If x and y be i.i.d. unif(0, 1), then the covariance of x + y and x - y is zero.
Yes, x y are x y independent studysoup.
Here x and y be i.i.d. unif(0, 1)
The i.i.d stands for independent and identically distributed
unif means that the uniformly distributed
Here x and y are i.i.d unif (0,1)
The covariance is defined as the relationship between the two random variables . Therefore, variance of one variable is equal to the change in another variable
Here we have to find the covariance of x + y and x - y
According to the properties of covariance
Cov(x+ y, x - y) = Cov (x, x) + Cov(y, x) - Cov(x, y) - Cov(y, y) = 0
Because Cov(x, x) = Cov (y, y)
Cov(y, x) = Cov(x, y)
Therefore, the covariance of x+y and x - y is 0.
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The domain of f(x,y) is the xy-plane, and values of f are given in the table below.
0 1 2 3 4
0 60 60 60 62 64
1 61 61 62 61 60
2 65 68 68 68 70
3 63 61 60 60 62
4 59 55 51 48 45
Find [tex]\int\limits_c grad f . dr[/tex] , where is
(a) A line from (0,1) to (1,2).
(b) A circle of radius 1 centered at (1,2) traversed counterclockwise.
Answer:
a) c[tex]\int\limits gradf.dr[/tex] = 1
b) c[tex]\int\limits gradf.dr[/tex] = 0
(the limit symbol has a circle in the center for part b)
Step-by-step explanation:
c[tex]\int\limits gradf.dr[/tex] = f(q) - f(p)
a) c[tex]\int\limits gradf.dr[/tex] = f(1, 2) - f(0, 1)
= 61 - 60
= 1
b) If C is the circle of radius 1 centered at the point beginning at point (1,2), we can think of C as both beginnings and ending at point ( 1, 3).
c[tex]\int\limits gradf.dr[/tex] = f(1, 3) - f(1, 3)
= 68 - 68
= 0
To accumulate $10,000 at the end of 3n years, deposits of $86 are made at the end of each of the first 2n years and 98 at the end of each of the next n years. The effective annual rate of interest is i. You are given (1+i)^n = 2.0. Determine i.
If you want to have $10,000 at the end of 3n years, you must deposit $86 at the end of the first 2n years and $98 at the end of each subsequent n years (12.25%).
How is the annual interest rate determined?Effective yearly interest rate is equal to (1 + (nominal rate divided by the number of compounding periods)). (Amount of compounding intervals) minus 1.
This would be: 10.47% = (1 + 10% x 12) x 12 - 1 for investment A.
It would be as follows for investment B: 10.36% = (1 + (10.1% 2)) 2 - 1.
The annual interest rate is it monthly or yearly?The interest payment that the borrower pays the lender is determined by the interest rate. Lenders' quoted interest rates are yearly rates. The interest payment on the majority of house mortgages is computed monthly. As a result, the rate is divided by 12 before the payment is determined.
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Please help I’m very confused on this question.
Skylar and Wyatt both play soccer. Wyatt scored 2 times as many goals as Skylar. Together, they scored 15 goals. Could Skyler have scored 3 goals? Why or why not?
No, because 3(2) + 3 ≠ 15
No, because 3(2) ≠ 15
Yes, because 3 goals is less than the total number of goals scored
Yes, because 3 goals is less than the number of goals Wyatt scored
Answer:
The correct answer is A. No, because 3(2) + 3 ≠ 15.
Step-by-step explanation:
because the total number of goals scored by Skylar and Wyatt was 15, and if Skylar scored 3 goals, then Wyatt would have scored 3 * 2 = 6 goals. Since the total number of goals scored by both players is 15, this scenario is not possible. Therefore, Skylar could not have scored 3 goals.
a clothing business finds there is a linear relationship between the number of shirts, n ,it can sell and the price, p , it can charge per shirt. in particular, historical data shows that 6000 shirts can be sold at a price of $55, while 8000 shirts can be sold at a price of $47. give a linear equation in the form p
So a linear equation in the form p is P(n) = -$0.004*n + $79.
We have two data points:
6000 shirts can be sold for $55 each.
8000 shirts can be sold for $47 each.
Then we can define the relation:P(n).
Where P is the price, and n is the number of shirts.
Now, we know that we can model this as a linear relationship that passes through the points (6000, $55) and (8000, $47)
A linear relationship can be written as:
y = a*x + b
where a is the slope and b is the y-axis intercept.
For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:
a = (y2 - y1)/(x2 - x1).
In this case, the slope is:
a = ($47 - $55)/(8000 - 6000) = -$0.004
Then our equation is:
P(n) = -$0.004*n + b
Now let's find the value of b, we know that:
P(6000) = $55= -$0.004*6000 + b
$55 = -$24 + b
b= $79
Our equation is:
P(n) = -$0.004*n + $79.
The linear equation P(n) = -$0.004*n + $79 has the form p.
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data are collected on the 35 students in a college history course. which of the following is not a variable for the data set? responses student birth month student birth month political affiliation of student political affiliation of student student age student age student address
The correct statement is-number of students in the data set
In research and data collecting, something that is being measured and whose values can change is referred to as a variable.
Given that we have a range of alternatives from January to December, the aforementioned example demonstrates how the student birth month is a variable. The student's ability to indicate whether they support particular political parties or not makes their political affiliation another variable. Since they are college students, the age of the student is another variable, with replies falling within a range of 20 to 25.
Since students will give different answers, whether they live at the same address or a different one, student address is also a variable.
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a coffee company wants to estimate the true proportion of the u.s. population that drinks its brand. it interviewed randomly selected people, and responded that they drink the company's brand. construct a percent confidence interval for the true proportion of people who drink the company's brand. group of answer choices
a) Construct a 95 percent confidence interval for the true proportion of people who drink the company’s brand.
b) How many individuals should be surveyed to be 95 percent confident of having the true proportion of people drinking the brand estimated to within 0.015?
You have a 5% chance of being wrong with a 95% confidence interval. With a 90 percent certainty span, you have a 10 percent chance of being incorrectly.
A confidence interval of ninety-nine percent would be larger than a confidence interval of ninety-five percent (for instance, plus or minus 4.5 percent as opposed to 3.5 percent).
We now have: 32 125 0.256 1 At a confidence level of 95 percent:
Zenit = 20.05/2 = 2196 95 percent confidence interval:
Considering the estimated proportion (P) = 0.256 n = = Zverit PL1-P) E 1.96 12 0.015) 0.256 (1-0.256) = 3251.94 3252
Zerit = 20.05/2 = 1.96 At 95%
confidence lovel margin of error (E) = 0.015
Now P = zerit / PL1-£) = 0.256 1.96 X 0.2561-0.256 125 = 0.256 0.0765.
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Identify the following distribution as binomial, geometric or neither.
Draw a card from a standard deck of 52 playing cards, observe the card, and replace the card within the deck. Count the number of times you draw a card in this manner until you observe a jack.
a) Binomial.
b) Geometric.
c) Neither.
A 52-card standard deck of playing cards is drawn, examined, and then placed back into the deck is a geometric distribution.
Given that,
A 52-card standard deck of playing cards is drawn, examined, and then placed back into the deck.
We have to find count how many times a card is drawn in this way until a jack is seen.
We know that,
What is the geometric distribution?A discrete probability distribution known as a geometric distribution can be used to describe the likelihood of experiencing success for the first time following a string of failures. Up until the first success, a geometric distribution can undergo an infinite number of trials.
So,
P(X=x)=qˣp.
Therefore, A 52-card standard deck of playing cards is drawn, examined, and then placed back into the deck is a geometric distribution.
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Fran has a drawer containing 4 black T-shirts, 3. orange T-shirts, and 5 blue T-shirts. If she picks one T-shirt at random from the drawer, what are the chances that it will NOT be orange?
Answer: 75%
Step-by-step explanation: 4+3+5=12. 12-3=9. So the probability of not picking orange is 9/12, or 3/4. 3/4=75%. So the answer would be 75%
Which equation of a line has a slope of -3 and passes through point (2, 4)?
y = 3x + 10
y = -3x + 10
y = 3x – 10
y = -3x – 10
Answer:
B
Step-by-step explanation:
y-4 = -3(x-2)
y - 4 = -3x + 6
y = -3x + 6 + 4
y = -3x + 10
Answer:
y = -3x+10
Step-by-step explanation:
Using the slope-intercept equation of a line
y = mx+b where m is the slope and b is the y intercept
y = -3x+b
Substitute the point into the equation
4 = -3(2) +b
4 = -6+b
10 =b
y = -3x+10
How is the quotient of 3,419 and 11 determined using an area model?
Answer:
37609
Step-by-step explanation:
You'd put the 3,419 inside the box and 11 outside the box.
But you'd need around 4 spaces inside the box for each number to be on top. And you'd need 2 spaces on the side. (Hopefully that makes sense)
Your answer should be 37609
which of the following are calculated metrics for understanding analytical data? session and time on page entry and exit average session duration and page bounce rate revenue and revenue per session
From the provided topics to select, For the analytical data we need something where we can extract information from it. The answer is page bounce rate revenue and revenue per session.
What do you mean by data?In mathematics, data is a collection of facts and figures that can take any shape, whether it be numerical or not. You can calculate numerical data, which is always gathered in number form and includes things like student test results, employee salaries, football team member heights, etc.
What do you mean by analytical data?The systematic computational analysis of data or statistics is known as analytics. It is employed for the identification, explanation, and dissemination of significant data patterns. It also involves using data patterns to make smart decisions.
Page bounce rate revenue and revenue per session are the topics which contains some data and facts from which an analysis process can extract some information for other uses.
So, The metrics for understanding analytical data are page bounce rate revenue and revenue per session.
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Give the POINT SLOPE equation of the line
that contains:
f(-1) = -7 and f(3) = -6.
This is one way we can represent a linear equation:
[tex]y-y_1=m(x-x_1)[/tex]
[tex](x_1,y_1)[/tex] is a point that falls on the line[tex]m[/tex] is the slopeSolving the QuestionWe're given:
Line contains [tex]f(-1) = -7[/tex] and [tex]f(3) = -6[/tex]These functions give us information on two points, as they are represented as [tex]f(x)=y[/tex]:
[tex]f(-1) = -7[/tex] ⇒ (-1, -7)[tex]f(3) = -6[/tex] ⇒ (3,-6)First, solve for the slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
⇒ Plug in the two points:
[tex]m=\dfrac{-6-(-7)}{3-(-1)}\\\\m=\dfrac{-6+7}{3+1}\\\\m=\dfrac{1}{4}[/tex]
⇒ Plug this into [tex]y_2-y_1=m(x_2-x_1)[/tex]:
[tex]y-y_1=\dfrac{1}{4}(x-x_1)[/tex]
Now, there are two ways we can write this equation, as we are given two points:
(-1, -7)
⇒ [tex]y-(-7)=\dfrac{1}{4}(x-(-1))\\y+7=\dfrac{1}{4}(x+1)[/tex]
(3,-6)
⇒ [tex]y-(-6)=\dfrac{1}{4}(x-(3))\\y+6=\dfrac{1}{4}(x-3)[/tex]
Answer[tex]y+7=\dfrac{1}{4}(x+1)[/tex] or [tex]y+6=\dfrac{1}{4}(x-3)[/tex]
PROOF Complete the flow proof to prove that if MN = PQ, MN = 5x - 10, and
PQ = 4x + 10, then MN = 90.
PQ,MN=5x-10 , and PQ=4x+10 , then MN=90. x-10=10 this equation is proved.
What is congruent addition?Congruent Addition is the Substitution segments to have the Property of Equality Given equally lengths; Substitutions Property's of Equality.
As per the given question
MN≈PQ
MN = 5x-10
And PQ=4x+10
Congruent segment have equal lengths:; Substitution property of equality
MN=PQ
5x-10 = 4x+10
Substitution property of equality
X-10=10
Addition property of equality
X=20
Substitution property of equality
MN= 5(20)-10
MN= 90
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1. On the map below, the fire department and the hospital have one matching coordinate. Determine the proper
order of the ordered pairs in the map, and write the correct ordered pairs for the locations of the fire departmer
and hospital. Indicate which of their coordinates are the same.
10
9
8
7
6
3
2
The ordered pairs for the coordinates of the fire department and of the hospital are given as follows:
Fire department: (6,7).Hospital: (10,7).The y-coordinate is the same for both the fire department and for the hospital.
How to define the ordered pairs?The notation of an ordered pair is given as follows:
(x-coordinate,y-coordinate).
From the graph, each coordinate is defined as follows:
x-coordinate: horizontal coordinate.y-coordinate: vertical coordinate.For this problem, as the fire department and the hospital are aligned horizontally, it means that they have the same y-coordinate.
Missing InformationThe image containing the location of the fire department and of the hospital is shown at the end of the answer.
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A bag contains 4 red marbles, 7 blue marbles and 8 green marbles. If two marbles are drawn out of the bag, what is the probability, to the nearest 10 th of a percent, that both marbles drawn will be green?
Answer=_____
The probability, to the nearest 10 th of a percent, that both marbles drawn will be green is 0.02
Given :
A bag contains 4 red marbles, 7 blue marbles and 8 green marbles. If two marbles are drawn out of the bag
Probability :
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.
total number of possible outcomes = 4 + 7 + 8
= 11 + 8
= 19
Total number of favourable outcomes = 7
probability = favourable outcomes / possible outcomes
= 7 / 19 * 18
= 7 / 342
= 0.02
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Statement
Reason
The coordinates of point D are (4, 5) and coordinates of point E are (5, 3) By the midpoint formula
Length of segment DE is √5 and length of segment AC is 2√5
Segment DE is half the length of segment AC
Slope of segment DE is -2 and slope of segment AC is -2
Segment DE is parallel to segment AC
Which of the following completes the proof? (6 points)
By the addition property
O By the distance formula
By construction
Given
By substitution
By the slope formula
Slopes of parallel lines are equal
By the slope formula, it is proved that the slope of AC is the same as the slope of DE thus option (D) is correct.
What is the slope?A slope is a tangent or angle at a point and a slope is the intensity of inclination of any geometrical lines.
Slope = Tanx where x will be the angle from the positive x-axis at that point.
The slope associated with two points (x₁, y₁) and (x₂, y₂) is given by
Slope = (y₂ - y₁)/(x₂ - x₁)
As per the given,
D(4,5) and E(5,3)
Slope = (3 - 5)/(5 - 4) = -2
A(6,8) and C(8,4)
Slope = (4 - 8)/(8 - 6) = -4/2 = -2
Since the slope of DE = slope of AC
Therefore, both lines will be parallel by the slope formula.
Hence "It is established using the slope formula that the slopes of AC and DE are the same".
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The complete diagram is below,
let y1 and y2 are independent random variables that are both uniformly distributed on the interval (0,1). find p(y1 <1
the probability of y1 being less than 1 is 1, regardless of the actual value of y1. P(y1 < 1) = 1
Since y1 and y2 are both uniformly distributed on the interval (0,1), this means that all possible values of y1 and y2 are equally likely. Therefore, the probability of y1 being less than 1 is 1, since any value of y1 between 0 and 1 has the same probability of occurring. This means that the probability of y1 being less than 1 is 1.
Since y1 and y2 are both uniformly distributed on the interval (0,1), this means that all possible values of y1 and y2 between 0 and 1 are equally likely to occur. This means that the probability of y1 being less than 1 is the same as the probability of y1 being equal to any value between 0 and 1. Since this probability is the same for all possible values of y1, the probability of y1 being less than 1 is 1, regardless of the actual value of y1. This is because any value of y1 between 0 and 1 has the same probability of occurring, meaning that the probability of y1 being less than 1 is 1.
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At time t=0, a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4
grams. The maximum weight of the culture is 20 grams.
(a) Write a logistic equation that models the weight of the bacterial culture.
(b) Find the culture's weight after 5 hours.
(c) When will the culture's weight reach 18 grams?
(d) Write a logistic dierential equation that models the growth rate of the culture's weight. Then
repeat part (b) using Euler's method with a step size of h=1.
(e) After how many hours is the culture's weight increasing most rapidly? Show work.
part (a) =the model's necessary solution is [tex]y=\frac{20}{1+19e^{0.779t} }[/tex] in logistic function.
part (b) =The weight of the culture is 14.425 grams after five hours.
part (c)=After 6.6 years, the culture has an 18 gram weight.
part (d)= The logistic differential equation used to simulate how quickly a culture's weight increases is [tex]\frac{dy}{dy}= 0.779y(1-\frac{y}{20})[/tex]
5. hours later, the culture weighs 11.571 grams.
part (e)= According to portion (c) of this question, the culture's weight will be 18 grams when t=6.6, which is why the weight of the culture increases most quickly at that time.
what is logistic function?Given a growth rate, r, and a carrying capacity, K, the logistic equation is a straightforward differential equation model that can be used to connect the change in population, d P d t, to the existing population, P.
given
a bacterial culture weighs 1 gram in time(t)=0
the culture weighs 4 after 2 hours
culture maximum weight is 20 grams.
for part (a)
The general solution for the logistic differential equation is [tex]y(t)=\frac{L}{1+be^{-kt} }[/tex]
The culture can weigh up to 20 grams , so put value of L= 20 grams
The value of [tex]e^{0}[/tex]is 1.
1+b Equals 20 when multiplied on both sides.
on solving b=19
[tex]y(t)= \frac{20}{19e^{-kt} }[/tex]
The culture weights 4 grams at the end of two hours, or y(2)=4.
value of t and y in [tex]y(t)= \frac{20}{19e^{-kt} }[/tex]
on simplifying k≈0.779
b and k values should be substituted in the general solution [tex]y(t)= \frac{20}{19e^{-kt} }[/tex]
Therefore, the model's necessary solution is [tex]y=\frac{20}{1+19e^{0.779t} }[/tex] in logistic equation.
for part (b)
The weight of the culture is 14.425 grams after five hours.
for part (c)
After 6.6 years, the culture has an 18 gram weight.
for part (d)
The logistic differential equation used to simulate how quickly a culture's weight increases is [tex]\frac{dy}{dy}= 0.779y(1-\frac{y}{20})[/tex]
5. hours later, the culture weighs 11.571 grams.
for part (e)
According to portion (c) of this question, the culture's weight will be 18 grams when t=6.6, which is why the weight of the culture increases most quickly at that time.
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Rick deposits $3 into his bank account. The following week, he deposits 3 times that amount. Each week after that, he deposits 3 times the amount he did the previous week. How much does he deposit in all during the first 3 weeks?
Answer:
answer on the picture......
What is the product of any integer and -1?
Answer: Its the same number with the sign changed..
Step-by-step explanation:
5 multiplied by negative 1 would be -5
And
negative 5 multiplied by negative 1 would be 5
Ana Maria needed to collect data to see why people at her college do not seem to want to join her book club. She decides to walk around the campus and ask people that she meets about the book club.
What data collection method is Ana Maria using?
Convenience sample, simple random sample, cluster sample, or voluntary response sample?
The data collection method used by Ana Maria for sampling is; simple random sample
What is the method of sampling?There are different methods of sampling such as;
Simple random sampling whereby each individual is chosen completely by chance & each member of the population has an equal chance, or probability, of being selected.
Systematic sampling whereby the Individuals are being selected at regular intervals from the sampling frame.
Stratified sampling whereby the population is first of all divided into subgroups or called strata who all share a similar characteristic.
Clustered sampling whereby subgroups of the population are used as the sampling unit, rather than individuals.
Convenience sampling whereby participants are selected based on availability and willingness to take part
Now, since she decides to walk around the campus and ask people that she meets about the book club. This will be an example of Simple random sampling.
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What expressions are equilvalent to 35+30s-45t?
Answer: 5(7+6s-9t)
Step-by-step explanation:
(note, this isn't the only expression that is equivalent but it's one of them)
You can notice that 35, 30, and 45 share a common factor of 5, and 5x7=35, 5x6=30, and 5x(-9)=-45, so:
5(7+6s-9t), which when you expand, it's equal to= 35+30s-45t
Solve:
b. m∠1=
m∠2=
m∠3=
The value of missing angles in the given figure are as follow:
m∠1= 56 ,m∠2= 56 and m∠3= 74.
As given in the question,
In the figure, MPQ is a triangle
We know that sum of the angles in the triangle is 180.
m∠ MPQ + m∠ PMQ + m∠1 =180
⇒58 + 66 + m∠1=180
⇒124 + m∠1= 180
⇒m∠1= 180 - 124
⇒m∠1=56
so m∠1=56.
∠1 and ∠2 are vertically opposite angles so ,
⇒m∠1= m∠2
therefore, m∠2=56.
Now in triangle QNO
∠QON + m∠2 + m∠3 =180
⇒50 + 56 +m∠ 3=180
⇒106 + m∠3= 180
⇒m∠3= 180 - 106
⇒m∠3=74
Therefore, the measure of the angles in the given figure are m∠1= 56 ,m∠2= 56 and m∠3= 74.
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consider a linear regression model where is a random vector with , , and no further assumptions are made about is an by deterministic matrix, and is invertible. is an unknown constant. let denote the least squares estimator of in this context. determine whether each of the the following statements is true or false. is the maximum likelihood estimator for .
False. The maximum likelihood estimator (MLE) is a method for estimating the parameters of a statistical model based on the maximization of the likelihood function. In the context of linear regression, the MLE would be the values of the parameters that maximize the likelihood of the observed data given the model.
The least squares estimator, on the other hand, is a method for estimating the parameters of a linear regression model by minimizing the sum of squared residuals, which is the difference between the observed response values and the fitted values predicted by the model. While the least squares estimator and the MLE can sometimes lead to similar estimates of the parameters, they are not the same thing and are derived using different principles.
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Write the equation in standard form. 4y−5x=3(4x−2y+1)
Answer:
To write the equation in standard form, we need to isolate the variables on one side of the equation and the constants on the other side. We can do this by combining like terms and using the distributive property.
First, we distribute the 3 on the right side of the equation:
4y - 5x = 12x - 6y + 3
Then, we combine like terms:
4y - 5x = 12x - 6y + 3
= -x + 6y + 3
Finally, we move all the constants to the right side of the equation and all the variables to the left side:
x - 6y = -3
This is the standard form of the equation. In standard form, the equation has the form ax + by = c, where a and b are coefficients, and c is a constant. In this case, the coefficients are 1 and -6, and the constant is -3.
Answer:
17x - 10y = - 3
Step-by-step explanation:
the equation of a line in standard form is
Ax + By = C ( A is a positive integer and B, C are integers )
4y - 5x = 3(4x - 2y + 1) ← distribute parenthesis
4y - 5x = 12x - 6y + 3 ( subtract 4y - 5x from both sides )
0 = 17x - 10y + 3 ( subtract 3 from both sides )
- 3 = 17x - 10y , that is
17x - 10y = - 3 ← in standard form
Solve each equation.
Square root of x+x^2=0
The solutions to the equation √( x + x² ) = 0 are x = 0 and x = -1.
What is the solution(s) to the quadratic equation?Given equation in the question;
√( x + x² ) = 0
First, remove the radical of the left side of the equation by squaring both sides.
(√( x + x² ))² = 0²
x + x² = 0²
x + x² = 0
Next, factor the left side of the equation.
The common factor between x and x² is x
x( 1 + x ) = 0
Hence
x = 0
1 + x = 0
Subtract 1 from both sides
1 - 1 + x = 0 - 1
x = 0 - 1
x = -1
Therefore, the values of x are 0 and -1.
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A hot air balloon rose at a constant rate. After 3 minutes, the balloon was 44 meters above
the ground. Then, 2 minutes later, the hot air balloon had risen to 70 meters above the
ground.
1A) How much did the hot air balloon rise each minute?
meters
___meters
1B) How far above the ground was the hot air balloon at the start of the ride?
___ meters
1C) Complete the equation that describes the relationship between the altitude of the hot air
balloon in meters, A, and the elapsed time in minutes, t.
Write your answer using whole numbers or decimals rounded to the nearest tenth.
A=___t+___
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1A) To find out how much the hot air balloon rose each minute, we need to find the rate at which it rose. The rate can be calculated by dividing the change in altitude by the change in time. In this case, the change in altitude is 70 meters - 44 meters = 26 meters, and the change in time is 2 minutes - 3 minutes = -1 minute (since the time is decreasing). The rate is therefore 26 meters/-1 minute = -26 meters/minute.
1B) The hot air balloon started at an altitude of 44 meters - (-26 meters/minute) * 3 minutes = 44 meters - (-78 meters) = 122 meters.
1C) To find the equation that describes the relationship between the altitude of the hot air balloon and the elapsed time, we can use the information that we have. The altitude at time t=0 is 122 meters, and the rate at which the altitude changes is -26 meters/minute. The equation is therefore A = -26t + 122.
The hot air balloon rise 13 m each minute.
The hot air balloon was 5 m above the ground at start.
The equation that describes the relationship between the altitude of the hot air balloon in meters is A = 13t + 5.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
We have, After 3 minutes, the balloon was 44 meters above the ground.
After 2 minutes later, the hot air balloon had risen to 70 meters above the ground.
1. The rate of change
= (70-44)/ (5-3)
= 26/ 2
= 13 m / min
2. At start the hot air balloon is
= (44-13 x 3)
= 5 m above the ground
3. The equation that describes the relationship between the altitude of the hot air balloon in meters is
A = 13t + 5.
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The Pacific halibut fishery has been modeled by the differential equation dy/dt= ky(1 - Y/M) where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 7 times 107 kg, and k = 0.78 per year. If y(0) = 2 times 107 kg, find the biomass a year later. (Round your answer to two decimal places.) times 107 kg How long will it take for the biomass to reach 4 times 107 kg? (Round your answer to two decimal places.) yr
The biomass a year later is [tex]& \ y \approx 3.23 \times 10^7 \mathrm{~kg}[/tex] and t=1.58 years to reach 4 times 107 kg of biomass.
The following differential equation: [tex]& \Rightarrow \frac{d y}{d t}=k y\left(1-\frac{y}{M}\right) \\[/tex]
[tex]& \Rightarrow \frac{d y}{d t}=k y\left(\frac{M-y}{M}\right)[/tex]
Using variable separable method:
[tex]$$\Rightarrow \frac{d y}{y(M-y)}=\left(\frac{k}{M}\right) d t$$[/tex]
Integrate both sides:
[tex]$$\begin{gathered}\Rightarrow-\int \frac{d y}{y(y-M)}=\int\left(\frac{k}{M}\right) d t \\\\\Rightarrow-\frac{1}{M} \int \frac{M}{y(y-M)} d y=\int\left(\frac{k}{M}\right) d t \\\\\Rightarrow-\frac{1}{M} \int \frac{y-(y-M)}{y(y-M)} d y=\int\left(\frac{k}{M}\right) d t \\\\\Rightarrow-\frac{1}{M} \int\left(\frac{1}{y-M}-\frac{1}{y}\right) d y=\int\left(\frac{k}{M}\right) d t\end{gathered}$$[/tex]
[tex]\Rightarrow-\frac{1}{M}(\ln (y-M)-\ln y)=\frac{k t}{M}+C \\[/tex]
Let C1 be a constant such that C1 = MC, therefore,
[tex]$$\begin{gathered}\Rightarrow-\ln (y-M)+\ln y=k t+C_1 \\\Rightarrow \ln \frac{y}{y-M}=k t+C_1 \\\Rightarrow \frac{y}{y-M}=e^{k t+C_1} \\\\\Rightarrow y=y e^{k t+C_1}-M e^{k t+C_1} \\\\\Rightarrow y e^{k t+C_1}-y=M e^{k t+C_1} \\\\\Rightarrow y=\frac{M e^{k t+C_1}}{e^{k t+C_1}-1} \\\\\Rightarrow y=\frac{M}{1-e^{-k t-C_1}} \\\Rightarrow y=\frac{M}{1-e^{-k t} e^{C_1}}\end{gathered}$$[/tex]
Let A be a new constant such that:
[tex]\Rightarrow y=\frac{M}{1-A e^{-k t}}[/tex]
Substitute the values of M and k, we'll get:
[tex]\Rightarrow y=\frac{7 \times 10^7}{1-A e^{-0.76 t}}[/tex]
Substituting the given initial condition:[tex]\Rightarrow y(0)=2 \times 10^7[/tex]
[tex]\Rightarrow 2 \times 10^7=\frac{7 \times 10^7}{1-A e^{-0.76(0)}} \\\\\\Rightarrow 2=\frac{7}{1-A} \\[/tex]
=2-2A=7
=A=-2.5
Therefore, [tex]\Rightarrow y=\frac{7 \times 10^7}{1+2.5 e^{-0.76 t}}[/tex]
On simplification, we'll get:
[tex]$$\begin{aligned}& \Rightarrow y \approx 32270466.08 \mathrm{~kg} \\& \Rightarrow y \approx 3.23 \times 10^7 \mathrm{~kg}\end{aligned}$$[/tex]
(b). Substitute the given value in the equation
[tex]$$\begin{gathered}\Rightarrow 4 \times 10^7=\frac{7 \times 10^7}{1+2.5 e^{-0.76 t}} \\\Rightarrow 4=\frac{7}{1+2.5 e^{-0.76 t}} \\\Rightarrow 4+10 e^{-0.76 t}=7\end{gathered}$$[/tex]
On simplification, we'll get t=1.58 years
Therefore, it will take for the biomass to reach 4 times 107 kg is 1.58 years.
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edpuzzle
YOUR TURN: FIND THE VALUE OF X AND Y:
!!!
MULTIPLE CHOICE QUESTION
Find the value of x:
3√4
8
4√3
4√8
8√4
The value of x is 8.
What is Pythagoras theorem?Pythagoras theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.
Given that, a right triangle,
Solving for x,
y² = 12×4
y = 4√3
y²+4² = x²
(4√3)² +4² = x²
x² = 64
x = 8
Hence, The value of x is 8.
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