Initial value that take the LaPlace transform of both sides of the given differential equation to create the corresponding algebraic equation is [tex]y(t) = 3t + sin4t + 3cos4t[/tex]
What is The Laplace transform?In terms of its usefulness in resolving physical issues, the Laplace transform is an important transform that is probably second only to the Fourier transform. The Laplace transform is especially helpful in solving linear ordinary differential equations, such as those that arise in the analysis of electronic circuits.
The initial value-
[tex]y" + 16y = 48t\\y(0)=3,y'(0)=7\\L( f^{n}(t) ) = s^{n}L(f(t)) - s^{n - 1}(f(0)) - s^{n - 2}(f(0)).........................................f^{n - 1} (0)\\[/tex]
By the formula,
[tex]s^{2}Y(s) - sy(0) - y'(0) + 16Y(s) = 48L(t)\\L(t^{n}) = \frac{n!}{s^{n+1} } \\\\On substitutinf the values,\\Y(s)(s^{2} + 16) - 3s - 7 = \frac{48}{s^{2} }\\Y(s)(s^{2} + 16) = \frac{48}{s^{2} } +3s + 7\\Y(s) = \frac{48 + s^{3} -7s^{2} }{s^{2}((s^{2} + 16) }\\Y(s) = \frac{3}{s^{2} } + \frac{4}{(s^{2} + 16} + \frac{3s}{(s^{2} + 16}[/tex]
On taking LaPlae inverse both side,
[tex]Y(t) = L( \frac{3}{s^{2} } ) + L( \frac{4}{(s^{2} + 16)} ) + L( \frac{3s}{(s^{2} + 16)} )\\Y(t) = 3t + sin4t + 3cos4t\\[/tex]
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If 5x + 2 = 7x - 7, then x =
Answer:
if 5x + 2 = 7x - 7 then the answer is x = 9/2.
the mean score of 888 players in a basketball game was 14.514.514, point, 5 points. if the highest individual score is removed, the mean score of the remaining 777 players becomes 121212 points. what was the highest score?
The highest score is 32 points.
Given,
Mean score of 8 players =14.5
After removing highest score then,
Mean score of 7 players =12
let
the sum of all score be 'x'
highest score be 'y'
At first,
[tex]Mean=\frac{sum\ of\ all\ scores}{total\ number\ of\ scores}\\\\14.5=\frac{x}{8}\\\\x=116[/tex]
after removing the highest score, mean will be
[tex]Mean=\frac{sum\ of\ all\ scores-highest\ score}{number\ of\ scores}\\\\12=\frac{x-y}{7}\\\\x-y=84[/tex]
substituting 'x'
[tex]116-y=84\\\\116-84=y\\\\y=32[/tex]
Thus, the highest score is 32 points.
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the base of a right triangle is 6 and the perpendicular height is 8, what is the length of a hypotenuse
The length of hypotenuse is 10. It can be obtained by using Pythagorean Theorem.
What is Pythagorean theorem?
It is a theorem which gives the relationship between the length of base, perpendicular height, and hypotonus. This relationship is valid only for a right angled triangle.
The length of hypotenuse is given by the following formula:
[tex]h=\sqrt{b^2+p^2}[/tex]
Here, h is the length of hypotenuse, b is the length of base, and p is the length of perpendicular.
Substitute 6 for b and 8 for p in the above equation.
[tex]h=\sqrt{6^2+8^2}\\=\sqrt{36+64}[/tex]
Now, take the square root of 100.
[tex]h=\sqrt{100}\\=10[/tex]
Hence, the length of hypotenuse is 10.
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Help help help help help I will give 20 points
Answer:
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at a deli counter, there are 9 sandwiches with meat and 10 vegetarian sandwiches. dante is at the counter buying sandwiches for a picnic. in how many ways can he choose 9 sandwiches if 8 or more must be vegetarian sandwiches?
There are two different ways in which he can choose 9 sandwitches. It can be solve by the concept of permutation and combination.
What is permutation and combination?
A permutation is an arrangement of several items taken one at a time or all at once in a specific order.
Combinations are based entirely on grouping. Combinations can be used to determine how many different groups that can be created from the available items.
Since all vegitarian sandwiches looks like same and all sandwitches with meat looks like same hence, their order will not affect the number of choice.
If 8 sandwitches are vegitarian then 1 sandwitch must be with meat. So that total sandwithches becomes 9.
If all 9 sindwitches are vegetarian then there is no option to choose another sandwitch. Because he can choose only 9 sandwitches.
Hence, there are two different ways in which he can choose 9 sandwitches.
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Identify the expression or expressions equivalent to -5 + 4(-3x+6) -4x
The equivalent expression of -5 + 4(-3x+6) -4x is -29 - 16x
How to determine the equivalent expression?From the question, we have the following parameters that can be used in our computation:
-5 + 4(-3x+6) -4x
Open the bracket in the above expression
This gives
-5 + 4(-3x+6) -4x = -5 - 12x - 24 -4x
Collect the like terms in the above expression
This gives
-5 + 4(-3x+6) -4x = -5 - 24 - 12x -4x
Evaluate the like terms in the above expression
This gives
-5 + 4(-3x+6) -4x = -29 - 16x
Hence, the equivalent expression is -29 - 16x
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what is the probability of rolling a die ten times and obtaining odd digits on eight or less of the rolls?
The probability of rolling a die 10 times and obtaining odd digit 8 or less of the rolls is 529/512 = 1.033
Let us call, getting an odd no. on a roll of die, Success.
The, clearly, the probability p of Success is 3/6 = 1/2.
Hence, q = 1 −p = 1/2
If, X = x denotes the no. of success in n trials, then, X is a
Binomial Random Variable, with parameters
n = 10, and, p = 1/2.
Binomial Random Variable:
This is a specific type of discrete random variable. A binomial random variable counts how often a particular event occurs in a fixed number of tries or trials. For a variable to be a binomial random variable, ALL of the following conditions must be met:
There are a fixed number of trials (a fixed sample size).On each trial, the event of interest either occurs or does not.The probability of occurrence (or not) is the same on each trial.Trials are independent of one another.Then, the Probability of x success out of n trials, i.e.,
P (X = x), is,
P (X = x) = p(x) =ⁿCₓ pˣ , qⁿ⁻ˣ, x = 0,1,2, ........, n
In our case,
P (X = x) = ¹⁰Cₓ (1/2)ˣ (1/2)¹⁰⁻ˣ ,x = 0,1,2,..., 10, i.e.,
P (X = x) = p(x) = ¹⁰Cₓ (1/2)¹⁰ = ¹⁰Cₓ /1024, x = 0, 1, ..., 10.
Hence,
The Required Probability = P( X< 8),
⇒ P (X = 0)+ P( X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) +P(X = 6) + P(X= 7) + P(X =8)
⇒ 1/1024 ( 1 + 10 + 45 + 120 +210 + 252 + 210 + 120 +90)
⇒ 1/1024 ×1058
⇒ 529 / 512
⇒ 1.033
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in a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. if 10 boys and 14 girls are competing, how many different ways could the six medals possibly be given out?
answer: 72 or 1572480 depends on how you do the math
suppose that a random sample of 25 measurements is selected from a population with mean 70 and standard deviation 10. what is the mean and standard deviation of the sampling distribution of the sample mean?
The mean and standard deviation of the sampling distribution is 70 and 2 respectively.
Given a random sample of 25 measurements means
n = 25.
Population with mean(µ) 70 means
µ = 70.
Population with standard deviation(σ) 10 means
σ = 10.
mean of sampling distribution is itself same as mean of population
so mean of sampling distribution is 70.
standard deviation of sampling distribution is
σ/[tex]\sqrt{n}[/tex].
put the value of σ= 10 and n =25.
10/[tex]\sqrt{25}[/tex] = 10/5 = 2
So mean of sampling distribution is 70 and standard deviation of sampling distribution is 2.
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which term below is a number that describes how the data is dispersed across the entire range of data? a.) shape b.) outlier c.) spread d.) center
the spread data is dispersed across the entire range of data.
What is range of data?
In statistics, we must put the given values, set of data, or set of observations in ascending order in order to determine the range. That indicates that you should start by writing the observations from lowest to highest value. The formula must now be used to determine the range of observations. As a result, the range may alternatively be thought of as the distance between the highest and lowest observation. The range of observation is the name given to the outcome. Statistics' range reflects the variety of observations.
When we plot data, we are considering the shape of the data. We plot the data based on the histogram, for example. The data's shape is Shape.
Outliers are different observations from the majority of the observations and are clustered in the center. Spread is the dispersion of the data throughout the entire range.
Hence, the spread data is dispersed across the entire range of data.
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Find the distance between (2, 3) and (−4, −9). Round your answer to the nearest hundredth, if necessary.
Answer:
I think it would be 13.42 because round 13.41640786… would be 13.42
I hope this helps
B. The President uses technology to produce the following random numbers.
26 17 58 68 2 17 1 29 33 51
Determine the numbers for 9 faculty members who will be included in the sample.
The numbers for 9 faculty members who will be included in the sample are 26, 17, 58, 68, 2, 1, 29, 33, 51
What is a simple random sample?
A subset of a statistical population called a simple random sample is one in which each member has an equal chance of being chosen. A straightforward random sample is intended to offer an unbiased representation of a group.
Given, the President uses technology to produce the following numbers:
26 17 58 68 2 17 1 29 33 51
Now, 9 faculty members will be included in the sample is:
26, 17, 58, 68, 2, 17, 1, 29, 33, 51
In the above data, 17 occurs with repeated frequency then we will make 17 once.
Hence, the required answer is 26, 17, 58, 68, 2, 1, 29, 33, 51
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C. If this is Thursday, then I do not go to church.
Converse:
Inverse:
Contrapositive:
Zoe goes to a restaurant and the subtotal on the bill was xx dollars. A tax of 5% is applied to the bill. Zoe decides to leave a tip of 15% on the entire bill (including the tax). Write an expression in terms of xx that represents the total amount that Zoe paid.
The expression in terms of x that represents the total amount that Zoe paid is 1.2x
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
A tax of 5% is applied to the bill.
Zoe decides to leave a tip of 15% on the entire bill (including the tax).
let the bill cost $x.
So, the price after tax and tip
= x+ 5% of x+ 15% of x
= x + 0.05x + 0.15x
= x + 0.20x
= 1.2x
Hence, the expression in terms of x that represents the total amount that Zoe paid is 1.2x
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one sample has n = 8 scores and m = 2. a second sample has n = 4 and m = 8. if the two samples are combined, then what is the mean for the combined sample?
The combined sample mean for sample one, whose size equals 8 scores and mean equals 2, and for sample second, whose size equals 4 scores and mean equals 8, is 4..
A combined sample mean is calculated by taking the mean of each group and combining (adding) the results of each average.
We have given that,
For First sample :
Sample size ,n₁ = 8 scores
sample mean, m₁ or X₁-bar = 2
Sample 2 :
Sample size, n₂ = 4
sample mean , m₂ or X₂-bar = 8
Combined sample mean is given by,
M or X-bar =( n₁× x₁-bar + n₂ × x₂-bar)/(n₁+n₂)
plug all the known values in above formula
=> M = (8×2 + 4×8 )/12
=> M = 48/12 = 4
Hence, combined sample mean is 4 .
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if n people of different heights are lined up in a queue, what is the probability that at least one person is just behind a taller person?
Probability of at least one people behind taller person = n-1/n.
From the question, we have
n people of different heights are lined up in a queue
Probability of at least one people behind taller person = 1 - 1/n =n-1/n
Probability:
The term probability refers to possibility. The occurrence of a random event is the focus of this mathematical field. There is a 0 to 1 value range. To determine the possibility of certain events, probability has been employed in mathematics. The likelihood of anything happening is often referred to as probability. You can better understand the outcomes of a random experiment by applying this fundamental theory of probability, which also holds true for the probability distribution. Utilize probability to gauge a situation's likelihood of happening. Numerous factors are difficult to forecast with absolute precision. Using it, we can only speculate on whether or how likely it is that an event will occur.
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The following inequalities form a system. y is greater than or equal to two-thirds times x plus 1 y is less than negative one-fourth times x plus 2 Which ordered pair is included in the solution to this system? (−6, 3.5) (−6, −3) (−4, 3) (−4, 4)
The inequalities are y ≥ (2/3)x + 1 and y < (-1/4)x + 2 and (−6, −3) satisfied the inequality.
What is inequality?A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.
In other words, inequality is just the opposite of equality for example 2 =2 then it is equal but if I say 3 =6 then it is wrong the correct expression is 3 < 6.
As per the given phrase,
y is greater than or equal to two-thirds times x plus 1 ⇒ y ≥ (2/3)x + 1
y is less than negative one-fourth times x plus 2 ⇒ y < (-1/4)x + 2
Substitute, point (−6, −3)
-3 ≥ (2/3)(-6) + 1
-3 ≥ -3
Substitute in second inequality,
-3 < (-1/4)(-6) + 2
-3 < 7/2
Hence "The inequalities are y ≥ (2/3)x + 1 and y < (-1/4)x + 2 and (−6, −3) satisfied the inequality".
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The dimensions of a box have been doubled. How many times greater is the volume of the box?
8, 6, 4, or 2
Answer:
The volume increases 8-fold or by 700%.
Step-by-step explanation:
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How much would you pay monthly on a $6,000 loan if you pay 12. 5 percent for 10 years?.
If we pay 12.5 percent for 10 years, our monthly payment for a $6000 loan will be about (C) $55.58.
What is the percentage?In mathematics, a percentage is a number or ratio that is expressed as a fraction of 100. Although "pct," "pct," and occasionally "pc" are also used as abbreviations, the percent symbol "%" is most usually used to denote it.
A % is a number that has neither dimensions nor a defined unit of measurement.
To calculate the percentage, we must divide the value by the entire amount and then multiply the result by 100.
So, if the interest rate is 12.5%, our total loan payment would be:
112.5/100 × 6,000 = $6,750.
The monthly payment would be if there was no buildup and other agreement in payment:
6750/12*10
6750/120
$56.25
Therefore, if we pay 12.5% for 10 years, our monthly payment for a $6000 loan will be about (C) $55.58.
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Complete question:
How much would you pay monthly on a $6,000 loan if you pay 12.5 percent for 10 years?
a. $87.90
b. $70.32
c. $55.58
d. $125.00?
an online gardening magazine wants to understand why its subscriber numbers have been increasing. what kind of reports can a data analyst provide to help answer that questio
Reports that describe how many customers shared positive comments about the gardening magazine on social media in the past year. Reports that compare past weather patterns to the number of people asking gardening questions to their social media. Reports that examine how a recent 50%-off sale affected the number of subscription purchases. Reports that predict the success of sales leads to secure future subscribers.
What is data analyst?In order to find relevant information, support inferences, and help decision-making, data analysis is the process of analyzing, cleaning, manipulating, and modeling data. A data analyst examines data to find significant consumer insights and potential uses for the information. They also advise the company's management and other stakeholders of this information. Although it is recommended that data analysts have a basic understanding of statistics and mathematics, many of their tasks may be completed without it. However, data analysts often need to be familiar with statistics, linear algebra, and calculus.
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dominic has a family history of hypertension. according to the chronic disease risk reduction intakes (cdrr) for sodium, dominic should lower or maintain his sodium intake to below mg/day to reduce his risk of chronic disease, including hypertension
Dominic should lower or maintain his sodium intake to below 2,300 mg/day.
Chronic disease :The Chronic Disease Risk Reduction (CDRR) consumption of 2,300 mg/day was recently developed due to connections between high sodium intake and the risk of hypertension. It is anticipated that sodium intake below this level will lower the risk of chronic diseases.
Adults may be at risk for chronic diseases if their sodium intake exceeds their CDRR intake. The population's cardiovascular health may be improved by efforts to reduce sodium consumption. Among healthy individuals, lowering daily sodium intake to CDRR is anticipated to lower the risk of developing chronic diseases, primarily by lowering blood pressure.
Hence, sodium intake to below 2,300 mg/day can reduce his risk of chronic disease.
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y+5-1/3x=7 find y and then x to find the corresponding values
Answer:
y+5-1/3x=7 find y and then x to find the corresponding values
y = 5 - 1/3x + 7
y = 5 - 1/3(x) + 7
y = 5 - (1/3)x + 7
y = (5 - 1/3x) + 7
y = (5 + 7) - (1/3)x
y = 12 - (1/3)x
For y = 12,
12 - (1/3)x = 12
(1/3)x = 0
x = 0
Therefore, when y = 12, x = 0.
what can you change in 3 (x + 4) - 2(x+4) + 0.5(x + 4) to get 1.5
Please Help!
To make the expression 3(x + 4) - 2(x + 4) + 0.5(x + 4) equals 1.5, we make x in the linear equation equals -3
Linear EquationA linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line. This is the reason why it is named as a 'linear equation'.
There are linear equations in one variable and linear equations in two variables.
In this problem given we have to simplify and equation the expression to 1.5 to make it an equation.
3(x + 4) - 2(x + 4) + 0.5(x + 4) = 1.5
3x + 12 -2x -8 + 0.5x + 2 = 1.5
3x - 2x + 0.5x + 12 - 8 + 2 = 1.5
1.5x + 6 = 1.5
Collect like terms
1.5x = 1.5 - 6
x = -3
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a robotic machine fills containers with an acid solution at the rate of 140 3 t milliliters ( ml ) per second, where t is in seconds and 0 ≤ t ≤ 60 . how many ml are put into a container in 60 seconds? round to the nearest whole number.
The container put 320 ml amount of the acid in 60 seconds.
Given, a robotic machine fills containers with an acid solution at the rate of 140 + 3t milliliters ( ml ) per second, where t is in seconds and 0 ≤ t ≤ 60 .
we have to find the amount of acid solution put into the container in 60 sec
As, Amount of acid be A(t)
A(t) = 140 + 3t
A(60) = 140 + 3(60)
A(60) = 140 + 180
A(60) = 320
So, the container put 320 ml amount of the acid in 60 seconds.
Hence, the container put 320 ml amount of the acid in 60 seconds.
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A certain type of car has room for 4 passengers.
How many passengers could fit in 78 cars?
How many cars would be needed to fit 78 passengers?
Answer:
1) 312
2) 20
Step-by-step explanation:
4x78=312
78 divided by 4=19.5 rounded up =20
if the roots of the characteristic equation of a second order differential equation that models a unforced circuit are 2 and -3
A polynomial equation is used to analyze the behavior of the problem's solutions and is known as the characteristic equation of a second-order differential equation.
It is frequently used to simulate real-world events, such as the operation of electrical circuits. The behavior of the differential equation's solutions may be predicted when the characteristic equation's roots are 2 and -3. The roots of the characteristic equation can be used to identify the properties of an unforced circuit when a second-order differential equation is used to represent it.
The circuit will behave oscillatory when the roots are 2 and -3. As a result, the output of the circuit will oscillate frequently, with the frequency of the oscillations being defined by the variation between the roots. The oscillations in this instance will occur at a frequency of 5 Hz.
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The correct question is-
What behavior of the differential equation's solutions can be determined when the roots of the characteristic equation are 2 and -3?
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identify the type i error and the type ii error that corresponds to the given hypothesis. the proportion of people who write with their left hand is equal to 0.29 .
Type I error is:
" Reject the claim that the proportion of people who write with their left hand is 0.29 when the proportion is actually 0.29 "
Type II error is:
"Fail to reject the claim that the proportion of people who write with their left hand is 0.29 when the proportion is actually different from 0.29 "
What is proportion?
An equation in which two ratios are made equal is called a proportion. For instance, you could write the ratio as 1: 3 (for every one boy, there are three girls), which means that 14 of the population is made up of boys and 34 of the population is made up of girls.
In Statistics, a type I error is the rejection of a true null hypothesis
and a type II error is the non-rejection of a false null hypothesis
Given that p = 0.29 is accepted
Now type I error is
C. Reject the claim that the proportion of people who write with their left hand is 0.29 when the proportion is actually 0.29
i.e. (false positive)
and type II error here is
B. Fail to reject the claim that the proportion of people who write with their left hand is 0.29 when the proportion is actually different from 0.29.
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samantha is considering job offers from two companies. company a offered her a starting salary of $48,000 with a $2100 raise at the end of each year. company b offered her a starting salary of $48,000 with a 3.7% raise at the end of each year.
Let f(t) represent Rosa's salary at Company A, t years after accepting a position at Company A, and let g(t) represent Rosa's salary at Company B t years after accepting a position at Company B.
Complete the table of values below:
t f(t) g(t)
0 50000 50000
1
2
3
The value of f(t) are 48000, 52200, 54300, 56400, and g(t) are 48000, 49776, 51617.71, and 53527.56.
She was given a starting wage of $48,000 from Company A, with annual raises of $2100.
The formula for Company A is
f(t) = 48000 + 2100t
She was given a starting wage of $48,000 from Company B, with an annual rise of 3.7%.
The formula for Company B is
g(t) = 48000 × [tex](1 + 0.037)^t[/tex]
The value of f(t) and g(t),
t ------- f(t) ------- g(t)
0 ----- 48000 ----- 48000
1 ------ 52200 ----- 49776
2 ----- 54300 ----- 51617.71
3 ----- 56400 ----- 53527.56
The graphical representation of both the linear equations of company A and company B.
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x + 2y = -6
6x +2y =4
Step-by-step explanation:
um, I'm going to use elimination method to solve this simultaneous equation
x+2y=-6.......... equation 1
6x+2y=4........ equation 2
subtract equation 2 from equation 1
-5x = -10
x = -10/-5
x=2
substitute x=2 in equation 1
therefore equation 1 becomes
2+2y=-6
2y=-6-2
2y=-8
y=-8/2
y=-4
the sum of three nonnegative numbers is 36, and one of the numbers is twice one of the other numbers. what is the maximum value of the product of these three num- bers?
324 is the maximum value of the product of these three numbers
What is maxima and minima?
The curve of a function has peaks and troughs called maxima and minima. A function may have any number of maxima and minima. Calculus allows us to determine any function's maximum and lowest values without ever consulting the function's graph. Maxima will be the curve's highest point within the specified range, and minima will be its lowest.
The extrema of a function are the maxima and minima. The maximum and minimum values of a function inside the specified ranges are known as maxima and minima, respectively. Absolute maxima and absolute minima are terms used to describe the function's maximum and minimum values, respectively, over its full range.
Let the three nonnegative numbers are x,y,z
According to the question
x+y+z=36
and y=2x
therefore, 3x+z=36--------------------------------------------------(1)
z=36-3x
Let 3xz= u--------------------------------------------------------------(2)
differentiating equation 2
du/dx=3z + 3xdz/dx
or
du/dx= z + xdz/dx
differentiating equation 1
3 + dz/dx = 0
dz/dx = -3
du/dz = 36-3x + x(-3)
du/dx = 12 - 2x--------------------------------------------------------------(3)
for maxima put du/dx = 0
x=6-------------------------------------------------------------------------------(4)
again differentiating du/dx = 12 - 2x
d2u/dx2=-2
which means 3xz= u is maximum at x=6
from equation 1 and 4
we get 18+z=36
z=18
Therefore 3xz= 3(6)(18)=324
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