In calculus, the derivative is a fundamental concept that measures how a function changes with respect to its input variable. It is a mathematical tool used to study the behavior of functions and is used in many areas of mathematics, physics, engineering, and economics.
When t = 3, the velocity of the object is 27 meters per second, and its acceleration is -2 meters per second squared.
When the velocity and acceleration have opposite signs, the speed of the object is decreasing. The acceleration of an object can be calculated using the derivative of the velocity of the object with respect to time.
v(t) = 36 - t², 0 ≤ t ≤ 6
Let us take the derivative of the velocity, v(t), to obtain acceleration, a(t) = v'(t) = -2t.When t = 3, v(3) = 36 - 3² = 27 meters per second.Therefore, when t = 3, the velocity of the object is 27 meters per second. When t = 3, a(3) = -2(3) = -6 meters per second squared. The object's acceleration is -6 meters per second squared.
The speed of the object is decreasing when the velocity and acceleration have opposite signs.
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Rectangle EFGH is dilated by a scale
factor of 3 to form rectangle E'F'G'H'.
Side EF measures 22. What is the
measure of side E'F'?
The measure of side E'F' of rectangle EFGH when dilated by a scale factor of 3 to form rectangle E'F'G'H' is 66 units.
What is scale factor?Scale factor is used in mathematics, particularly in geometry and measurement, to describe the transformation of one figure into another through dilation or resizing. It is represented by a number or a ratio, such as 2:1 or 1/2, which indicates how many times larger or smaller the new figure is compared to the original. For example, if the side length of a square is doubled, the resulting square is four times larger in area, because the area of a square is proportional to the square of its side length. In this case, the scale factor of the dilation is 2, because the new square is twice as large as the original square. Scale factor is also used in maps, blueprints, and architectural drawings to represent the ratio between the actual size of an object or space and its representation on paper or screen.
Here,
If rectangle EFGH is dilated by a scale factor of 3 to form rectangle E'F'G'H', then all corresponding sides of the two rectangles are in the ratio of 3:1. Since EF measures 22, then the corresponding side E'F' of the dilated rectangle is:
E'F' = EF × scale factor
E'F' = 22 × 3
E'F' = 66 units
Therefore, the measure of side E'F' is 66.
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Schools have different ways of fund raising. The parents and the SGB of Progress High School agree that each learner should donate an amount to the school. The money is payable during the first month of the year. 1.1 Use TABLE 1 to answer the questions that follow. Write down the donation per leamer. 1.2 TABLE 1: INCOME IN RANDS OF FUND RAISING Number of learners that paid Income (R) 1 200 1.3 1.5 Calculate the missing value A. 10 2 000 20 45 215 4 000 9 000 A [Adapted from original school financial books ] Use TABLE 1 and write down the dependent variable. 1.4 Write the income received from 10 leamers to the income received from 45 learners, in a ratio in its simplest form. (3) (2) (2) (2) have The SGB chairperson claims that if 80% of the leamers paid, the school would raised more than R170 000. There are 1 100 learners enrolled at the school. Verify, by showing ALL calculations, whether his statement is valid. (4)
Answer: 1.1. The donation per learner cannot be determined from the given table.
1.2. TABLE 1: INCOME IN RANDS OF FUNDRAISING
Number of learners that paid Income (R)
1 200
1.3. To calculate the missing value A, we need to add up all the given incomes and subtract it from the total income for 45 learners, which is 45 x A. Then we can solve for A:
Total income = 200 + 150 + 10(2,000) + 20(45) + A + 4,000 + 9,000
Total income = 45A
45A = 25,150
A = 558.89
Therefore, the missing value A is R558.89.
1.4. The dependent variable in the table is the income received from fundraising.
To find the ratio of income received from 10 learners to income received from 45 learners, we need to divide the income received from 10 learners by the income received from 45 learners and simplify the fraction:
Income from 10 learners = R1,100 (since each learner donates R110)
Income from 45 learners = R215
Ratio = Income from 10 learners : Income from 45 learners
= 1,100 : 215
= 20 : 3 (in its simplest form)
The total number of learners enrolled at the school is 1,100. If 80% of the learners paid, then the number of learners who paid is:
80% of 1,100 = 0.8 x 1,100 = 880 learners
The minimum income that the school can raise if 80% of the learners paid is when each of the 880 learners paid the minimum donation, which is R150:
Minimum income = 880 x 150 = R132,000
Since R132,000 is less than R170,000, the SGB chairperson's statement is not valid.
Step-by-step explanation:
Please please please help me!!!!!
The volume of the sphere which is equivalent to the lung capacity is approximately =2,571 cm³
How to calculate the volume of the sphere?To calculate the volume of a sphere the formula used = V = 4/3 πr³
Radius = 8.5 cm
First cube the radius = 8.5³ = 614.125
The, multiply r³ by π = r³×π = 614.125× 3.14= 1928.3525
Take this answer and multiply it by 4 = 4×1928.3525= 7713.41
Last, divide this answer by 3 = 7713.41/3 = 2571.136666
Therefore the volume of the balloon = 2,571 cm³(approximately)
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Please help with the highlighted red on part(c) as well as part(d).
Use the formula (2) above and the results from part (b) to write the general solution of our system (4). Write this solution in your document. What happens to the system as t gets large?
Consider the system of differential equations
dx/dt = 3x+4y
dy/dt = -x - 2y
The solution of the system goes to infinity as t gets large.
Consider the given system of differential equations: dx/dt = 3x + 4ydy/dt = −x − 2yNow, find the solution of the system:(2) dx/dt + 2 dy/dt = 7xObserve the second equation and multiply it by 2:2 dy / dt = −2x − 4y(3) dx/dt + 2 dy/dt = 3x − 4ySubstitute the value of dy/dt from (3) into (2):(2) dx/dt + 3x − 4y = 7xSimplify this equation:dx/dt − 4x = 4yTake the laplace transform of both sides: Laplace{dx/dt − 4x} = Laplace{4y} ⇒ sX(s) − x(0) − 4X(s) = 4Y(s) ⇒ X(s) = {4Y(s)}/{s − 4}Now, find the Laplace transform of y(s):(1) dy/dt = −x − 2y ⇒ Laplace{dy/dt} = −Laplace{x} − 2Laplace{y} ⇒ sY(s) − y(0) = −X(s) − 2Y(s) ⇒ Y(s) = {−X(s)}/{s + 2}Substitute the value of X(s) in the above equation, we get:Y(s) = {−4Y(s)}/[(s − 4)(s + 2)]Simplify this equation:Y(s) = {A}/{s − 4} + {B}/{s + 2}Now, find the values of A and B:Multiplying the above equation by (s − 4) and (s + 2), we get:Y(s) = A(s + 2) + B(s − 4)Simplify this equation:Y(s) = (A + B)s + 2A − 4BAs per the equation (3), we can say A + B = 0, 2A − 4B = −4On solving, we get A = 2, B = −2Therefore, the value of Y(s) is:Y(s) = {2}/{s − 4} − {2}/{s + 2}Now, substitute the value of X(s) and Y(s) in the following equation:dx/dt − 4x = 4yThe solution of the above differential equation is:x(t) = {2}/{3}e^{4t} − {5}/{3}e^{-2t}The above is the general solution of the given system of differential equations.Now, find what happens to the system as t gets large?As t gets large, the term {5}/{3}e^{-2t} will tend to zero and {2}/{3}e^{4t} will tend to infinity.
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A grading machine can grade 96 multiple choice test in 2 minutes. if a teacher has 300 multiple choice test to grade, predict the number of minutes it will take the machine the grade the tests. please answer with details please
Answer:
6.25 minutes
Step-by-step explanation:
So, if there are 96 tests being graded in 2 minutes, the average per minute (using the equation 96/2 = average) would be 48 per minute. Next you would do 300/48 to get the number of minutes that it would take to grade that amount. The answer to that would be 6.25, So it would take approximately 6.25 minutes.
GIVING BRAINLIST SO HURRY!!!!
What is the median of the data set represented by the dot plot?
Answer:
14
Step-by-step explanation:
scores 10 10 12 13 14 15 18 19 19 Median score is the one in the middle
fifty-three percent of all persons in the u.s. population have at least some college education. choose 10 persons at random. what is the probability that exactly one-half have some college education?
The probability that exactly one-half have some college education is 0.0439 (rounded to four decimal places)
Step by step explanation: We can use the binomial probability distribution formula for this problem, where:p = probability of success (i.e., having some college education) = 0.53q = probability of failure (i.e., not having some college education) = 0.47n = number of trials (i.e., persons chosen at random) = 10x = number of successes (i.e., persons with some college education) = [tex]5P(x = 5) = C(10,5)p^5q^5[/tex] where [tex]C(n,r)[/tex] is the combination function that gives the number of ways to choose r items from a set of n items.
It is given by:[tex]C(n,r) = n! / (r!(n-r)!)[/tex] Thus, we can substitute the given values and compute:[tex]P(x = 5) = C(10,5)p^5q^5 = 252(0.53)^5(0.47)^5= 0.0439[/tex] (rounded to four decimal places)Therefore, the probability that exactly one-half have some college education is 0.0439.
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If Karel starts at Street 1 and Avenue 1, facing East, where will Karel be, and what direction will Karel be facing after running the following code? (Assume the world is 10x10 in size)move();turnLeft();putBall();turnLeft();turnLeft();turnLeft();move();turnLeft();
After running the given code, Karel will be at Street 2 and Avenue 2, facing North.
Here is a step-by-step explanation of what the code does:
move(); - Karel moves one block east, to Street 1 and Avenue 2.
turnLeft(); - Karel turns left to face north.
putBall(); - Karel puts a ball at Street 1 and Avenue 2.
turnLeft(); turnLeft(); turnLeft(); - Karel turns left three times to face south.
move(); - Karel moves one block south to Street 2 and Avenue 2.
turnLeft(); - Karel turns left to face east.
So after executing this code, Karel will be at Street 2 and Avenue 2, facing North.
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I need help with this question (PLEASE IM BEGGING YOU)
Answer:
The correct options are A and B.
For option A:
Sy = y = 3x - 10
Substituting x = -8 and y = 4:
4 = 3(-8) - 10
4 = -24 - 10
4 = -34
Since the left-hand side does not equal the right-hand side, (-8,4) is not a solution to the system.
For option B:
y = -1x + 10
Substituting x = -8 and y = 4:
4 = -1(-8) + 10
4 = 8 + 10
4 = 18 - Incorrect
For option C:
y = 5x + 24
y = 5(-8) + 24
y = -40 + 24
y = -16 - Incorrect
For option D:
y = 6x + 68
Substituting x = -8 and y = 4:
4 = 6(-8) + 68
4 = -48 + 68
4 = 20 - Incorrect
For option E:
y = -3x + 7
9 = -3(-8) + 7
9 = 24 + 7
9 = 31 - Incorrect
Therefore, options A and B are incorrect.
A phlebotomist measured the cholesterol levels of a sample of 25 people between the ages of 35 and 44 years old. Here are summary statistics for the samples:
The 90% confidence interval for [tex]$\mu_W - \mu_M$[/tex] is approximately [tex]$-13.2 \pm 27.1$[/tex], or (-40.3, 13.9).
The answer is (c) [tex]13.2 \pm 33.1[/tex], which is not correct.
Which is the confidence interval?
A confidence interval is a range around a measurement that conveys how precise the measurement is.
We can use the two-sample t-interval formula to find the confidence interval for the difference in means:
[tex]$\bar{x}_W - \bar{x}M \pm t{\alpha/2, \nu} \sqrt{\frac{s_W^2}{n_W} + \frac{s_M^2}{n_M}}$[/tex]
where [tex]$\bar{x}_{W}$[/tex] and [tex]$\bar{x}M$[/tex] are the sample means, [tex]$s_W$[/tex] and [tex]$s_M$[/tex] are the sample standard deviations, [tex]$n_W$[/tex] and [tex]$n_M$[/tex] are the sample sizes, [tex]$\nu$[/tex] is the degrees of freedom, and [tex]$t{\alpha/2, \nu}$[/tex] is the critical value from the t-distribution with [tex]$\nu$[/tex]degrees of freedom and a level of significance of [tex]$\alpha=0.1$[/tex] (since we want a 90% confidence interval, which corresponds to a 10% level of significance).
Plugging in the values, we get:
[tex]$\bar{x}_W - \bar{x}M \pm t{0.05/2, 23} \sqrt{\frac{s_W^2}{n_W} + \frac{s_M^2}{n_M}}$[/tex]
[tex]$\begin{aligned} &= 213.6 - 226.8 \pm t_{0.025, 23} \sqrt{\frac{45.3^2}{14} + \frac{49.4^2}{11}} \ &= -13.2 \pm 2.074 \times 13.102 \ &= -13.2 \pm 27.115 \end{aligned}$[/tex]
Therefore, the 90% confidence interval for [tex]$\mu_W - \mu_M$[/tex] is approximately [tex]$-13.2 \pm 27.1$[/tex], or (-40.3, 13.9).
The answer is (c) [tex]13.2 \pm 33.1[/tex], which is not correct.
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complete question:
A phlebotomist measures the cholesterol levels of a sample of 25 people between
the ages of 35 and 44 years old. Here are summary statistics for the samples.
Cholesterol levels (mg per 100mL)
Women 35-44 years old
Men 35-44 years old
Sample mean
tilde x_{W} = 213.6
overline x_{M} = 226.8
Sample standard deviation
s_{W} = 45.3
s_{M} = 49.4
Sample size
n_{W} = 14
nu = 11
Assume that the conditions for inference have been met. Let mu_{w} ^ (- mu_{u}) be the difference in mean
cholesterol levels (in milligrams per 100 ml) in women and men of those ages.
Which of the following is a 90% confidence interval for mu_{W} ^ (- mu_{M})
Use a calculator to calculate the interval. (State your calculator settings for partial credit)
(a) - 13.2 plus/minus 33.1
(b) -13.2±29.7
(c) 13.2 plus/minus 33.1
(d) 13.2 plus/minus 29.7
(e) 13 plus/minus 33.1
Answer:
-13.2+- 33.1
Step-by-step explanation:
a chess club with 80 memberd is electing a new presidentm jose received 52 votes. what percentage of the club members voted for Jose?
Answer:
Step-by-step explanation:
(52 divided by 80) multiplied by 100
that will give you the answer
How could 1/5 + -4/5 be modeled on a number line
The expression 1/5 + -4/5 has plotted on the number line
To model 1/5 + (-4/5) on a number line, we can start by placing a point at 0 on the number line. Then we can represent 1/5 by placing another point to the right of 0, one-fifth of the distance from 0 to 1. We can represent -4/5 by placing a point to the left of 0, four-fifths of the distance from 0 to -1.
To add these two points together, we start at the point representing 1/5 and move four-fifths of the distance towards the left, since -4/5 is negative. The resulting point represents the sum of 1/5 and -4/5.
To solve this problem algebraically, we can add the two fractions together by finding a common denominator:
1/5 + (-4/5) = 1/5 - 4/5
= (1-4)/5
= -3/5
= -0.6
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Jaxon is flying a kite, holding his hands a distance of 3.25 feet above the ground and
letting all the kite's string play out. He measures the angle of elevation from his hand
to the kite to be 24°. If the string from the kite to his hand is 105 feet long, how many
feet is the kite above the ground? Round your answer to the nearest tenth of a foot if
necessary.
Answer: 46.0 ft
Step-by-step explanation:
[tex]\text{sin} \ 24^o=\dfrac{x}{105}[/tex]
[tex]x=105 \ \text{sin 24}^o[/tex]
So, the distance above the ground is [tex]\text{105 sin} \ 24^o+3.25\thickapprox \boxed{46.0 \ \text{ft}}[/tex]
a woman with blood type a has three children with a man who has blood type ab. the first child has blood type b. what is the probability that the second child born to the couple will have blood type ab?
The probability that the second child born to the couple will have blood type AB is 1/2.
What is Blood Grouping?Blood grouping is the categorization of blood into groups depending on the type of antigen and antibody present on the surface of red blood cells (RBCs).
The ABO blood grouping system, which includes A, B, AB, and O blood types, is the most well-known blood grouping system.
Blood group Rh (Rhesus) grouping system is another well-known blood grouping system that can be determined.
Therefore, the probability that the second child born to the couple will have blood type AB is 1/2.
50 percent of the offspring of this couple will inherit the gene for the ABO blood type A from the mother, while the remaining 50% will inherit the gene for the blood type B from the father.
The probability of inheriting the gene for blood type AB is thus 1/2 for a child born to these parents.
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juan mcdonald is willing to pay $900 for a new ipad. he offers to pay $800 for an ipad at the apple store. it costs apple $700 to produce this ipad. a voluntary economic transaction between juan and apple occur because would be better off due to the transaction. a. will not; only apple b. will; both juan and apple c. will not; only juan d. will; neither juan nor apple
In the following question, option B, Juan McDonald is willing to pay $900 for a new iPad. he offers to pay $800 for an iPad at the apple store. it costs apple $700 to produce this iPad. A voluntary economic transaction between "Juan and Apple" will occur because it would be better off due to the transaction.
What is a voluntary economic transaction? Voluntary economic transactions involve willing buyers and willing sellers, and they usually involve the exchange of goods and services in return for money. In a free-market economy, people have the freedom to make voluntary economic transactions, and government intervention is minimal. What are the benefits of voluntary economic transactions? The benefits of voluntary economic transactions include the following: They are beneficial to both parties involved in the transaction.
The buyer obtains what they require, while the seller obtains the money they require. The exchange of goods and services for money encourages individuals to create, produce, and sell more. The economy is stimulated by increased economic activity, which leads to more job creation, more employment opportunities, and more revenue for the government. Juan McDonald is willing to pay $900 for a new iPad. He offers to pay $800 for an iPad at the Apple store. It costs Apple $700 to produce this iPad.A voluntary economic transaction between Juan and Apple will occur because it would be better off due to the transaction. So, the answer is option B: will; both Juan and Apple.
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kuta software infinite algebra 2 logarithmic equationsSolve each equation.1) log 5x = log (2x + 9)2) log (10 − 4x) = log (10 − 3x)3) log (4 p − 2) = log (−5 p + 5) 4) log (4k − 5) = log (2k − 1)5) log (−2a + 9) = log (7 − 4a) 6) 2log 7−2r = 07) −10 + log 3(n + 3) = −10 8) −2log 57x = 29) log −m + 2 = 4 10) −6log 3(x − 3) = −2411) log 12 (v2+ 35) = log 12 (−12v − 1) 12) log 9(−11x + 2) = log 9(x2 + 30)
The solutions of provide logarithmic equations are present in below :
1) x = 9 ; 2) x = 0 ; 3)p = 7/9 ; 4) k= 2 ; 5) a= -1 ; 6) r = -1/2 ; 7) n = 2 ; 8) x = 1/35 ; 9) m = -2 ; 10) x = 84 ; 11) v = -6, -6 ; 12) x = -4, -7
The logarithmic number is associated with exponent and power, such that if xⁿ = m, then it is equal to logₓ m = n. That is exponential value are inverse of logarithm values. Some basic properties of logarithmic numbers:
Product property : logₐ mn = logₐ m + logₐ n Quotient property : logₐ m/n = logₐ m - logₐ n Power property : logₐ mⁿ = n logₐ m Change of base property : log꜀a = (logₙ a) / (logₙ b) log꜀a = n <=> cⁿ = aNow, we solve each logarithm equation one by one. Assume that 'log' is the base-10 logarithm where absence of base.
1) log (5x) = log (2x + 9)
Exponentiate both sides
=> 5x = 2x + 9
=> 3x = 9
=> x = 9
2) log (10 − 4x) = log (10 − 3x)
Exponentiate both sides,
=> 10 - 4x = 10 - 3x
simplify, => x = 0
3) log (4p − 2) = log (−5p + 5)
Exponentiate both sides,
=> 4p - 2 = - 5p + 5
simplify, => 9p = 7
=> p = 7/9
4) log (4k − 5) = log (2k − 1)
Exponentiate both sides,
=> 4k - 5 = 2k - 1
simplify, => 2k = 4
=> k = 2
5) log (−2a + 9) = log (7 − 4a)
Exponentiate both sides,
=> - 2a + 9 = 7 - 4a
simplify, => 2a = -2
=> a = -1
6) 2log₇( −2r) = 0
=> log₇( −2r) = 0
using the property, log꜀a = n <=> cⁿ = a
=> ( 7⁰) = - 2r
=> -2 × r = 1 ( since a⁰ = 1 )
=> r = -1/2
7) −10 + log₃(n + 3) = −10
=> log₃(n + 3) = −10 + 10 = 0
using the property, log꜀a = n <=> cⁿ = a
=> 3⁰ = n + 3
=> 1 = n + 3
=> n = 2
8) −2log₅ ( 7x ) = 2
=> log₅ 7x = -1
=> 5⁻¹ = 7x
=> x = 1/35
9) log( −m) + 2 = 4
=> log( −m) = 2
Exponentiate both sides,
=> -m = 2
=> m = -2
10) −6log₃ (x − 3) = −24
simplify, log₃ (x − 3) = 4
=> (x - 3) = 3⁴ ( since log꜀a = n <=> cⁿ = a )
=> x - 3 = 81
=> x = 84
11) log₁₂ (v²+ 35) = log₁₂ (−12v − 1)
=> log₁₂ (v²+ 35) - log₁₂ (−12v − 1) = 0
Using the quotient property of logarithm,
[tex]log_{12}( \frac{v²+ 35}{-12v-1}) = 0 [/tex]
[tex]\frac{v²+ 35}{-12v - 1} = {12}^{0} = 1 [/tex]
[tex]v²+ 35 = −12v − 1[/tex]
[tex]v²+ 35 + 12v + 1 = 0[/tex]
[tex]v²+12v + 36 = 0[/tex]
which is a quadratic equation, and solve it by middle term splitting method,
[tex]v²+ 6v + 6v + 36= 0[/tex]
[tex]v(v + 6) + 6(v + 6)= 0[/tex]
[tex](v + 6) (6 + v)= 0[/tex]
so, v = -6, -6
12) log₉(−11x + 2) = log₉ (x²+ 30)
=> log₉ (x²+ 30) - log₉(−11x + 2) = 0
Using the quotient property of logarithm,
[tex]log₉(\frac{x²+ 30 }{−11x + 2}) = 0[/tex]
[tex] \frac{x²+ 30}{-11x + 2} ={9}^{0} = 1 [/tex]
=> x² + 30 = - 11x + 2
=> x² + 11x + 30 -2 = 0
=> x² + 11x + 28 = 0
Factorize using middle term splitting,
=> x² + 7x + 4x + 28 = 0
=> x( x + 7) + 4( x + 7) = 0
=> ( x + 4) (x+7) = 0
=> either x = -4 or x = -7
Hence, required solution is x = -4, -7.
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Complete question:
kuta software infinite algebra 2 logarithmic equationsSolve each equation.
1) log 5x = log (2x + 9)
2) log (10 − 4x) = log (10 − 3x)
3) log (4 − 2) = log (−5 p + 5)
4) log (4k − 5) = log (2k − 1)
5) log (−2a + 9) = log (7 − 4a)
6) 2log₇ −2r = 0
7) −10 + log₃(n + 3) = −10
8) −2log₅ 7x = 2
9) log −m + 2 = 4
10) −6log₃ (x − 3) = −24
11) log₁₂ (v²+ 35) = log₁₂ (−12v − 1)
12) log₉(−11x + 2) = log₉ (x²+ 30)
I'll give Brainliest to whoever answers correctly and the answer is for 25 points
Answer:
y= (5/3)x -4/3
Step-by-step explanation:
Slope obtained from the two points of the graph.
m=5/3
Points: A(-1,-3). B(2,2)
m=(y2-y1)/(x2-x1)
m=(2-(-3))/(2-(-1))
m=5/3
Replacing in the equation form: y=mx+b
y=(5/3)x+b
2=(5/3)2+b
2=10/3 +b
2-10/3= b
6/3 - 10/3 = b
b= -4/3
Joining all the terms we obtain:
y= (5/3)x -4/3
Answer: The slope is 5/3
Step-by-step explanation:
the poin at (2,2) is 5 up and 3 right of the point at (
-1,-3)
For a plant having the transfer function G(s) 74109 it is proposed to use a controller in a unity feedback system and having the transfer function D(s) Solve for the parameters of this controller so that the closed loop will have the characteristic equation (s 6) (s + 3)(s2 + 3s + 9) = 0.1 s(s+di) { Answer: c2 = 18, ci = 54. co = 162, di = 9} Exercise. Show that if the reference input to the system of the above exercise is a step of amplitude A, the steady-state error will be zero.
At steady-state, the Laplace transform of the output is given by [tex]lim s->0 sC(s)[/tex]. If this limit exists, then the steady-state error is zero.
The transfer function of a plant with transfer function G(s) = 74109 is given.
It is proposed to use a controller in a unity feedback system with the transfer function D(s). The task is to determine the parameters of the controller so that the closed-loop will have the characteristic equation (s 6) (s + 3)(s2 + 3s + 9) = 0. The parameters of the controller can be determined by comparing the coefficients of the open-loop transfer function with the characteristic equation.The open-loop transfer function of the system is given by G(s)D(s). The characteristic equation of the closed-loop system is given by 1 + G(s)D(s) = 0.We proceed in the following manner:
Step 1: Write the open-loop transfer function.[tex]G(s)D(s) = 74109 * (s+ci)/(s+c1) * K/(s+di)[/tex]
Step 2: Write the characteristic equation.(s + 6) (s + 3) (s2 + 3s + 9) = 0
Step 3: Compare the coefficients of the open-loop transfer function with the characteristic equation.
c1 + ci + di = 9c1ci + c1di + ci(di + 3) + K * 74109 = 27c1ci(di + 3) + K * 74109 * c1 = 81K * 74109 * di = 6 * 3 * 9 * (-74109)
Step 4: Solve for the parameters of the controller.The solution is obtained by solving the above equations.c2 = 18ci = 54co = 162di = 9
Step 5: Show that if the reference input to the system of the above exercise is a step of amplitude A, the steady-state error will be zero.The steady-state error can be calculated using the final value theorem. If the system is subjected to a step input of amplitude A, then the Laplace transform of the input is A/s.
The output of the system is given by[tex]C(s) = G(s)D(s) R(s)[/tex], where R(s) is the Laplace transform of the reference input. The steady-state error is given by the difference between the input and the output at steady-state. This can be calculated using the final value theorem.
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Please help me with my math
Answer:
[tex]\textsf{$\boxed{\checkmark}\;\;y$-value\;of\;vertex\;is\;$-1$}[/tex]
[tex]\textsf{$\boxed{\checkmark}$\;\;Minimum\;value\;occurs\;at\;$y = -1$}[/tex]
Step-by-step explanation:
Given equation:
[tex]y=x^2+8x+15[/tex]
As the given equation is quadratic with a positive leading coefficient, it is a parabola that opens upwards. Therefore, its vertex is its minimum point. This means that the minimum value of the range is the y-value of the vertex.
The x-value of the vertex of a parabola in the form y = ax² + bx + c is x = -b/2a. Therefore, the x-value of the vertex of the given equation is:
[tex]\implies x=\dfrac{-8}{2(1)}=-4[/tex]
To find the y-value of the vertex, substitute x = -4 into the equation:
[tex]\begin{aligned}\implies y&=(-4)^2+8(-4)+15\\&=16-32+15\\&=-16+15\\&=-1\end{aligned}[/tex]
Therefore, the minimum y-value of the function is y = -1, so the range is y ≥ -1.
Therefore, the following are true statement about the given equation:
y-value of vertex is -1Minimum value occurs at y = -1Solve for b. a = 9b²c
b=±√ac÷9
b=±√a÷3c
b=±√ac÷3c
b=±3√ac
Step-by-step explanation:
9 b^2 c = a divide both sides of the equation by 9c
b^2 = a / (9c) now take the sqrt of both sides
b = ± sqrt ( a/(9c) ) = ± 1/3 sqrt ( a/c) <====I do not see the correct answer posted.... perhaps one of the choices was transcribed incorrectly....
solve the differential equation by variation of parameters, subject to the initial conditions y(0)
The differential equation by variation of parameters, subject to the initial conditions y(0) can be found by homogeneous equation and by integrating it.
The solution to the differential equation by variation of parameters, subject to the initial conditions y(0), can be found by following these steps:
1. Separate the equation into homogeneous and particular parts.
2. Solve the homogeneous equation for its particular solution.
3. Construct a new auxiliary equation from the particular solution, integrating the particular solution with respect to the parameters.
4. Solve the auxiliary equation, again with respect to the parameters, and then integrate the result with respect to x.
5. Substitute the particular solution and the new parameters back into the original equation to solve it completely.
6. Substitute the initial condition y(0) to solve for the constants of integration.
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solve for x and graph the solution on the number line below
We can graph this solution on the number line by placing a point at 7.6.
What is graph ?Graphs are visual representations of data or information. They are used to show relationships between different pieces of data and display numerical data in a more meaningful way. Graphs are made up of individual elements called nodes which are connected by edges. Nodes represent individual data points or pieces of information. Edges represent the relationship between two nodes and can represent either a physical or a logical link. Graphs can be used to represent many different types of data or information, such as social networks, transportation networks, and even biological relationships. Graphs are a powerful tool for understanding complex data sets, making them an essential tool for data analysis.
80∠ 10x + 4 = 20
80 10x + 4 = 20
80 - 4 = 10x
76 = 10x
7.6 = x
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We can graph this solution on the number line by placing a point at 7.6.
What is graph ?Graphs are visual representations of data or information. They are used to show relationships between different pieces of data and display numerical data in a more meaningful way. Graphs are made up of individual elements called nodes which are connected by edges. Nodes represent individual data points or pieces of information. Edges represent the relationship between two nodes and can represent either a physical or a logical link. Graphs can be used to represent many different types of data or information, such as social networks, transportation networks, and even biological relationships. Graphs are a powerful tool for understanding complex data sets, making them an essential tool for data analysis.
80∠ 10x + 4 = 20
80 10x + 4 = 20
80 - 4 = 10x
76 = 10x
7.6 = x
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Radioactive decay tends to follow an exponential distribution; the half-life of an isotope is the time by which there is a 50% probability that decay has occurred. Cobalt-60 has a half-life of 5.27 years. (a) What is the mean time to decay? (b) What is the standard deviation of the decay time? (c) What is the 99th percentile? (d) You are conducting an experiment which first involves obtaining a single cobalt-60 atom, then observing it over time until it decays. You then obtain a second cobalt-60 atom, and observe it until it decays; and then repeat this a third time. What is the mean and standard deviation of the total time the experiment will last?
The half-life of an isotope is the time by which there is a 50% probability that decay has occurred if Cobalt-60 has a half-life of 5.27 years then mean time to decay is 7.65 years , standard deviation of the decay time is 3.82 years , the 99th percentile is 36.4 years and the mean and standard deviation of the total time the experiment will last is 6.61 years.
(a) The mean time to decay can be found using the formula: [tex]mean = half-life / ln(2)[/tex].
Therefore, for cobalt-60 with a half-life of 5.27 years, the mean time to decay is:
[tex]mean = 5.27 / ln(2) \approx7.65 years[/tex]
(b) The standard deviation of the decay time can be found using the formula:
standard deviation = [tex]half-life /(\ sqrt{(ln(2))}).[/tex]
Therefore, for cobalt-60 with a half-life of 5.27 years, the standard deviation of the decay time is:
standard deviation = [tex]5.27 / (\sqrt{(ln(2))}) \approx 3.82 years[/tex]
(c) The 99th percentile can be found using the cumulative distribution function (CDF) of the exponential distribution. For cobalt-60 with a half-life of 5.27 years, the CDF is:
[tex]CDF(t) = 1 - e^{(-t/5.27)}[/tex]
Setting the CDF equal to 0.99 and solving for t, we get:
[tex]0.99 = 1 - e^{(-t/5.27)}[/tex]
[tex]e^{(-t/5.27)} = 0.01[/tex]
[tex]-t/5.27 = ln(0.01)[/tex]
[tex]t = -5.27 * ln(0.01)[/tex]
[tex]t\approx 36.4 years[/tex]
Therefore, the 99th percentile of the decay time for cobalt-60 is approximately 36.4 years.
(d) Three cobalt-60 atoms, then the total time the experiment will last is the sum of the decay times of each atom. Since the decay times are independent and identically distributed, the mean and standard deviation of the total time can be calculated by adding the means and variances of each individual decay time.
The mean of the total time is:
[tex]mean(total) = mean(atom1) + mean(atom2) + mean(atom3)[/tex]
[tex]mean(total) = 7.65 + 7.65 + 7.65[/tex]
[tex]mean(total) = 22.95 years[/tex]
The variance of the total time is:
[tex]variance(total) = variance(atom1) + variance(atom2) + variance(atom3)[/tex]
[tex]variance(total) = (3.82)^2 + (3.82)^2+ (3.82)^2[/tex]
[tex]variance(total) \approx43.67 years[/tex]
The standard deviation of the total time is the square root of the variance:
standard deviation(total) [tex]= \sqrt{(variance(total))}[/tex]
[tex]standard deviation(total) \approx6.61 years[/tex]
Therefore, the mean and standard deviation of the total time for observing three cobalt-60 atoms until they decay are approximately 22.95 years and 6.61 years, respectively.
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compute the determinants in exercises 9-14 by cofactor expansions. at each step, choose a row or column that involves the least amount of computation. [\begin{array}{ccc}6&3&2&4&0\\9&0&-4&1&0\\8&-5&6&7&1\\3&0&0&0&0\\4&2&3&2&0\end{array}\right]
Answer:
Step-by-step explanation:
The alpha level is set _____ the analysis of the data. Use the following Distributions tool to identify the boundaries that separate the extreme samples from the samples that are more obviously consistent with the null hypothesis. Assume the null hypothesis is nondirectional, meaning that the critical region is split across both tails of the distribution. The z-score boundaries at an alpha level alpha = .05 are: z = 1.96 and z = -1.96 z = 3.29 and z = -3.29 z = 2.58 and z = -2.58 To use the tool to identify the z-score boundaries, click on the icon with two orange lines, and slide the orange lines until the area in the critical region equals the alpha level. Remember that the probability will need to be split between the two tails. To use the tool to help you evaluate the hypothesis, click on the icon with the purple line, place the two orange lines on the critical values, and then place the purple line on the z statistic. The critical region is _____. The z-score boundaries for an alpha level a = .001 are: z = 2.58 and z = -2.58 z = 1.96 and z = -1.96 z = 3.29 and z = -3. 29 Suppose that the calculated z statistic for a particular hypothesis test is 2.64 and the alpha is .001. This z statistic is _____ the critical region. Therefore, the researcher _____ reject the null hypothesis, and he ______ conclude the alternative hypothesis is probably correct.
The researcher should reject the null hypothesis, and he can conclude the alternative hypothesis is probably correct.
The alpha level is set prior to the analysis of the data. To use the tool to identify the z-score boundaries, the user must slide the orange lines until the area in the critical region equals the alpha level. The critical region is between the two z-score boundaries. The z-score boundaries for an alpha level a = .001 are: z = 2.58 and z = -2.58, z = 1.96 and z = -1.96, z = 3.29 and z = -3.29. With an alpha of .001, the calculated z statistic of 2.64 is within the critical region. Therefore, the researcher should reject the null hypothesis, and he can conclude the alternative hypothesis is probably correct.
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Find the first five terms of the sequence a(n)=3n-1
Answer:
-1, 2, 5 , 8 , 11 or 2, 5, 8, 11 , 14
Step-by-step explanation:
Assume n starts from 0
a(0) = 3 (0) -1 = -1
a(1) = 3 (1) -1 = 2
a(2) = 3 (2) -1 = 5
a(3) = 3 (3) -1 = 8
a(4) = 3 (4) -1 = 11
notice If we Assume n starts from 1
a(1) = 3 (1) -1 = 2
a(2) = 3 (2) -1 = 5
a(3) = 3 (3) -1 = 8
a(4) = 3 (4) -1 = 11
a(5)= 3 (5) -1 = 14
Notice a sequence can start from any N. But the most common ones are n=0 or n=1
a tank is being filled with water at the rate of gallons per hour with t>0 measured in hours. if the tank is originally empty, how many gallons of water are in the tank after 5 hours?
After 5 hours, the tank will have 50 gallons of water.
To solve this problem, use the following equation:
Gallons = Rate x Time
In this case, the rate is 5 gallons per hour and the time is 5 hours, so the equation becomes:
Gallons = 5 x 5
Therefore, the tank will have 50 gallons of water after 5 hours.
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Explain why it makes sense that 0 is the domain of y = √x?
It makes sense that 0 is the domain of y = √x because the square root of any number can never be negative, and 0 is the smallest positive number. Therefore, any values of x less than 0 would result in an undefined answer.
Help help please brainlist please ill mark
Answer:
Second choice
∠BCA ≅ ∠DCA
Step-by-step explanation:
This logically follows from the fact that both ∠BCA and ∠DCA are right angles from the previous step. And the reason given is "All right angles are ≅)
So they are congruent
Find m ∠ JKL using the picture
The value of m∠JKL is 34°
How to find the value of m∠JKL?An angle formed by a tangent and a secant intersecting outside a circle is equal to one-half the difference of the measures of the intercepted arcs.
Based on the theorem above, we can say:
m∠JKL = 1/2 * (159 - 91)
m∠JKL = 1/2 * 68
m∠JKL = 34°
Therefore, the value of m∠JKL in the circle is 34°.
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