[tex]\underbrace{\begin{bmatrix}-2&-1&2\\-2&2&3\\-4&1&3\end{bmatrix}}_A\underbrace{\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}}_x=\underbrace{\begin{bmatrix}-1\\-1\\4\end{bmatrix}}_b[/tex]
Multiply [tex]A[/tex] on the left side with the following elimination matrix [tex]E_1[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\-1&1&0\\-2&0&1\end{bmatrix}}_{E_1}A=\begin{bmatrix}-2&-1&2\\0&3&1\\0&3&-1\end{bmatrix}[/tex]
Multiply [tex]E_1A[/tex] on the left by another elimination matrix [tex]E_2[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\0&1&0\\0&-1&1\end{bmatrix}}_{E_2}(E_1A)=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}[/tex]
[tex]\implies\boxed{U=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}}[/tex]
Multiply on the left by the inverse of [tex]E_2E_1[/tex]:
[tex](E_2E_1)^{-1}(E_2E_1)A=(E_2E_1)^{-1}U[/tex]
[tex]A=\underbrace{({E_1}^{-1}{E_2}^{-1})}_LU[/tex]
We have
[tex]{E_1}^{-1}=\begin{bmatrix}1&0&0\\1&1&0\\2&0&1\end{bmatrix}[/tex]
[tex]{E_2}^{-1}=\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix}[/tex]
[tex]\implies\boxed{L=\begin{bmatrix}1&0&0\\1&1&0\\3&1&1\end{bmatrix}}[/tex]
Heart failures are due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances (73%) or foreign objects (27%). Natural occurrences are caused by arterial blockage (56%), disease (27%), and infection (e.g., staph infection) (17%).(a) Determine the probability that a failure is due to induced substance.(b) Determine the probability that a failure is due to disease or infection.
Answer:
(a) The probability is 9.49%
(b) The probability is 38.28%
Step-by-step explanation:
The probability that a failure is due to induced substance is calculated as a multiplication as:
(13%) * (73%) = 9.49%
Where 13% is the percentage of heart failures that are due outside factors and 73% is the percentage of outside factors that are due induced substances.
On the other hand, the probability that a failure is due to disease or infection is the sum of the probability that a failure is due to disease and the probability that a failure is due to infection.
Then, the probability that a failure is due to disease is calculated as:
(87%) * (27%) = 23.49%
Where 87% is the percentage of heart failures that are due natural factors and 27% is the percentage of natural factors that are due disease.
At the same way, the probability that a failure is due to infection is calculated as:
(87%) * (17%) = 14.79%
So, the probability that a failure is due to disease or infection is:
23.49% + 14.79% = 38.28%
Exactly find sin(x+y) if sinx= 1/3 where x ends in the
2ndquadrant and cosy= 1/5 where y ends in the first quadrant.
Answer: [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]
Step-by-step explanation:
Since we have given that
[tex]\sin x=\dfrac{1}{3}\\\\so,\\\\\cos x=\sqrt{1-\dfrac{1}{9}}=\sqrt{\dfrac{8}{9}}=\dfrac{2\sqrt{2}}{3}[/tex]
Since x ends in the 2 nd quadrant,
So, [tex]\cos x=\dfrac{-2\sqrt{2}}{3}[/tex]
Similarly,
[tex]\cos y=\dfrac{1}{5}\\\\So,\\\\\sin y=\sqrt{1-\dfrac{1}{25}}=\sqrt{\dfrac{24}{25}}=\dfrac{2\sqrt{6}}{5}[/tex]
So, sin(x+y) is given by
[tex]\sin x\cos y+\sin y\cos x\\\\\\=\dfrac{1}{3}\times \dfrac{1}{5}+\dfrac{2\sqrt{6}}{5}\times (-)\dfrac{2\sqrt{2}}{3}\\\\\\=\dfrac{1}{15}-\dfrac{8\sqrt{3}}{15}\\\\\\=\dfrac{1-8\sqrt{3}}{15}[/tex]
Hence, [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]
Cory invests $4000 at 3.5%. How much will he have in 6 years if the interest is compounded monthly ?
Answer:
$ 4933.2 ( approx )
Step-by-step explanation:
∵ Future value formula is,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where,
P = principal amount,
r = annual rate,
n = number of periods,
t = number of years,
Given,
P = $ 4,000, r = 3.5 % = 0.035, t = 6 years n = 12 ( number of months in 1 year = 12 ),
Hence, the future value would be,
[tex]A=4000(1+\frac{0.035}{12})^{72}=4933.20414683\approx \$ 4933.2[/tex]
Digoxin (LANOXIN) pediatric elixir contains 0.05 mg (milligram) of digoxin in each milliliter (mL) of elixir. If there are
1000 μg (micrograms) in each milligram, how many micrograms of digoxin would be delivered in each dose of 0.6 mL?
Answer:
30 micro grams
Step-by-step explanation:
1 ml contains 0.05 mg (milligram) of digoxin
So, 0.6 ml contains digoxin = [tex]0.6 \times 0.05[/tex]
= [tex]0.03 mg[/tex]
Now 1 mg contains 1000 μg (micro grams)
So, 0.03 mg contains micro grams= [tex]0.03 \times 1000[/tex]
= [tex]30[/tex]
Hence 30 micro grams of digoxin would be delivered in each dose of 0.6 ml .
If alc and ab = c, prove that alb
Answer:
This isn't true.
Step-by-step explanation:
Think of the case a=2, b=3 and c=6. We have that a|b, since 2|6.
We also have that ab=c, since 2*3=6. However, it is NOT true that a|b, as 2 does NOT divide 3. As this you can construct many other examples where a|c and ab=c BUT a does NOT divide b.
Other counterexamples:
a=2, b=5, c=10
a=2, b=7, c=14
a=2, b=11, c=22
a=2, b=13, c=26
oco serves a tennis ball at vs = 50 m/s and charges the net at vc = 10 m/s. The opponent, x = 25 m away on the other side of the court, returns the ball with a speed half that of the serve. How close does Coco get to the net (x/2 away) before she meets the return?
Answer:
3.055 m
Step-by-step explanation:
In this solution we will use next notation:
[tex]t_1[/tex]= time elapsed since oco serves the ball until it reaches its opponent.
[tex]t_2[/tex]= time elapsed since the opponent returns the ball until it reaches oco.
d= Total distance traveled by Oco since serving the ball until meeting the return.
We know that oco serves at vs = 50 m/s and her opponent is x=25 m away. Then, t_1 is given by
[tex]t_1=\frac{25m}{50m/s}=0.5s[/tex]
To compute t_2 observe that the return speed is 12.5 m/s and the distance that the ball will travel is [tex]25-(10t_1+10t_2)[/tex]. Then,
[tex]t_2=\frac{25-10t_1-10t_2}{12.5}=\frac{20-10t_2}{12.5}\implies t_2=\frac{20}{22.5}=\frac{8}{9}s[/tex].
Therefore,
[tex]d=10(t_1+t_2)=10(0.5+\frac{8}{9})=10(\frac{17}{18})=\frac{85}{9}m[/tex]
Finally, as Oco started 12.5m away from the net, when she meets the return she will be
[tex]12.5-\frac{85}{9}=\frac{55}{18}=3.055m[/tex]
away from the net.
A quadratic function is given.
f(x) = 2x^2 + 4x + 3
(a) Express the quadratic function in standard form.
Answer:
2x² + 4x + 3 = 0
Step-by-step explanation:
The function is said to be quadratic if it has highest degree = 2.
Further, The standard form of Quadratic Equation is:
ax² + bx + c = 0
where, a ≠ 0
a, b and c are constants
and x is unknown variable.
Thus, The Standard form of given Quadratic Equation is 2x² + 4x + 3 = 0
To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, complete the square to get f(x) = 2(x + 1)^2 - 5, which reveals the vertex of the parabola at (-1, -5).
Explanation:Expressing a Quadratic Function in Standard FormTo express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, also known as vertex form, we need to complete the square. The standard form of a quadratic function is typically written as f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola formed by the graph of the quadratic equation.
Here's how we can rewrite the given quadratic function:
Divide the coefficient of the x term by 2, which is 4/2 = 2, and then square it to get 4.Add and subtract this value inside the parentheses to complete the square: f(x) = 2(x^2 + 2x + 4 - 4) + 3.Rewrite the equation by grouping the perfect square trinomial and combining the constants: f(x) = 2((x + 1)^2 - 4) + 3.Finally, distribute the coefficient and combine like terms to complete the process: f(x) = 2(x + 1)^2 - 5.In this form, it is clear that the vertex of the parabola is at (-1, -5).
Find the point on the sphere (x+5)^2 + y^2 + (z−9)^2 = 99 nearest to
(a) the xy-plane.
(b) the point (−9,0,9).
Answer:
a) Since the sphere intersects the xy-plane then the set of points of the sphere nearest to the xy-plane is the set of points in the circumference [tex](x+5)^2+y^2=18[/tex].
b)(-14.9, 0, 9 )
Step-by-step explanation:
a) The centre of the sphere is (-5,0,-9) and the radio of the sphere is [tex]\sqrt{99} \sim 9.9[/tex]. Since |-9|=9 < 9.9, then the sphere intersect the xy-plane and the intersection is a circumference.
Let's find the equation of the circumference.
The equation of the xy-plane is z=0. Replacing this in the equation of the sphere we have:
[tex](x+5)^2+y^2+9^2=99[/tex], then [tex](x+5)^2+y^2=18[/tex].
b) Observe that the point (-9,0,9) has the same y and z coordinates as the centre and the x coordinate of the point is smaller than that of the x coordinate of the centre. Then the point of the sphere nearest to the given point will be at a distance of one radius from the centre, in the negative x direction.
(-5-[tex]\sqrt{99}[/tex], 0, 9)= (-14.9, 0, 9 )
Use a truth table to determine whether the following statement is a contradiction, a tautology or neither. If it is a contradiction or a tautology, verify your answer using logical equivalences. ((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q
Answer:
The statement [tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q[/tex] is a tautology.
Step-by-step explanation:
A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it's constructed.
We can see from the truth table that the last column contains only true values. Therefore, the statement is a tautology.
Logical equivalences are a type of relationship between two statements or sentences in propositional logic. To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. Continue doing this until you have achieved the desired statement form.
[tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q \\\equiv \neg[(p \lor q) \land (p \implies r) \land (\neg r)] \lor q[/tex] by implication law
[tex]\equiv \neg[(p \lor q) \land (\neg p \lor r) \land (\neg r))] \lor q[/tex] by implication law
[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor \neg(\neg r) \lor q[/tex] by de Morgan’s law
[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor r \lor q[/tex] by Double Negative
[tex]\equiv [(\neg p \land \neg q) \lor (p \land \neg r)] \lor r \lor q[/tex] by de Morgan’s law
[tex]\equiv [(\neg p \land \neg q) \lor q] \lor [(p \land \neg r) \lor r][/tex] by commutative and associative laws
[tex]\equiv [(\neg p \lor q) \land (\neg q \lor q)] \lor [(p \lor r) \land (\neg r \lor r)][/tex] by distributive laws
[tex]\equiv (\neg p \lor q) \lor (p \lor r)[/tex] by negation and identity laws
[tex]\equiv (\neg p \lor p) \lor (q \lor r)[/tex] by communicative and associative laws
[tex]\equiv T[/tex] by negation and domination laws
Therefore, the statement is a tautology.
The given logical statement is a tautology, as confirmed by a truth table and verified through logical equivalences, specifically equivalent to "q OR ~q," demonstrating its truth in all possible scenarios.
To determine whether the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" is a contradiction, a tautology, or neither, we can create a truth table. The statement has three propositional variables: p, q, and r, so we need a truth table with 2^3 = 8 rows to cover all possible combinations of truth values for these variables.
p | q | r | (p OR q) | (p -> r) | (~r) | ((p OR q) AND (p -> r) AND (~r)) | (((p OR q) AND (p -> r) AND (~r)) -> q)
--|---|---|----------|----------|-----|---------------------------------|-----------------------------------------
T | T | T | T | T | F | F | T
T | T | F | T | F | T | F | T
T | F | T | T | T | F | F | T
T | F | F | T | F | T | F | T
F | T | T | T | T | F | F | T
F | T | F | T | T | T | T | T
F | F | T | F | T | F | F | T
F | F | F | F | T | T | F | T
In the last column, we evaluate the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" for each row.
Now, let's analyze the results:
- The statement is True in all rows. Therefore, it is a tautology because it is always true, regardless of the truth values of p, q, and r.
We can also verify this using logical equivalences. The statement "((p OR q) AND (p -> r) AND (~r)) -> q" is logically equivalent to "q OR ~q," which is always true by the law of excluded middle. This confirms that the original statement is a tautology.
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I need help quick please!!!
Solve the system of inequalities:
2x−1 < x+3
5x−1>6−2x
x−5<0
Final answer:
To solve the system of inequalities, first, solve each inequality separately. Then, combine the solutions to find the common range of values for x that satisfy all the inequalities.
Explanation:
To solve the system of inequalities:
2x - 1 < x + 3
5x - 1 > 6 - 2x
x - 5 < 0
First, let's solve the first inequality:So, the solution to the system of inequalities is: x < 4, x > 1, x < 5
Which ratio is NOT equivalent to the other choices? A) 6:15 B) 6 to 15 C) 6 15 D) 15 6
Answer:
D
Step-by-step explanation:
Because ordering in ratios is important, so it must stay constant like 6,15.
Answer:
The answer is: D) 15/6
Step-by-step explanation:
The ratio of two given numbers such as X and Y is expressed by the symbol ':' Therefore, the ratio of X and Y or X:Y can be referred to as X is to Y and can also be expressed as a fraction X/Y or X÷Y.
Therefore, the ratio can be expressed in a number of ways, 6:15 = 6 to 15 = 6/15
Whereas, 15/6 = 15:6 ≠ 6:15
Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.
Answer:
Not always we can use a calculator to determine if a number is rational or irrational.
Step-by-step explanation:
Consider the provided information.
Can you ever use a calculator to determine if a number is rational or irrational.
Irrational number: A number is irrational if it cannot be expressed by dividing two integers. The decimal expansion of Irrational numbers are neither terminate nor periodic.
The calculators gives the approximate answer, whether the number is irrational or rational.
If it shows the terminating decimal then number is rational but otherwise, it is not possible to identify whether the number is rational or irrational as you can only see a few digits.Calculator shows the terminating decimal while the decimal expansion of an irrational number is not terminating.So, it would be difficult to identify whether a large number produced by the calculator is irrational or not. As we know that many rational numbers can be incredibly large.
So, we can say that not always we can use calculator to determine if a number is rational or irrational.
Thus, Not always we can use a calculator to determine if a number is rational or irrational.
A survey of 85 families showed that 36 owned at least one DVD player. Find the 99% confidence interval estimate of the true proportion of families who own at least one DVD player. Place your limits, rounded to 3 decimal places, in the blanks. Place the lower limit in the first blank
The 99% confidence interval estimate for the genuine proportion of families who own at least one DVD player is:
Lower bound: 0.287.
Upper bound: 0.563.
To find the 99% confidence interval estimate of the true proportion of families who own at least one DVD player, follow these steps:
Step 1. Determine the sample proportion [tex](\( \hat{p} \))[/tex]:
Number of families surveyed ( n ) = 85
Number of families owning at least one DVD player ( x ) = 36
Sample proportion [tex](\( \hat{p} \)) = \( \frac{x}{n} = \frac{36}{85} = 0.4247 \)[/tex]
Step 2. Find the standard error (SE) of the sample proportion:
Standard error formula: [tex]\( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)[/tex]
Plug in the values: [tex]\( SE = \sqrt{\frac{0.4247 \times (1 - 0.4247)}{85}} = \sqrt{\frac{0.4247 \times 0.5753}{85}} = \sqrt{\frac{0.2443}{85}} = \sqrt{0.002875} = 0.0536 \)[/tex]
Step 3. Determine the z-value for a 99% confidence interval:
The z-value for a 99% confidence interval is approximately 2.576.
Step 4. Calculate the margin of error (ME):
Margin of error formula: [tex]\( ME = z \times SE \)[/tex]
Plug in the values: [tex]\( ME = 2.576 \times 0.0536 = 0.1381 \)[/tex]
Step 5. Determine the confidence interval
Lower limit: [tex]\( \hat{p} - ME = 0.4247 - 0.1381 = 0.2866 \)[/tex]
Upper limit: [tex]\( \hat{p} + ME = 0.4247 + 0.1381 = 0.5628 \)[/tex]
Therefore, the 99% confidence interval estimate of the true proportion of families who own at least one DVD player is:
Lower limit: 0.287
Upper limit: 0.563
Complete Question:
A survey of 85 families showed that 36 owned at least one DVD player. Find the 99 \% confidence interval estimate of the true proportion of families who own at least on DVD player. Place your limits, rounded to 3 decimal places, in the blanks. Do not use any labels or symbols other than the decimal point. Simply provide the numerical values. For example, 0.123 would be a legitimate entry.
Lower limit (first blank) [tex]$=$ $\qquad$[/tex] ______ , Upper limit (second blank) = _______
The yield of strawberry plants depends on the amount of fertilizer fed to the plants. Agricultural research shows that an acre of strawberry plants will yield 770 pounds of strawberries when 70 cubic feet of fertilizer are applied. If 100 cubic of feet of fertilizer are applied, the yield will be 1100 pounds of strawberries. Use linear interpolation to estimate the yield if 75 cubic feet of fertilizer are applied. Select an answer
By using the principles of linear interpolation, the yield of strawberries with 75 cubic feet of fertilizer can be calculated as approximately 616.25 pounds.
Explanation:The yield of strawberries based on the amount of fertilizer fed to the plants can be estimated using linear interpolation. We can establish two points based on the given information: (70, 770) and (100, 1100), where the first number represents the amount of fertilizer and the second one, the yield. The interpolation line equation can be formulated as y = mx + c where m = (y2 - y1) / (x2 - x1); as such, m = (1100 - 770) / (100 - 70) = 8.25.
To find the value of c (y-intercept), we use the equation with one of the known points and solve c = y1 - m * x1 = 770 - 8.25 * 70 = -5.
The yield, y at 75 cubic feet of fertilizer can be calculated as y = 8.25 * 75 - 5 = 616.25. Therefore, the estimated yield of strawberries when 75 cubic feet of fertilizer is applied is approximately 616.25 pounds.
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Estimate the product. Round the first factor to the nearest whole number, round the second factor to the nearest hundred, and then multiply.
9 1/6 * 645 3/5
The product is approximately blank.
Answer:
The product is approximately 5400.
Step-by-step explanation:
9 1/16 rounded to the nearest whole number is 9.
645 3/5 rounded to the nearest hundred is 600.
Multiplying these values gives a product of 9·600 = 5400.
The product is approximately 5400.
Answer:
5,400.
Step-by-step explanation:
9 1/6 = 9 to the nearest whole number.
645 3/5 = 600 to the nearest hundred.
The produce = 9 * 600 = 5400.
A cubic function generally has the form f(x) = ax3 + bx2 + cx + d. If we know that for some x-value x = p we have f(p) = 0, then it must be true that x − p is a factor of f(x). Since we are told that f(3) = 0, we know that _____ is a factor.
Hi!
You know that if f(p) = 0, then (x-p) is a factor of the polynomial f(x)
Then, f(3)=0 is the case p=0, son the factor is (x-3)
Answer: Since we are told that f(3) = 0, we know that (x-3) is a factor.
Need help fast please!!!!
Answer:
∠DBC = 25°∠DCB = 65°∠ACD = 25°Step-by-step explanation:
All the right triangles are similar, so all will have the same angles.
The missing angle (B) in ΔABC is the complement of the given one:
∠DBC = 90° - 65° = 25°
The missing angles in the smaller triangles are the complements of the known acute angles in those triangles.
A diagram can help you see this.
A company makes t-shirts and their research shows that that price and demand are related linearly: p = mx + b. They know that in order to sell 10 shirts they need to set the price at $26, and in order to sell 20 shirts they need to set the price at $6. Find the linear equation relating price to demand. p= c. Preview
Answer:
P= -2x +46
Step-by-step explanation:
the relation between price and demand is
P= mx +b ........................1
when demand is 10 shirts price is $26
when demand is 20 shirts price is $6
firstly put P= 26 and x= 10 in 1
26= 10m + b.......................2
secondly put x= 20 and P= 6 in 1
6= 20m + b ............................3
solving 2 and 3 we get
m = -2
putting this value of m in either of 2 and 3 to get b
b= 46
so the final relation obtained by putting m= -2 and b= 46 in 1 we get
P= -2x +46
Determine the value (or values) of h such that the matrix: 2 - 3 h - 6 9 5 is the augmented matrix of a consistent linear system.
Answer:
In order to have a consistent linear system represented by the augmented matrix:
[tex]\left[\begin{array}{ccc}2&-3&h\\-6&9&5\end{array}\right][/tex]
the value of h must be:
[tex]h=-\frac{5}{3}[/tex]
Step-by-step explanation:
A system is consistent if it has a solution, this solution can be unique or a set of infinite solutions.
First, you take the augmented matrix and find the equivalent row echelon form using Gaussian-Jordan elimination:
To do this, you have to multiply the 1st row by 3 and add it to the 2nd row, the resulting matrix is:
[tex]\left[\begin{array}{ccc}2&-3&h\\0&0&5+3h\end{array}\right][/tex]
Now, write the system of equations:
[tex]2x_1-3x_2=h\\0x_1+0x_2=5+3h[/tex]
The only way this system has a solution is if 5+3h=0, then, to satisfy this, the value of h must be:
[tex]h=-\frac{5}{3}[/tex]
S is the set of current U.S. Senators from states that begin with A
Write each set using the roster method. Pay attention to repeated elements and think about why you don't need to list the same element more than once.
The List of Senators is below:
John Boozman
Doug Jones
Martha McSally
Lisa Murkowski
Tom Cotton
Richard C. Shelby
Kyrsten Sinema
Dan Sullivan
Answer:
see below
Step-by-step explanation:
The "roster method" means you simply list them all:
{John Boozman, Doug Jones, Martha McSally, Lisa Murkowski, Tom Cotton, Richard C. Shelby, Kyrsten Sinema, Dan Sullivan}
_____
There are no senators from these states with the same name, so repeated elements is not an issue here.
The set S includes the Senators John Boozman, Tom Cotton from Arkansas, Richard C. Shelby, Doug Jones from Alabama, and Lisa Murkowski, Dan Sullivan from Alaska. Each state has two unique senators, thus there are no repeated elements in the set.
The set S of current U.S. Senators from states that begin with 'A' using the roster method can be written as follows:
John Boozman (Arkansas)
Tom Cotton (Arkansas)
Richard C. Shelby (Alabama)
Doug Jones (Alabama)
Lisa Murkowski (Alaska)
Dan Sullivan (Alaska)
Each state beginning with 'A' (Alabama, Alaska, and Arkansas) contributes two senators to the set. S, as defined, would not have repeated elements since senators are unique to each state they represent, and no senator represents more than one state.
Find the inverse Laplace transforms, as a function of x, of the following functions:
2s^2/ (s – 1)(s^2 +1)
Answer: The required answer is
[tex]f(x)=e^x+\cos x+\sin x.[/tex]
Step-by-step explanation: We are given to find the inverse Laplace transform of the following function as a function of x :
[tex]F(s)=\dfrac{2s^2}{(s-1)(s^2+1)}.[/tex]
We will be using the following formulas of inverse Laplace transform :
[tex](i)~L^{-1}\{\dfrac{1}{s-a}\}=e^{ax},\\\\\\(ii)~L^{-1}\{\dfrac{s}{s^2+a^2}\}=\cos ax,\\\\\\(iii)~L^{-1}\{\dfrac{1}{s^2+a^2}\}=\dfrac{1}{a}\sin ax.[/tex]
By partial fractions, we have
[tex]\dfrac{s^2}{(s-1)(s^2+1)}=\dfrac{A}{s-1}+\dfrac{Bs+C}{s^2+1},[/tex]
where A, B and C are constants.
Multiplying both sides of the above equation by the denominator of the left hand side, we get
[tex]2s^2=A(s^2+1)+(Bs+C)(s-1).[/tex]
If s = 1, we get
[tex]2\times 1=A(1+1)\\\\\Rightarrow A=1.[/tex]
Also,
[tex]2s^2=A(s^2+1)+(Bs^2-Bs+Cs-C)\\\\\Rightarrow 2s^2=(A+B)s^2+(-B+C)s+(A-C).[/tex]
Comparing the coefficients of x² and 1, we get
[tex]A+B=2\\\\\Rightarrow B=2-1=1,\\\\\\A-C=0\\\\\Rightarrow C=A=1.[/tex]
So, we can write
[tex]\dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s+1}{s^2+1}\\\\\\\Rightarrow \dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}.[/tex]
Taking inverse Laplace transform on both sides of the above, we get
[tex]L^{-1}\{\dfrac{2s^2}{(s-1)(s^2+1)}\}=L^{-1}\{\dfrac{1}{s-1}\}+L^{-1}\{\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}\}\\\\\\\Rightarrow f(x)=e^{1\times x}+\cos (1\times x)+\dfrac{1}{1}\sin(1\times x)\\\\\\\Rightarrow f(x)=e^x+\cos x+\sin x.[/tex]
Thus, the required answer is
[tex]f(x)=e^x+\cos x+\sin x.[/tex]
If we changed our speed limit signs to metric, what would probably replace 45 mi/h? (Please round your answer to the nearest 1 km/h.)
km/h
Ken has 7 baseballs. Each baseball weighs 0.3 pound . Ken wants to use this model to find the total weight of the baseballs. Each hundredths block in the model represents 1 whole .
Answer:
Step-by-step explanation:
Given that,
Ken has 7 base ball
And each base ball have a weight of 0.3 pounds
If 1 ball = 0.3 pounds
Then, 7 baseball = 7 × 0.3 pounds
So, 7 baseball = 2.1 pounds
Then, the total weight of the 7base balls is 2.1 pounds
But, we want to to use the box to represent this data following the given information in the attachment
Note: since, the weight of the baseball is in decimal points
Let 0.1 pounds be 1 Square box.
Answer
1. To represent the weight of one base ball (0.3 pounds), ken should shade 3 Square box.
Since 1 pounds is 1 Square box
2. To represent the weight of all the seven base ball (2.1pounds), he should shade this amount seven times.
This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question the first part of the question
3. The shaded part of the model will represent the expression 0.3 × 7
4. The total weight of the base ball is 2.1 pounds.
Given that ;
Ken has no. of baseballs = 7
And each baseball have a weight is = 0.3 pounds
By unity method;
If 1 ball = 0.3 pounds
Then, 7 baseball = 7 × 0.3 pounds
So, 7 baseball = 2.1 pounds
Then, the total weight of the 7 baseballs is 2.1 pounds.
As per given in the question ;
We want to use the box to represent this data given in the following question .
Since, the weight of the baseball is in decimal points
Let 0.1 pounds be 1 Square box.
The weight of one base ball (0.3 pounds), ken should shade 3 Square box.Since 1 pounds is 1 Square box.
The weight of all the seven base ball (2.1pounds), he should shade this amount seven times.This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question .
The shaded part of the model will represent the expression 0.3 × 7 = 2.1 The total weight of the baseball is 2.1 pounds.For the more information about the weight management follow the link given below .
https://brainly.com/question/25124568
2 boats leave the same port at the same time.
1 traveled at a speed of 30 mph heading N 50 E
The other traveled at a speed of 26 mph heading S 70 E
How far apart are the two boats after 1 hour?
Answer: Hi!
First, if you think that a compass has degrees as units, then N50E would be
50 degrees from north in the direction of the east, so if you put our 0 in east and count counterclockwise this will be an angle of 40 degrees.
If you think north has te Y axis positive direction, and east as the X axis positive direction. then the first boat has an angle of 40° counterclockwise from the +x
so the velocity in y is Vy=30mph*sin(40°) and in x is Vx= 30mph*cos(40°)
then the total displacement will be 22.98m to east and 19.28 north
the second one goes to s 70 e, so using the same notation as before, you can write this has -20° degrees count counterclockwise.
so decomposing the velocity will give us
Vy = 26*sin(-20°) and the displacement in Y is -8.89m
Vx = 26*cos(-20°) and the displacement in X is 24.43m
so the distance between the boats in y will be 19.28m - (-8.99)m = 28.27m
and in x: 24.43m - 22.98m = 1.45m
and the total distance is [tex]D^{2} = 1.45^{2} + 28.27^{2}[/tex]
so D = 28.30 m
How many phone numbers are possible in the (770) area code if:
For the form ABC-XXXX, A is restricted to numbers 2-9. B, C, and X can be digit 0-9. Also, the number 867-5309 is not used.
a. 6,999,999
b. 7,000,000
c. 7,999,999
d. 8,000,000
Answer:
C
Step-by-step explanation:
A can be from 2- 9 ( 8 digits)
B can be 0 to 9 (10 digits)
C can be 0 to 9 (10 digits)
Each of the X's can be 0 to 9 (10 digits)
To get the number of possibilities, we multiply them to get:
8 * 10 * 10 * 10 * 10 * 10 * 10 = 8,000,000
But now, 1 number (867-5309) is restricted, so the number of possibilities decrease by 1:
8,000,000 - 1= 7, 999, 999
Correct answer is C
Suppose that 88% of bolts and 83% of nails meet specifications. One bolt and one nail are chosen independently. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. What is the probability that at least one of them meets specifications? (Round the final answer to four decimal places.)
Answer:
0.9796
Step-by-step explanation:
Given that 88%of bolts and 83% of nails meet specifications.
This implies that for a randomly selected bowl the prob that it meets specifications = P(A) = 0.88
Similarly, for a randomly selected bolt, it meets specifications is P(B) = 0.83
We know that bolt and nail are independent of each other.
Hence [tex]P(A \bigcap B) = P(A)P(B)\\\\=0.88*0.83=0.7304[/tex]
Required probability = Probability that atleast one of them meets specifications)
[tex]= P(AUB)\\=P(A)+P(B)-P(A \bigcap B)\\=0.88+0.83-0.7304\\=1.71-0.7304\\=0.9796[/tex]
Final answer:
To find the probability that at least one of the chosen bolt or nail meets specifications, use the complement rule. Multiply the probabilities that each does not meet specifications and subtract from 1 to find the probability that at least one meets specifications.
Explanation:
To find the probability that at least one of the chosen bolt or nail meets specifications, we can use the complement rule. The complement of the event that at least one meets specifications is the event that none of them meet specifications. The probability that the bolt does not meet specifications is 1 - 0.88 = 0.12, and the probability that the nail does not meet specifications is 1 - 0.83 = 0.17. Since the events are chosen independently, we can multiply these probabilities together to get the probability that both do not meet specifications: 0.12 x 0.17 = 0.0204.
Using the complement rule, we subtract this probability from 1 to find the probability that at least one meets specifications: 1 - 0.0204 = 0.9796. Therefore, the probability that at least one of the bolt and nail meets specifications is approximately 0.9796.
Rewrite the following system of linear equations in matrix equation form and in vector equation form. Solve the system.
a - b + 2x - 8y + z =3
2a - b - 4x + y - 2z = 1
-4a + b + 4x - 3x - z = -1
Answer:
The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]
Step-by-step explanation:
The matrix associated to the problem is [tex]A=\left[\begin{array}{ccccc}1&-1&2&-8&1\\2&-1&-4&1&-2\\-4&1&4&-3&-1\end{array}\right][/tex] and the vector of independent terms is (3,1,-1)^t. Then the matrix equation form of the system is Ax=b.
The vector equation form is [tex]a\left[\begin{array}{c}1\\2\\-4\end{array}\right]+b\left[\begin{array}{c}-1\\-1\\1\end{array}\right] + x\left[\begin{array}{c}2\\-4\\4\end{array}\right]+y\left[\begin{array}{c}-8\\1\\-3\end{array}\right] + z\left[\begin{array}{c}1\\-2\\-1\end{array}\right]=\left[\begin{array}{c}3\\1\\-1\end{array}\right][/tex].
Now we solve the system.
The aumented matrix of the system is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\2&-1&-4&1&-2&1\\-4&1&4&-3&-1&-1\end{array}\right][/tex].
Applying rows operations we obtain a echelon form of the matrix, that is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\0&1&-8&-15&-4&-5\\0&0&1&-80&-9&-4\end{array}\right][/tex]
Now we solve for the unknown variables:
x-80y-90z=-4 then x=-4+80y+90zb-8x-15y-4z=-5, b-8(-4+80y+90z)-15y-4z=-5 then b=-37+655y+724z.a-b+2x-8y+z=3, a-(-37+655y+724z)+2(-4+80y+90z)-8y+z=3, then a=-26+503y+543zSince the system has two free variables then has infinite solutions.
The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]
please help
tap the picture there are two problem I'm stuck on
Answer:
1. g(x)=2x+1-3 --> g(x)=2x-2, which is also y=2x-2, so you can graph it.
Step-by-step explanation:
Question 1: If f(x) = 2x+1, then you can see that all you have to do is substitute the equation for f(x) into the g(x) equation because g(x)= f(x)-3. So, if you substitute it, the equation will be g(x) = (2x+1) -3, then you just solve the rest of the equation. Put it into slope intercept form, y=mx+b, and then graph the equation.
Sorry, I don't really understand number 2 myself, so hopefully I could help with he first one.
A lidocaine drip is infusing at 30 mL/hr on an infusion device. The drug is mixed 2g in 500mL D5W. How many mg/minute is the patient receiving?
Answer:
2 mg/minute.
Step-by-step explanation:
We have been given that a drug is mixed 2 g in 500 ml.
First of all, we will convert 2 grams to milligrams. 1 gram equals 1000 milligrams.
2 grams = 2,000 mg.
Now, we will find amount of mg per ml as:
[tex]\text{The amount of mg per ml}=\frac{2000\text{ mg}}{\text{500 ml}}[/tex]
[tex]\text{The amount of mg per ml}=\frac{4\text{ mg}}{\text{ml}}[/tex]
We have been given that a lidocaine drip is infusing at 30 mL/hr on an infusion device, so amount of mg per hr would be:
[tex]\text{Amount of mg per hour}=\frac{30\text{ ml}}{\text{ hr}}\times \frac{4\text{ mg}}{\text{ml}}[/tex]
[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}[/tex]
We know 1 hour equals 60 minutes.
[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}\times \frac{\text{1 hour}}{\text{60 minutes}}[/tex]
[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{60 minutes}}[/tex]
[tex]\text{Amount of mg per hour}=\frac{2\text{ mg}}{\text{minute}}[/tex]
Therefore, the patient is receiving 2 mg/minute.
To calculate the mg/minute that the patient is receiving from the lidocaine drip, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams.
Explanation:To calculate the mg/minute that the patient is receiving, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams. Here's how:
Convert the flow rate from mL/hr to mL/minute: Since there are 60 minutes in an hour, the flow rate is 30 mL/hr ÷ 60 = 0.5 mL/minute.Convert the drug concentration from grams to milligrams: There are 1000 milligrams in a gram, so 2 grams is equal to 2 × 1000 = 2000 milligrams.Calculate the mg/minute: Multiply the flow rate in mL/minute by the drug concentration in milligrams: 0.5 mL/minute × 2000 milligrams = 1000 mg/minute.Learn more about lidocaine here:https://brainly.com/question/33462292
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If 10 millimeters (10 mm) equals 1 centimeter (1 cm), then 10 square millimeters (10 sq mm) equals: Select one: A 100 sq cm B. 0.10 sq cm C. 0.0010 sq cm D. 0.010 sq cm E. 1 sq cm
Answer:
The correct answer is B. : 10 sq mm = 0.1 sq cm
Step-by-step explanation:
It is just a matter of changing the units. The equivalence we need to know is 1cm = 10 mm. Also, we need to have in mind that we can write 10 sq mm as 10 mm*mm, because : 10 sq mm = 10 mm² = 10 mm*mm
Now we multiply two times by the fraction (1cm / 10 mm), which does not alter our measurement because the fraction is the same as multiplying by 1.
10 sq mm = 10 mm* mm = (10 mm*mm)*(1 cm / 10 mm)*(1 cm / 10 mm) = (10 mm*mm*cm*cm/ 10*10 mm*mm) =10/100 cm*cm = 0.1 cm² = 0.1 sq cm
Therefore, we have the equivalency : 10 sq mm = 0.1 sq cm