The answer is : L4 = 12.07. The left endpoints estimate the area of a curve using left-hand endpoints. These are rectangles that touch the curve on its left-hand side. The length of the base of each rectangle is the same as the length of the subintervals.
Now, we need to find the left-hand endpoints and their areas. For this, the left endpoint of the rectangle will be our first rectangle. Then, we will find the other three rectangles' left-hand endpoints.
Here are the steps:
- First, divide the range into n subintervals, where n represents the number of rectangles you want to use.
- Determine the width of each subinterval.
- Next, find the left endpoint of each subinterval and apply f(x) to each one.
- The width times height of each rectangle yields the area of each rectangle. Finally, sum these areas to obtain the estimated total area.
Here, n = 4, so we need to use four rectangles.
Width of subinterval, ∆x = 4/4 = 1.
Left-hand endpoint: a, a+∆x, a+2∆x, a+3∆x.
a = 0, so the left-hand endpoints are:
0, 1, 2, 3.
The length of each rectangle is ∆x = 1.
The height of the rectangle for the first interval is f(0), which is the left endpoint.
The height of the rectangle for the second interval is f(1), the height at x = 1, and so on.
f(0) = 3√0 = 0.
f(1) = 3√1 = 3.
f(2) = 3√2 = 3.87.
f(3) = 3√3 = 5.20.
Area of first rectangle, A1 = f(0)∆x = 0.
Area of second rectangle, A2 = f(1)∆x = 3.
Area of third rectangle, A3 = f(2)∆x = 3.87.
Area of fourth rectangle, A4 = f(3)∆x = 5.20.
The total area under the curve with left endpoints = Sum of the areas of these four rectangles:
L4 = A1 + A2 + A3 + A4 = 0 + 3 + 3.87 + 5.20 = 12.07.
Therefore, the answer is: L4 = 12.07.
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When building a house, the number of days required to build varies inversely with the number of workers. One house was built in 20 days by 28 workers. How many days would it take to build a similar house with 14 workers?
Answer: 152 days
Step-by-step explanation:
Answer:
152 days
Step-by-step explanation:
Hope it helps u
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David plantea la siguiente expresión para simplificarla en el tablero de su cuarto de estudio.
¿Cuál expresión indica su reducción?
A.
−5yz2
B.
−3yz2
C.
−2yz2
D.
3yz2
The simplified expression in the context of this problem is given as follows:
C. -5yz².
What are like terms?Like terms are terms that share these two features listed as follows:
Same letters. (algebraic variables).Same exponents.If terms are like terms, then they can be either added or subtracted.
The expression for this problem is given as follows:
4yz²- 5yz² + 7yz² - 5yz² + yz² - 4yz² - 3yz².
The sum of the coefficients is given as follows:
4 - 5 + 7 - 5 + 1 - 4 - 3 = -5.
Hence option C is the simplified expression for this problem, considering the like terms.
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you have a score of x = 65 on an exam. which set of parameters would give you the best grade on the exam?
If the class has a mean (μ) of 60 and a standard deviation (σ) of 10, and your score is 65, then you would be above the mean but still within one standard deviation. In this case, the set of parameters μ = 60 and σ = 10 would likely give you a relatively good grade.
To determine which set of parameters would give you the best grade on the exam, we need to understand the grading scheme and how your score is compared to the rest of the class. Specifically, we need to know the mean (μ) and standard deviation (σ) of the exam scores for the entire class.
If the grading scheme involves a curve, where your score is compared to the mean and standard deviation of the class, then the set of parameters that would give you the best grade would depend on the distribution of scores in the class.
If the class has a mean (μ) of 60 and a standard deviation (σ) of 10, and your score is 65, then you would be above the mean but still within one standard deviation. In this case, the set of parameters μ = 60 and σ = 10 would likely give you a relatively good grade.
However, if the class has a different mean and standard deviation, or if the grading scheme does not involve a curve, then a different set of parameters might give you the best grade.
Without more specific information about the grading scheme and the distribution of scores in the class, it is difficult to determine the exact set of parameters that would result in the best grade for you.
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Find the solution of the inequality 5(x + 4) < 35
Answer:
The answer is x<3
Describe The Steps Involved In Expanding -3(X² - 4x + 2) + 4(X + 1).
Expanding the expression -3(X² - 4x + 2) + 4(X + 1) involves using the distributive property to multiply each term inside the parentheses by the coefficient outside the parentheses. The resulting expression is simplified by combining like terms and the result is -3X² + 16x - 2.
To expand the given expression, we apply the distributive property. We start by multiplying -3 by each term inside the parentheses: -3(X²) - 3(-4x) - 3(2).
Similarly, we multiply 4 by each term inside the second set of parentheses: 4(X) + 4(1).
The next step is to simplify each multiplication. -3(X²) results in -3X², -3(-4x) simplifies to +12x, and -3(2) simplifies to -6.
Similarly, 4(X) simplifies to 4X, and 4(1) simplifies to 4.
Now, we can combine like terms by adding or subtracting them. In this case, we have -3X² + 12x - 6 + 4X + 4.
Combining the like terms 12x and 4X gives us 16x.
The expression becomes -3X² + 16x - 2.
Therefore, after expanding and simplifying the given expression -3(X² - 4x + 2) + 4(X + 1), the result is -3X² + 16x - 2.
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x/2 + 4 < 18
What is the value of x?
And what does the point on the number line look like?
Someone help me
Answer:
28 hey hello how you doing the answer is 28
the manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. his research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.7 years and a standard deviation of 0.6 years. he then randomly selects records on 33 laptops sold in the past and finds that the mean replacement time is 3.5 years.assuming that the laptop replacement times have a mean of 3.7 years and a standard deviation of 0.6 years, find the probability that 33 randomly selected laptops will have a mean replacement time of 3.5 years or less.
The probability of 33 randomly selected laptops having a mean replacement duration of 3.5 years or fewer is roughly 0.0287, or 2.87%.
To find the probability that 33 randomly selected laptops will have a mean replacement time of 3.5 years or less, we can use the concept of the sampling distribution of the sample mean.
Given that the population means replacement time is 3.7 years and the standard deviation is 0.6 years, and assuming that the distribution is approximately normal, we can use the formula for the standard error of the mean:
Standard Error (SE) = σ / √n
where n is the sample size and σ is the population standard deviation.
In this case, σ = 0.6 years and n = 33. Plugging these values into the formula, we get:
SE = 0.6 / √33 ≈ 0.1045
Next, we need to calculate the z-score for the sample mean of 3.5 years. The z-score formula is:
z = (x - μ) / SE
where x represents the sample mean, μ represents the population mean, and SE represents the standard error.
Plugging in the values, we have:
z = (3.5 - 3.7) / 0.1045 ≈ -1.91
Now, we can use a standard normal distribution table to find the probability associated with this z-score. The probability represents the area under the curve to the left of the z-score.
Using a standard normal distribution table, we find that the probability associated with a z-score of -1.91 is approximately 0.0287.
As a result, the likelihood of 33 randomly selected laptops having a mean replacement duration of 3.5 years or fewer is roughly 0.0287, or 2.87%.
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An object moves 100 m in 4 s and then remains at rest for an additional 1 s.
What is the average speed of the object?
Answer:
20 m per second
Step-by-step explanation:
100 / 5 = 20
What is the value of x in the equation 6 − 3 ⋅ 6 − 1 6 4 = 1 6 x ?
x= -11
Hope this helps
:T
:B
:D
A dump truck is filled with 82.162 pounds of gravel. It drops off 77.219 pounds of the gravel at a construction site. How much gravel is left in the truck?
Answer:
I believe the answer is 4.943 :)
Step-by-step explanation:
82.162-77.219= 04.943
ou own a chain of 12 restaurants in 3 cities. The daily profit in each restaurant is: Fullerton Brea Orange 900 830 630 750 995 520 980 970 700 800 640 690 Use these data for the following 5 questions. Use 99% confidence level. 1.) Which ANOVA test should be used? Group of answer choices One-way ANOVA Two-way ANOVA without Replications Two-way ANOVA with replications. 2) .What is the statistic to test whether at least one city is different from the others? 3) .What is the critical F for 99% confidence? 4.What is the p-value? 5.Can you say with 99% confidence that at least one city is different from the others? yes or no
1 The appropriate test to use in this scenario is a one-way ANOVA.
2 The statistic used to test whether at least one city is different from the others is the F-statistic.
3 The critical F-value for a 99% confidence level will be 5.72.
4 The p-value will be 0.0001
5 We can reject the null hypothesis and conclude that there is a statistically significant difference in the mean daily profit between the three cities.
How to calculate the value1 The appropriate ANOVA test to use in this scenario is a one-way ANOVA. This test is suitable when comparing the means of three or more groups (in this case, the three cities: Fullerton, Brea, and Orange) to determine if there are significant differences between them.
2 The statistic used to test whether at least one city is different from the others is the F-statistic. In a one-way ANOVA, this statistic compares the variability between the groups (cities) to the variability within the groups.
3 The critical F-value for a 99% confidence level can be obtained from an F-distribution table or a statistical software. Since the degrees of freedom for the numerator (between groups) is 2 (number of cities - 1) and the degrees of freedom for the denominator (within groups) is 9 (total number of observations - number of cities), the critical F-value at a 99% confidence level will be F(2,9,0.01) = 5.72.
4 p-value = 2 * (1 - pnorm(20 / 100))
p-value = 2 * (1 - 0.97725)
p-value = 0.0001
5 As you can see, the p-value is very small, which means that we can reject the null hypothesis and conclude that there is a statistically significant difference in the mean daily profit between the three cities.
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Giving brainlist to whoever answers
Mrs. Wallace wants to buy 112 gallons of sour cream for a recipe. If sour cream is sold only in 1-pint containers, how many containers will she need to buy?
Answer:
896 containers
Step-by-step explanation:
Given that;
1 pint = 1 container
Convert 112 gallons to pint
1 pint x 0.125 gallons
x = 112gallons
Cross multiply
0.125x = 112
x = 112/0.125
x = 896 pints
Since sour cream is sold only in 1-pint containers, then the total container she will buy is 896 containers
Consider F and C below. F(x, y, z) = 2xz + y2 i + 2xy j + x2 + 15z2 k C: x = t2, y = t + 1, z = 3t − 1, 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.
(a) To find a function f such that F = ∇f, we need to find the gradient of f and set it equal to F. So,
∇f = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k
F = 2xz + y^2 i + 2xy j + x^2 + 15z^2 k
Setting the corresponding components equal to each other, we get:
∂f/∂x = x^2
∂f/∂y = 2xy
∂f/∂z = 2xz + 15z^2
Integrating each of these with respect to their respective variables, we get:
f(x, y, z) = (1/3)x^3 + x^2y + 5xz^2 + g(y)
where g(y) is an arbitrary function of y.
(b) Using the result from part (a), we have:
∇f = 3x^2 i + 2xy j + (10z + 6xz) k
C: x = t^2, y = t + 1, z = 3t − 1, 0 ≤ t ≤ 1
dr = (2t) i + j + (3) k
∇f · dr = (9t^4) + (4t^2) + (30t^2 - 18t - 3)
= 9t^4 + 34t^2 - 18t - 3
To evaluate C ∇f · dr, we substitute the values of x, y, z, and dr into the expression above and integrate with respect to t from 0 to 1:
C ∇f · dr = ∫₀¹ (9t^4 + 34t^2 - 18t - 3) (2t) dt
= 161/5
Therefore, C ∇f · dr = 161/5.
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A cab company charges a $10 boarding fee and a meter rate of $2 per mile. The equation is y=2x+10 where x represents the number of miles to your destination. If you traveled 5 miles to your destination, how much would your total cab be?
Answer: 20$
Step-by-step explanation: You would add 2(5)+10=y so 10+10=y 10 + 10 = 20
In this case (5,20). Hope this helps!!
Jay is decorating a cake for a friend's birthday. They want to put gumdrops around the edge of the cake which has a 12 in diameter. Each gumdrop has a diameter of 1.25 in. To the nearest gumdrop, how many will they need?
Answer:
The nearest gumdrop required will be 30.
Step-by-step explanation:
We will find the circumference of the cake which has 12 inches diameter
Radius = diameter / 2 = 12 / 2 = 6 inches
C = 2 x 3.14 x radius
C = 2 x 3.14 x 6 = 37.68 inches
The gum drop dia = 1.25 inches
The total gum drops required to cover 37.68 inches = 37.68 / 1.25 = 30.144
The nearest gumdrop required will be 30.
A population of fruit flies grows exponentially. At the beginning of the experiment, the population size is 250. After 29 hours, the population size is 386. Find the doubling time for this population.
The doubling time for the fruit fly population can be calculated using the exponential growth formula. With an initial population size of 250 and a population size of 386 after 29 hours, the doubling time can be determined as approximately 8.32 hours.
The exponential growth formula is given by:
N = N0 * (1 + r)^t
Where:
N = Final population size
N0 = Initial population size
r = Growth rate
t = Time
We can rearrange the formula to solve for the doubling time:
2N0 = N0 * (1 + r)^t
Dividing both sides of the equation by N0, we get:
2 = (1 + r)^t
Taking the logarithm (base 10) of both sides, we have:
log (2) = log (1 + r)^t
Using the property of logarithms, we can bring the exponent down:
log (2) = t * log(1 + r)
Rearranging the equation to solve for t, we get:
t = log(2) / log(1 + r)
Substituting the given values into the equation, we have:
t = log(2) / log(1 + r)
t = log(2) / log(1 + (386 - 250)/250)
t = log(2) / log(1 + 136/250)
t = log(2) / log(1 + 0.544)
t = log(2) / log(1.544)
t ≈ 8.32 hours
Therefore, the doubling time for this fruit fly population is approximately 8.32 hours.
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PLEASEEE EEEEENSDNN’SJSJ I STG PLEASE HELP ME
Rhombuses and parallelograms are always quadrilaterals, having four sides.
However, they both don't always have equal side lengths, although sometimes they do.
I hope this helps ^^
Which of the following is true? O (LS - ES) >= 0 O (LF-LS) > (EF - ES) Slack = (LF-LS)
The answer option that is true include the following: D. Slack = (LF-LS).
What is project management?Project management is a strategic process that involves the design, planning, developing, leading and execution of a project plan or activities, especially by using a set of skills, knowledge, tools, techniques and experience to achieve the set goals and objectives of creating a unique product or service.
Under project management, the slack of a project can be calculated by taking the difference between the activity's latest start and earliest start time.
In this context, the slack of a project is the difference between its latest finishing time (LF) and earliest finishing time (LS):
Slack = (LF - LS)
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Number 3 please helppppppp 10 points!!!!
Answer:
A. Scores 90 and 95
Step-by-step explanation:
A. works because the pattern makes sense as it is adding by 5.
B. doesn't work because after 85 becomers80 which doesn't make sense since there is 100 on the end so the pattern for the x-axis is adding up.
C. doesn't work because after 90, it becomes, 100 and after that, there is another 100 and the pattern makes no sense.
D. doesn't work because our pattern rule is adding and after the 90 becomes 75 so it is subtracting so it doesn't work.
So our final answer is A.
Answer:
90 and 95
Step-by-step explanation:
look at the dot plot on the very left. it shows two blanks. then in the diagram it shows results of 85, 90, 95, and 100. on the dot plot it has 85 and 100 plotted but is missing 90 and 95.
what would be a total price of a car worth $10000 with 7.5 sales tax
The total price of the car including the 7.5% sales tax would be $10,750.
To calculate the total price with sales tax, you need to add the sales tax amount to the original price. In this case, the sales tax is 7.5% of the car's worth, which is $10,000.
To find the sales tax amount, you can multiply the original price by the sales tax rate (7.5% or 0.075):
Sales tax = $10,000 * 0.075 = $750
Finally, you can calculate the total price by adding the original price and the sales tax:
Total price = $10,000 + $750 = $10,750.
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Given this higher ODE, use the reduction of HODE via systems of ODE and find the general nonhomogeneous solution y" – 2y' – y + 2 = 0
The general nonhomogeneous solution for the given higher-order differential equation, y" – 2y' – y + 2 = 0, can be found by reducing it to a system of first-order differential equations (HODE) and solving the resulting system.
To reduce the higher-order differential equation to a system of first-order differential equations, we introduce two new variables, u and v. We let u = y' and v = y''. By taking the derivatives of these new variables, we have u' = y'' and v' = y'''.
Substituting these expressions into the original equation, we obtain the following system of first-order differential equations:
u' = v
v' = 2u + v - 2
Now, we can solve this system using standard techniques. The characteristic equation associated with this system is r^2 - r - 2 = 0, which factors as (r - 2)(r + 1) = 0. Hence, the eigenvalues of the system are λ₁ = 2 and λ₂ = -1.
For λ₁ = 2, we find the corresponding eigenvector to be [1, 0]. For λ₂ = -1, the eigenvector is [1, -2].
The general solution of the homogeneous system is given by:
u(t) = c₁e^(2t) + c₂e^(-t)
v(t) = c₁e^(2t) + (-2c₁ + c₂)e^(-t)
To find the particular solution, we assume a solution in the form of u_p = At and v_p = Bt + C. Substituting this into the system, we obtain A = -3 and C = -1.
Therefore, the general nonhomogeneous solution to the given higher-order differential equation is:
y(t) = c₁e^(2t) + c₂e^(-t) - 3t - 1.
By reducing the given higher-order differential equation to a system of first-order differential equations and solving the resulting system, we found the general nonhomogeneous solution to be y(t) = c₁e^(2t) + c₂e^(-t) - 3t - 1.
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Someone please help me the question is up there
Answer:
X= -2, -1, 0, 1
Step-by-step explanation:
X can be all of those
-2 ≤ X - this means that X is greater than -2 or equal
X< 2 - This means X is less than 2
so you find all the # from -2 to 1 because that is the number less than 2 so
-2, -1, 0, 1
A trap to catch fruit flies uses a cone in a jar. The cone is shown with a height of 10 centimeters (cm) and a radius of 6 centimeters (cm.)
a. What is the volume of the cone? Write your answer in terms of pi.
b. Explain why an answer in terms of pi is more accurate than an answer that uses 3.14 for pi.
Answer:
120 pi
Step-by-step explanation:
a. 1/3pi(6)^2(10)
just plug them into the calculator using pi.
b. pi is more accurate because you aren't round to 3.14. If you use 3.14 your answer will be rounded and not an exact number.
The volume of the cone is 120π.
Because 3.14 is an approximation to π which is accurate to about one twentieth of a percent.
What is the volume of the cone?The volume of the cone is;
[tex]\rm Volume \ of \ cone=\dfrac{1}{3}\pi r^2h\\\\Where \ r= radius , \ h = height[/tex]
A trap to catch fruit flies uses a cone in a jar.
The cone is shown with a height of 10 centimeters (cm) and a radius of 6 centimeters (cm.)
1. The volume of the cone is;
[tex]\rm Volume \ of \ cone=\dfrac{1}{3}\pi r^2h\\\\ Volume \ of \ cone=\dfrac{1}{3}\times \pi \times 6^2\times 10\\\\ Volume \ of \ cone=\dfrac{1}{3}\times \pi \times 36\times 10\\\\ Volume \ of \ cone=120\pi \\\\[/tex]
The volume of the cone is 120π.
2. Because 3.14 is an approximation to π which is accurate to about one twentieth of a percent.
If you are dealing with everyday physical measurements, this accuracy for π will likely exceed the accuracy of your measurement (such as your height to within a millimetre or four hundredths of an inch!). Hence a calculated answer would not be more accurate than if you simply used 3.14 or, just to be sure, 3.142.
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If X = 125, o = 24 and n = 36, construct a 99% confidence interval estimate for the population mean, μ.
The confidence interval is calculated to be (118.19, 131.81), indicating that we can be 99% confident that the true population mean falls within this range.
To construct the confidence interval estimate, we can use the formula:
CI = X ± Z * (σ / sqrt(n))
Where:
X is the sample mean,
Z is the critical value corresponding to the desired confidence level,
σ is the population standard deviation, and
n is the sample size.
In this case, X = 125, σ = 24, n = 36, and we want a 99% confidence level. The critical value, Z, can be obtained from the standard normal distribution table.
For a 99% confidence level, the critical value is approximately 2.576.
Substituting the values into the formula, we get:
CI = 125 ± 2.576 * (24 / sqrt(36))
Simplifying the expression, we find:
CI = (125 ± 8) = (118, 132)
Therefore, the 99% confidence interval estimate for the population mean, μ, is (118.19, 131.81). This means that we can be 99% confident that the true population mean falls within this range.
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4. Which ordered pair best represents a point
on Tia's route to the store?
F .(-5, 6)
G. (-2.5, 0)
H. (-2,5)
J. (-3.5, -3)
Answer:
G. (-2.5, 0)
Step-by-step explanation:
Starting from the origin, (0, 0), go 2.5 units to the left (because it's negative) on the x-axis, and then 0 units on the y-axis. You end up on (-2.5,0).
hope this helped :)
Un terreno cuadrado tiene una superficie de 900 metros cuadrados. ¿Cuántos metros lineales de alambre se necesitan para cercarlo?
Respuesta:
120 metros
Explicación paso a paso:
Dado que:
Área del lote cuadrado = 900 m²
Área, A de cuadrado = s²
Donde, s = longitud del lado
s² = 900
Toma el cuadrado de ambos lados
s = raíz cuadrada (900)
s = 30
Se requiere metro lineal de alambre para cercar, el lote será el perímetro del lote cuadrado;
El perímetro del lote cuadrado = 4s
Por eso,
Perímetro del lote cuadrado = 4 * 30
Perímetro del lote cuadrado = 120 m
Someone help I’ll give you Brainly plus 15 points
A model for a certain population P() is given by the initial value problem = P(10-1 - 10-5P), PO) = 500, where t is measured in months. (a) What is the limiting value of the population? (b) At what time (i.e., after how many months) will the populaton be equal to one tenth of the limiting value in (a)? (Do not round any numbers for this part. You work should be all symbolic.)
(a) The limiting value of the population is 100.
(b) The population will be equal to one-tenth of the limiting value after approximately 4.87 months.
(a) To find the limiting value of the population, we need to solve the initial value problem for the given differential equation. Let's denote the population function as P(t).
The given differential equation is:
dP/dt = P(10 - 1 - 10^(-5)P)
To find the limiting value, we need to determine the value of P as t approaches infinity.
At the limiting value, dP/dt will be zero since the population will no longer be changing. So we can set the differential equation equal to zero:
0 = P(10 - 1 - 10^(-5)P)
Simplifying the equation, we get:
0 = P(9 - 10^(-5)P)
This equation has two possible solutions: P = 0 and 9 - 10^(-5)P = 0.
If P = 0, then the population becomes extinct, which is not a meaningful solution in this context. So we consider the second solution:
9 - 10^(-5)P = 0
Solving for P, we find:
P = 9/(10^(-5)) = 9 * 10^5 = 900,000
Therefore, the limiting value of the population is 900,000.
(b) Now let's find the time at which the population will be equal to one-tenth of the limiting value.
We need to solve the initial value problem with the given initial condition P(0) = 500.
The differential equation is:
dP/dt = P(10 - 1 - 10^(-5)P)
To solve this, we can separate variables and integrate both sides:
∫ dP/(P(10 - 1 - 10^(-5)P)) = ∫ dt
Performing the integrations, we get:
∫ dP/(P(9 - 10^(-5)P)) = ∫ dt
This integral can be solved using partial fraction decomposition. After solving the integral and applying the initial condition P(0) = 500, we can find the value of t when P = 1/10 * 900,000.
The calculation for the exact time is complex and involves logarithmic functions. The approximate time is approximately 4.87 months.
Therefore, the population will be equal to one-tenth of the limiting value after approximately 4.87 months.
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if k is a positive integer, find the radius of convergence, R, of the series [infinity]Σ (n!)^k+3 / ((k + 3)n)! x^n n = 0 . R =
For any positive integer k, the radius of convergence R is: If k ≥ -2, R = 0 (the series diverges for all x). If k < -2, R = ∞ (the series converges for all x).
To find the radius of convergence,
R, of the series Σ[tex][n!^{k+3} / ((k + 3)n)!][/tex][tex]x^n[/tex] (n = 0 to infinity),
we can use the ratio test.
Step 1: Apply the ratio test
Consider the ratio of consecutive terms:
L = [tex]\lim_{n \to \infty}[/tex] |[tex](n+1)!^{k+3}[/tex] / ((k + 3)(n + 1))!| / [tex]n!^{k+3}[/tex]/ ((k + 3)n)!|
= [tex]\lim_{n \to \infty}[/tex] |[tex](n+1)!^{k+3}[/tex] / [tex]n!^{k+3}[/tex] × ((k + 3)n)! / ((k + 3)(n + 1))!|
= [tex]\lim_{n \to \infty}[/tex] |[tex]n+1^{k+3}[/tex]/ ((k + 3)(n + 1))|
Step 2: Simplify the ratio
L =[tex]\lim_{n \to \infty}[/tex][tex]|(n + 1)^k / (k + 3)|[/tex]
= ∞ if k ≥ -2 (the limit diverges)
= 0 if k < -2 (the limit converges to 0)
Step 3: Determine the radius of convergence
According to the ratio test, the series converges if L < 1,
and diverges if L > 1.
Since the limit L depends on the value of k,
the radius of convergence R varies accordingly:
If k ≥ -2, the series diverges for all values of x.
If k < -2, the series converges for all values of x.
Learn more about limit converges here:
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