is exist .
[tex]\lim_{x \to a+} f(x)= \lim_{x \to a-} f(x)=f(a)[/tex]
Given that,
[tex]\lim_{x \to 2-} f(x)= \lim_{x \to 2+} f(x)=7[/tex]
But f(2) is not defined.
It means that function f(x) has jump discontinuity at x = 2
A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph.
So that, Function f(x) has removable discontinuity at x = 2
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The false statement among the given options is that redefining f(2) to 7 will make the new function continuous at x = 2.
Explanation:The false statement among the options given is (c) If we re-define f so that f(2) = 7 then the new function will be continuous at x = 2.
The given information states that both the left and right limits as x approaches 2 are equal to 7, which suggests that the limit as x approaches 2 exists and is equal to 7. This means that option (a) lim x→2 f(x)=7 is true.
We know that f(2) is not defined in the original function, meaning there is a hole in the graph at x = 2. Therefore, option (d) f has a removable discontinuity at x = 2 is also true.
However, if we redefine f(2) = 7, the new function will still have a jump discontinuity at x = 2 since there will be a discontinuity between the values of f(2) before and after the redefinition. Therefore, option (c) is false.
So, the correct answer is (c) If we redefine f so that f(2) = 7 then the new function will be continuous at x = 2.
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A chemist wants to mix a 22% acid solution with a 36% acid solution to get 28 L of a 26% acid solution. How many liters of the 22% solution and how many liters of the 36% solution should be mixed?
Answer:
20 L of 22% solution and 8 L of 36% solution
Step-by-step explanation:
Volume of 22% solution + volume of 36% solution = volume of 26% solution
x + y = 28
Acid in 22% solution + acid in 36% solution = acid in 26% solution
0.22x + 0.36y = 0.26(28)
0.22x + 0.36y = 7.28
Solve the system of equations using either elimination or substitution. I'll use substitution:
x = 28 − y
0.22(28 − y) + 0.36y = 7.28
6.16 − 0.22y + 0.36y = 7.28
0.14y = 1.12
y = 8
x = 28 − y
x = 20
The chemist should use 20 L of 22% solution and 8 L of 36% solution.
Answer:
There should be mixed 20 L of the 22% acid solution with 8L of the 36% acid solution
Step-by-step explanation:
We are mixing two acids.
x = liters of 22% acid solution
y = liters of 36% acid solution
x + y = 28 (total liters)
0.22x +0.36y = 0.26* 28
Since x+y=28 means y = 28-x
Now we will use substitution to find x
0.22x + 0.36(28-x) = 0.26 * 28
0.22x + 10.08 - 0.36x = 7.28
0.14x = 2.8
x = 20
y = 28 - 20 = 8
⇒ We use 20 liters of the 22% solution to be mixed with 8 liters of the 36% solution to form 28l of a 26% acid solution.
What is the negation of the following statement: "I exercise and I feel tired."
A. I exercise and I feel tired.
B. I exercise and I feel envigorated.
C. I don't exercise or I feel tired.
D. I exercise or I feel tired.
E. I don't exercise and I feel tired.
F. I don't exercise or I feel envigorated.
G. I don't exercise and I feel envigorated.
H. I exercise or I feel envigorated.
Answer:
G. I don't exercise and I feel envigorated.
Step-by-step explanation:
In this sentence the I exercise and I feel tired you need to say in order tos ay the negation of this sentence would be:
I do not exercise nor feel tired, since there´s no option that says this, we can choose the one that says I don´t exercise, and envigorated is the opposite than tired, so the correct option would be I don´t exercise and I feel envigorated.
Final answer:
The correct negation of the statement "I exercise and I feel tired" is "I don't exercise or I feel envigorated," which corresponds to option F.
Explanation:
The negation of the compound statement "I exercise and I feel tired" involves negating both parts of the statement and changing the conjunction 'and' to 'or'. This is in line with De Morgan's laws which state that the negation of a conjunction is the disjunction of the negations. Therefore, the negated form of the statement would be "I don't exercise or I don't feel tired." From the given options, the one that best matches this structure is:
F. I don't exercise or I feel envigorated.
Envigorated is understood as the opposite of feeling tired in this context. So, statement F is the correct negation as it correctly captures the negation of both parts of the original statement.
Write the negation of: If we lose electricity, then the data will be lost.
Write the negation of: If we lose electricity, then the data will be lost. Proposedsolution: PA- I have proposed PQ but I need a proper negation and proof tables.
Step-by-step explanation:
Consider the provided statement.
If we lose electricity, then the data will be lost.
We are need to write the negation of the above statement.
First divide the whole statement in two parts
Let us consider p = We lose electricity and q = The data will be lost.
The symbol use for negation is tilde [tex]\sim[/tex]
[tex]\sim(p\rightarrow q)[/tex]
[tex]p\wedge \sim q[/tex]
Represent T for true and F for False.
The required table is shown below:
p q [tex](p\wedge \sim q)[/tex]
F F F
F T F
T F T
T T F
Hence the required table is shown above.
The investors club invests 500 at 6% simple interest. How much
is in the account afte 90 days?
Answer: There would be $507.39 in the account after 90 days.
Step-by-step explanation:
Since we have given that
Principal = $500
Rate of interest = 6%
Number of days = 90 days
As we know that "Simple interest":
[tex]Interest=\dfrac{P\times R\times T}{100}\\\\Interest=\dfrac{500\times 6\times 90}{100\times 365}\\\\Interest = \$7.39[/tex]
So, Amount = Principal + Interest
Amount = $500 + $ 7.39
Amount = $507.39
Hence, There would be $507.39 in the account after 90 days.
Which image (A'B'C'D') of ABCD cannot be produced using only reflections? A. B. C. D.
Answer:
the answer is D
Step-by-step explanation:
There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them. 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and 75 take both physics and chemistry. Only 30 take both physics and calculus. How many students take physics?
Answer:
150 students take physics.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the students that take calculus.
-The set B represents the students that take physics
-The set C represents the students that take chemistry.
-The set D represents the students that do not take any of them.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of students that take only calculus, [tex]A \cap B[/tex] is the number of students that take both calculus and physics, [tex]A \cap C[/tex] is the number of students that take both calculus and chemistry and [tex]A \cap B \cap C[/tex] is the number of students that take calculus, physics and chemistry.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C), D[/tex]
There are 360 people in my school. This means that:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) + D = 360[/tex]
The problem states that:
15 take calculus, physics, and chemistry, so:
[tex]A \cap B \cap C = 15[/tex]
15 don't take any of them, so:
[tex]D = 15[/tex]
75 take both calculus and chemistry, so:
[tex]A \cap C = 75[/tex]
75 take both physics and chemistry, so:
[tex]B \cap C = 75[/tex]
30 take both physics and calculus, so:
[tex]A \cap B = 30[/tex]
Solution:
The problem states that 180 take calculus. So
[tex]a + (A \cap B) + (A \cap C) + (A \cap B \cap C) = 180[/tex]
[tex]a + 30 + 75 + 15 = 180[/tex]
[tex]a = 180 - 120[/tex]
[tex]a = 60[/tex]
Twice as many students take chemistry as take physics:
It means that: [tex]C = 2B[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 75 + 30 + 15[/tex]
[tex]B = b + 120[/tex]
-------------------------------
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]C = c + 75 + 75 + 15[/tex]
[tex]C = c + 165[/tex]
----------------------------------
Our interest is the number of student that take physics. We have to find B. For this we need to find b. We can write c as a function o b, and then replacing it in the equations that sums all the subsets.
[tex]C = 2B[/tex]
[tex]c + 165 = 2(b+120)[/tex]
[tex]c = 2b + 240 - 165[/tex]
[tex]c = 2b + 75[/tex]
The equation that sums all the subsets is:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) + D = 360[/tex]
[tex]60 + b + 2b + 75 + 30 + 75 + 15 + 15 = 360[/tex]
[tex]3b + 270 = 360[/tex]
[tex]3b = 90[/tex]
[tex]b = \frac{90}{3}[/tex]
[tex]b = 30[/tex]
30 students take only physics.
The number of student that take physics is:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 75 + 30 + 15[/tex]
[tex]B = 30 + 120[/tex]
[tex]B = 150[/tex]
150 students take physics.
Final answer:
Using a Venn Diagram approach and the information given, we find that 45 students take physics at the school.
Explanation:
To find out how many students take physics at the school, we can use the Venn Diagram principle and the given data. We know that 15 students take calculus, physics, and chemistry together. Additionally, 180 students take calculus, and twice as many students take chemistry as take physics. With 75 students taking both calculus and chemistry, and another 75 taking both physics and chemistry, while only 30 take both physics and calculus, we can establish relationships and solve for the number of students taking each subject.
Let's denote the number of students taking physics as P. Then, the number of students taking chemistry would be 2P.
Total taking calculus and chemistry (C ∩ Ch) = 75Total taking calculus and physics (C ∩ P) = 30Total taking physics and chemistry (P ∩ Ch) = 75Total taking all three (C ∩ P ∩ Ch) = 15Students taking none = 15To avoid double counting, we must subtract those taking all three subjects once for each combination:
(C ∩ Ch) - (C ∩ P ∩ Ch) = 75 - 15 = 60 (only calculus and chemistry)(C ∩ P) - (C ∩ P ∩ Ch) = 30 - 15 = 15 (only calculus and physics)(P ∩ Ch) - (C ∩ P ∩ Ch) = 75 - 15 = 60 (only physics and chemistry)With twice as many students in chemistry as in physics, we can write the equation:
180 + 2P + P - (15 + 60 + 15 + 60) + 15 = 360
Solving for P:
P + 2P + 180 - 150 + 15 = 3603P = 360 - 45 - 1803P = 135P = 45Therefore, 45 students take physics at the school.
The number of bacteria in a flask grows according to the differential equation (dy)/(dt)= 0.06 y In this question, time is measured in hours and the number of bacteria, y, is measured in millions. The number of bacteria at time t = 0 is 4 million. Enter a formula for the number of bacteria at time t y = Click here to preview your answer. Incorrect: Your answer is incorrect. What is the value of the growth constant? Growth constant : per hour. How long does it take for the number of bacteria to double? (Enter your answer correct to two decimal places.) Doubling time : hours. How many million bacteria will be present after 9 hours have passed? (Enter your answer correct to one decimal place.) Number present after 9 hours : million.
Answer:
a) y = 4e^(0.06t)
b) 0.06
c) 11.55 hours
d) 6.9 million
Step-by-step explanation:
When the growth rate (millions per hour) is proportional to the number (millions), the relationship is exponential. The growth rate is the constant of proportionality.
a) Formula for y(t):
y = 4e^(0.06t)
__
b) The growth constant is 0.06, the multiplier of t in the exponential function. It is the constant of proportionality in the given differential equation:
y' = 0.06y.
__
c) The doubling time is found from ...
2 = e^(0.06t) . . . the multiplying factor is 2 to double the original number
ln(2) = 0.06t . . . . taking natural logs
ln(2)/0.06 = t ≈ 11.55 h . . . . doubling time
__
d) Put t=9 into the formula from part (a). After 9 hours, there will be ...
y(9) = 4e^(0.06·9) ≈ 6.9 . . . . million bacteria present
Answer:
y = 4e^(0.06t).
Step-by-step explanation:
dy/dt = 0.06y
Solving:
dy = 0.06y dt
dy/y = 0.06dt
Integrating both sides:
ln y = 0.06t + C
y = e^(0.06t + C)
y = Ae^(0.06t) where A is a constant.
At t = 0 , y = 4 million so
y = 4 = Ae^0 = A
So the formula is
y = 4e^(0.06t).