find the radius of convergence and interval of convergence of each series. (a) [infinity] qn=1 (3n)! 3n xn (b) [infinity] qn=1 (3x −1)n n2n (c) [infinity] qn=2 x2n n(ln n)2
Answers
The radius and interval of convergence for (a) are R=3 and (-3, 3), for (b) are R=∞ and (-∞, ∞), and for (c) are R=1 and (-1, 1).
(a) For the series ∑(3n)!/(3^n x^n), we can use the ratio test:
lim (n→∞) |a_n+1/a_n| = lim (n→∞) [(3n + 3)(3n + 2)(3n + 1))/(3^(n+1))] |x|
= lim (n→∞) (27n^3 + 81n^2 + 74n + 22)/(27*3^n) |x|
= |x| lim (n→∞) [(n + 1)^3/n^3] (27/3^n)
= |x| lim (n→∞) [(1 + 1/n)^3] (27/3^n)
= |x|/3
Since the limit exists and is finite, the series converges absolutely if |x|/3 < 1, i.e., |x| < 3, and diverges if |x|/3 > 1, i.e., |x| > 3. Thus, the radius of convergence is R = 3, and the interval of convergence is (-3, 3).
(b) For the series ∑(3x-1)^n/n^(2n), we can use the root test:
lim (n→∞) |(3x-1)/n^2|^(1/n) = lim (n→∞) |3x-1|/n^2
= 0
Since the limit is less than 1 for all values of x, the series converges absolutely for all x. Thus, the radius of convergence is R = ∞, and the interval of convergence is (-∞, ∞).
(c) For the series ∑(x^2n)/(n(ln n)^2), we can use the ratio test:
lim (n→∞) |a_n+1/a_n| = lim (n→∞) (x^2(n+1))/[(n+1)(ln(n+1))^2] [(n(ln n)^2)/(x^2)]
= lim (n→∞) (x^2/(ln(n+1))^2) [(ln n)^2/x^2]
= lim (n→∞) [(x^2 ln^2 n)/(ln^2(n+1) x^2)]
= 1
Since the limit is equal to 1 for all values of x, the ratio test is inconclusive, and we need to use another test. One option is the Cauchy condensation test:
∑(2^n x^(2^n))/(2^n (ln 2^n)^2) = ∑(x^(2^n))/(n^2 ln^2 2)
Using the ratio test again, we get:
lim (n→∞) |a_n+1/a_n| = lim (n→∞) (x^(2^(n+1)))/(x^(2^n)) [(ln 2^n)^2/(ln 2^(n+1))^2]
= lim (n→∞) x^(2^n) [(ln 2n)^2/(ln 2n + ln 2)^2]
= x^2
Thus, the Cauchy condensed series converges absolutely if |x^2| < 1, i.e., |x| < 1, and diverges if |x^2| > 1, i.e., |x| > 1.
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Related Questions
describe the evidence that allowed scientists to conclude that electrons occupy specific energy levels and are not just randomly distributed outside the nucleus
Answers
Scientists can conclude that electrons occupy specific energy levels and are not just randomly distributed outside the nucleus.
The evidence that allowed scientists to conclude that electrons occupy specific energy levels and are not just randomly distributed outside the nucleus comes from the application of the quantum mechanical model. The quantum mechanical model states that the energy of an electron is quantized, meaning it can only occupy discrete energy levels rather than any random energy level in between. This is expressed mathematically through the equation E n = nhf, which states that the energy of an electron is equal to the product of Plank's constant (h) and the frequency (f) of the electron's orbit multiplied by the quantum number (n). This equation demonstrates that electrons can only occupy certain energy levels and not any random energy level. Therefore, scientists can conclude that electrons occupy specific energy levels and are not just randomly distributed outside the nucleus.
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For a certain relationship, y varies inversely with x. When x is 4, y is equal to 30. What is the value of y when x is equal to 10?
Answers
Step-by-step explanation:
inversely means
y = k/x
therefore
30 = k/4
120 = k
y = 120/10 = 12
FYI :
directly would mean
y = kx
The value of y will be 12 when x is equal to 10.
What is inverse variation?The relationship between two variables known as "inverse variation" occurs when the value of one variable increases while the value of the other variable decreases.
Given that, y varies inversely with x.
Here, y∝1/x
y=k/x
xy=k
When x is 4, y is equal to 30, we get
k=4×30
k=120
When x=10, we get
10y=120
y=12
Therefore, the value of y will be 12 when x is equal to 10.
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An object is moving with velocity (in ft/sec) v(t)=t2−1t−12
Find the displacement and total distance travelled from t=0 to t=6
Answers
To find the displacement and total distance traveled by the object from t=0 to t=6, we need to integrate the velocity function over the given time interval.
The displacement can be found by integrating the velocity function v(t) with respect to t over the interval [0, 6]. The integral of v(t) represents the net change in position of the object during this time interval.
The total distance traveled can be determined by considering the absolute value of the velocity function over the interval [0, 6]. This accounts for both the forward and backward movements of the object.
Now, let's calculate the displacement and total distance traveled using the given velocity function v(t) = t^2 - (1/t) - 12 over the interval [0, 6].
To find the displacement, we integrate the velocity function as follows:
Displacement = ∫[0,6] (t^2 - (1/t) - 12) dt.
To find the total distance traveled, we integrate the absolute value of the velocity function as follows:
Total distance = ∫[0,6] |t^2 - (1/t) - 12| dt.
By evaluating these integrals, we can determine the displacement and total distance traveled by the object from t=0 to t=6.
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8. Sam packed 15 boxes in 5 days. Dylan packed 60 boxes 4 weeks and 2 days. Who packed the most
boxes in the least in the least amount of time? Find each unit rate.
Answers
Answer:
sam did the most
Step-by-step explanation:
let's find box packed per day
sam - 15 boxes in 5 days
so 3 boxes in 1 day
Dylan - 60 boxes in 4 weeks and 2 days
that is 60 in 30 days
that will make 2 boxes in 1 day
if the measure 3= (2y+20) degrees, and the measure of 5= (210-4y) degrees, what is the measure of 4?
Answers
Answer:
110°
Step-by-step explanation:
y = 25
Angle 5 equals Angle 4 because of the Alternate Angles Postulate.
Angle 4 equals 110°
The measure of angle ∠4 is 110 degrees.
What are supplementary angles?Two angles whose sum is 180° are called supplementary angles. If a straight line is intersected by a line, then there are two angles form on each of the sides of the considered straight line.
Given that the measure Angle 3= (2y+20) degrees, and the measure of Angle 5= (210-4y) degree
Then ∠3 + ∠5 = 180, By Co-interior angles are supplementary
2y + 20 + 210 - 4y = 180
2y - 4y + 20 + 210 = 180
-2y + 230 = 180
Now Subtract 230 from both sides;
-2y = 180 - 230
-2y = -50
Then Divide both sides by (-2);
y = -50/-2
y = 25
Now, ∠5 = 210 - 4y
= 210 - 100
∠5 = 110
∠4 = ∠5 { by Alternate interior angles are equal}
∠ 4 = 110°
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27. Write down the co-ordinates of the points where the line y = 4x - 3 cuts the x-axis
Answers
Step-by-step explanation:
it is a line, so that is 1 point, where it cuts the x-axis.
in general that means y = 0.
so,
0 = 4x - 3
4x = 3
x = 3/4
the point, where the line cuts the x-axis is
(3/4, 0)
Which two equations could represent the line that goes through the points (-3,3) and (7,5)
Answers
y=3x-3
y=7x-2/5
Express the ratio below in its simplest form
12:6
Answers
Answer:
12/6 simplified to lowest terms is 2/1.
Step-by-step explanation:
Divide both the numerator and denominator by the HCF
12 ÷ 6
6 ÷ 6
Reduced fraction:
12/6 simplified to lowest terms is 2/1.
Answer: 2:1
Step-by-step explanation:
12:6
Both left and right can be divided by 6, like a fraction, reduce.
= 2:1
The ability for nike to manufacture its own shoes and then build stores for distribution is an example of?
Answers
The ability for Nike to manufacture its own shoes and then build stores for distribution is an example of Forward Integration.
What is forward integration ?
In order to lower manufacturing costs and increase efficiency, a sort of vertical integration known as "forward integration" moves up the supply chain.
An operational competency is what?
Functional competences are determined by the tasks and commitments employees make to a certain position. An average of three to five functional skills are ascribed to a given job, depending on the complexity and level of responsibility of the position as well as the seniority of the occupational role.Learn more about Forward Integration
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is the average (arithmetic mean) of x and y greater than 20?1. the average (arithmetic mean) of 2x and 2y is 48.2. x
Answers
After answering the provided question, we can conclude that the average (arithmetic mean) of 2x and 2y is 48.
What is mean?The mean of a dataset is the sum of all values divided by the total number of values; this is also known as the arithmetic mean (as opposed to the geometric mean). Often referred to as the "mean," this is the most often used measure of central tendency. Simply dividing the total number of values in the dataset by the sum of those values yields this result. Calculations can be performed using both raw data and data that has been combined into frequency tables. The average of a number is referred to as its average. It is simple to calculate: Divide the total number of digits by the number of digits. the total divided by the count.
The average (arithmetic mean) of 2x and 2y is 48.
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Determine whether the value is a discrete random variable, continuous random variable, or not a random variable. a. The height of a randomly selected giraffe b. The number of points scored during a basketball game c. The gender of college students d. The amount of rain in City Upper B during April e. The number of fish caught during a fishing tournament f. The number of bald eagles in a country
Answers
The height of a randomly selected giraffe, the number of points scored during a basketball game, the gender of college students, the amount of rain in City Upper B during April, and the number of fish caught during a fishing tournament are all examples of discrete or continuous random variables, while the number of bald eagles in a country is not a random variable.
a. Continuous random variable
b. Discrete random variable
c. Discrete random variable
d. Continuous random variable
e. Discrete random variable
f. Not a random variable
a. The height of a randomly selected giraffe is a continuous random variable because it can take on any value within a certain range.
b. The number of points scored during a basketball game is a discrete random variable because it is a countable number.
c. The gender of college students is a discrete random variable because it is a categorical variable with two possible values.
d. The amount of rain in City Upper B during April is a continuous random variable because it can take on any value within a certain range.
e. The number of amount fish caught during a fishing tournament is a discrete random variable because it is a countable number.
f. The number of bald eagles in a country is not a random variable because it is a fixed number and not a random sample.
The height of a randomly selected giraffe, the number of points scored during a basketball game, the gender of college students, the amount of rain in City Upper B during April, and the number of fish caught during a fishing tournament are all examples of discrete or continuous random variables, while the number of bald eagles in a country is not a random variable.
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The ornaments below are mathematically
similar.
Calculate the surface area of the smaller
ornament.
If your answer is a decimal, give it to 1 d.p.
surface area = 21,840 cm²
140 cm
surface area =
43
35 cm
...
cm²
Not drawn accurately
Answers
The surface area of the smaller ornament is given as follows:
1365 cm².
How to obtain the surface area of the smaller ornament?The surface area of the smaller ornament is obtained applying the proportions in the context of the problem.
The ratio between the side lengths is given as follows:
35/140 = 1/4.
As the side lengths are measured in units while the surface area is measured in units squared, the ratio of the areas is given as follows:
(1/4)² = 1/16.
Meaning that the surface area of the smaller ornament is given as follows:
1/16 x 21840 = 1365 cm².
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2x+2= 4x solve this equation
Answers
Answer:
x = 1
Step-by-step explanation:
2x+2 = 4x
You're trying to solve for x
Subtract 2x from both sides
2 = 4x - 2x
Then, do 4x-2x
2=2x
Lastly, do 2 divided by 2 to solve for x
x = 1
which of the following is true of the number of hours of daylight? group of answer choices the number of hours of daylight varies depending on the latitude of the observer. the number of hours of daylight varies the least at higher latitudes. the number of hours of daylight varies the most along the equator. the number of hours of daylight includes the hours between dawn and twilight, not just the hours from sunrise to sunset.
Answers
The statement that is true about the number of hours of daylight is: "the number of hours of daylight varies depending on the latitude of the observer."
Hence, option (a) is the correct choice.
Day length will be roughly 12 hours at latitude 0° (the Equator). The Equator region has a steady 12 hour day light all year.
Day duration can be observed to extend to 24 hours or reduce to zero if latitude increases to 80° (polar circles - north or south) (depending on time of year).
The quantity of daylight received by an observer is determined by their latitude and the season.
The number of hours of daylight changes the greatest near the equator, whereas it varies the least towards higher latitudes.
The number of daylight hours covers the hours between dawn and twilight as well as the hours from sunrise to sunset.
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Four more than three times a number is 16
Find the number
Answers
Answer:
n = 4
Step-by-step explanation:
We can find the number, n, using the equation:
3n + 4 = 16
3n = 12
n = 4
Identification. Read carefully the following statements. Provide the best word that describes each statement. 1. This pattem in nature are also exhibited in the external appearances of the animals. 2. This kind of pattern is exhibited by objects when their similar parts are regularly arranged around a central axis and the pattern looks the same after a certain amount of rotation. 3. These are curved patterns made by series of circular shapes revolving around a central point 4. This kind of pattern is exhibited by objects which do not change its size and shape even if it moved to another location. 5. Describes as an exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a contor or an axis. It indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other.INSTRUCTION: Determine the value of term under the Fibonacci sequence. Let Fib (n) be the nth term of
the Fibonacci sequence, with Fib (1) = 1, Fib (2) = 1, Fib (3) = 2, and so on. 1.(3pts)- Find Fib (8) 2.(3pts)- Find
Fib (19) 3.(4pts)- If Fib (22) = 17, 711 and Fib (24) = 46, 368, what is Fib (23)?
Answers
1. The word that describes a pattern in nature that is also exhibited in the external appearances of animals is "symmetry."
Symmetry refers to the balanced arrangement of parts or elements. In nature, many animals display symmetry in their body structure, such as butterflies with their symmetrical wings.
2. The word that describes a pattern exhibited by objects when their similar parts are regularly arranged around a central axis and the pattern looks the same after a certain amount of rotation is "rotational symmetry."
Rotational symmetry occurs when an object can be rotated and still maintain its original appearance. For example, a circle has rotational symmetry because it looks the same after any amount of rotation.
3. The word that describes curved patterns made by a series of circular shapes revolving around a central point is "spiral."
Spirals can be seen in various aspects of nature, like the arrangement of petals in a sunflower or the shape of a seashell.
4. The word that describes a pattern exhibited by objects that do not change their size and shape even if they are moved to another location is "translation symmetry."
Translation symmetry means that an object can be moved without altering its characteristics. For instance, a rectangle has translation symmetry because it remains the same even if it is shifted horizontally or vertically.
5. The word that describes an exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a contour or an axis is "reflection symmetry" or "mirror symmetry."
Reflection symmetry means that an object can be divided into two equal halves, and each half is a mirror image of the other. For example, the letter "A" exhibits reflection symmetry when it is divided vertically down the middle.
Now, moving on to the second part of the question, which involves finding values in the Fibonacci sequence:
1. Fib(8) = 13
2. Fib(19) = 4181
3. Fib(23) = 28,657
1. To find Fib(8), we can use the Fibonacci sequence. Fib(1) = 1, Fib(2) = 1, and each subsequent term is the sum of the previous two terms. So, Fib(3) = Fib(1) + Fib(2) = 1 + 1 = 2.
Similarly, Fib(4) = 2 + 1 = 3, Fib(5) = 3 + 2 = 5, Fib(6) = 5 + 3 = 8, and Fib(7) = 8 + 5 = 13. Therefore, Fib(8) is 13.
2. To find Fib(19), we can continue using the Fibonacci sequence. By applying the same pattern of adding the previous two terms, we find that Fib(19) is 4181.
3. Given that Fib(22) = 17,711 and Fib(24) = 46,368, we can determine Fib(23) by subtracting the known terms
. Since Fib(23) = Fib(24) - Fib(22),
we have Fib(23) = 46,368 - 17,711 = 28,657.
In summary:
1. Fib(8) = 13
2. Fib(19) = 4181
3. Fib(23) = 28,657
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Which shows an equivalent decimal, fraction and percent
A-0. 125,1/8,12. 5%
B-0. 20,1/10,20%
C-0. 45,4/5,45%
D-0. 60,2/3,66%
Answers
An equivalent decimal, fraction and percent is C) 0.45, 4/5, 45%.
To understand why this is the correct choice, we need to look at how decimals, fractions, and percentages are related. Decimals are just another way of representing fractions, where the denominator is a power of 10. For example, 0.45 is the same as 45/100 or 9/20.
Percentages, on the other hand, are just a way of representing fractions with a denominator of 100. So, 45% is the same as 45/100 or 9/20.
The other options do not have equivalent representations of decimals, fractions, and percentages.
In conclusion, understanding the relationships between decimals, fractions, and percentages is essential to solving problems like these. The key is to remember that they are just different ways of representing the same value, and with a bit of practice, it becomes second nature to recognize equivalent representations.
Therefore, option C shows an equivalent decimal (0.45), fraction (4/5), and percent (45%).
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The inverse of x→y is:
Ox-y
O~x-y
y x
8~x~y
O~y~x
Answers
The correct relation which is the inverse of relation is,
⇒ y → x
We have to given that,
Relation is defined as,
⇒ x → y
Since we know that,
An inverse relation is, as the name implies, the inverse of a relationship. Let us review what a relation is. A relation is a set of ordered pairs. Consider the two sets A and B.
The set of all ordered pairings of the type (x, y) where x A and y B are represented by A x B is then termed the cartesian product of A and B. A relation is any subset of the cartesian product A x B.
Now, We can write the inverse of relation is,
⇒ x → y
⇒ y → x
Thus, The correct relation which is the inverse of relation is,
⇒ y → x
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Which statement describes when the plans are based on the same number of aerobic exercise sessions?
Each plan utilizes a combination of 2 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 3 aerobic exercise sessions per week.
Answers
The statement that describes when the plans are based on the same number of aerobic exercise sessions is:
Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week; option BWhat is the number of strength-training exercises and aerobic exercises per week?The number of strength-training exercises and aerobic exercises per week is calculated as follows:
Let a be the number of strength-training exercises and b be the number of aerobic exercises per week respectively.
For the beginner plan:
15a + 20b = 90 eqn. (1)
For the advanced plan:
20a + 30b = 130 eqn. (2)
Solving the simultaneous equation by elimination method:
Multiply eqn. (1) by 3 and eqn. (2) by 2
45a + 60b = 270 eqn. (3)
40a + 60b = 260 eqn. (4)
Subtract eqn. (4) from eqn. (3)
5a = 10
a = 2
Substitute a = 2 in eqn (2)
b = 3
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Complete question:
A personal trainer designs exercise plans based on a combination of strength-training and aerobic exercise. A beginner plan has 15 minutes per session of strength training and 20 minutes per session of aerobic exercise for a total of 90 minutes of exercise in a week. An advanced plan has 20 minutes per session of strength training and 30 minutes of aerobic exercise for a total of 130 minutes of exercise in a week.
Which statement describes when the plans are based on the same number of aerobic exercise sessions?
Each plan utilizes a combination of 2 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 3 aerobic exercise sessions per week.
question 3.. will give brainliest
Answers
Answer:
-0.5
Step-by-step explanation:
Answer:
-0.50
Step-by-step explanation:
4.75 + (-5.25)
Using a number line you can show the sum of points 4.75 and -5.25 is -0.50
1) Starting at 4.75 draw an arrow 5.25 units in the negative direction, arriving at -0.5.
Given J \\ m \\ n, find the value of X
Answers
Answer:
x would be 41°
Step-by-step explanation:
find volume of figure down below
Answers
The volume of figure down below is 470.996 cubic in.
The volume of the cone is the product of one-third of the height, pie, and square of the radius that is;
The volume of the cone = 1/3(πr²)(height)
Since we can see that there are two cones combined together to form one figure.
So, Volume of first cone = 1/3(πr²)(height)
= 1/3(π 5²)(7)
= 1/3(π 25)(7)
= 183.166
Volume of second cone = 1/3(πr²)(height)
= 1/3(π 5²)(11)
= 1/3(π 25)(11)
= 287.83
Therefore, the volume of the figure is;
287.83 + 183.166
470.996
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The total cost, C, of renting a van is $65 each day, d, the van is rented plus $.33 for
mile, m, the van is driven. What is the expression for C?
Answers
Answer:
.33m+65=C
Step-by-step explanation:
to solve the equation you will need to multipy .33 x the amount of miles drove after you get the answer for that you will need to add 65 then thats your final answer
A wooden roller is 1cm long and 8cm in diameter find its volume in cm³
Answers
The volume of the wooden roller is approximately equal to 50.27 cm³ (when rounded to two decimal places).
To find the volume of the wooden roller, we can use the formula for the volume of a cylinder:
Volume = π x (radius)^2 x height
First, we need to find the radius of the wooden roller. The diameter is given as 8cm, so the radius is half of that, or 4cm.
Now, we have the following dimensions:
Radius = 4cm
Height = 1cm
Plugging these values into the formula for the volume of a cylinder, we get:
Volume = π x (4cm)^2 x 1cm
= 16π cm^3
Therefore, the volume of the wooden roller is approximately equal to 50.27 cm³ (when rounded to two decimal places).
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if the number of times you take the test were independent of the chance you fail what could that mean?
Answers
if the number of times you take the test were independent variable of the chance you fail what could that mean the difficulty of the test is consistent and unchanging.
The difficulty of the test is consistent and unchanging, making it so that the chance of failing is solely determined by the individual's performance on the test. Factors such as knowledge of the subject and the ability to focus can play a role in a person's success, but the actual chance of failing is not affected by how many times the test is taken. This means that those who fail the test will have to work harder and prepare better in order to pass it on the next attempt. Taking the test multiple times does not guarantee a higher chance of success, as the difficulty remains the same each time due to independent variable.
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Please answer! 50 points
Answers
Answer:
my best choice would be C because it seems more accurate to me.
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
Shana bought sodas and popcorn for the movies. Sodas cost $3 each and popcorn costs $4 per bag. Shana bought 7 things from the concession, all either sodas or bags of popcorn. Shana spent a total of $26. Write a system of equations involving the number of sodas, s, and the bags of popcorn, b. Solve the system to see how many of each Shana bought.
Answers
Answer: $18s & $8p
Explanation: 3s * 6 = 18s
4p * 2 = 8p
18s + 8p = 26sp
After p practice sessions, a subject could perform a task in T(p)=36(p+1)-1/3 minutes for 0≤p≤10. Find T′ (7) and interpret your answer.
Answers
The value of T'(7) obtained after taking the first differential of the function is 36.
Given the T(p) = 36(p + 1) - 1/3
Diffentiate with respect to p
T'(p) = d/dp [36(p + 1) - 1/3]
= 36 × d/dp (p + 1) - d/dp (1/3)
= 36 × 1 - 0
= 36
This means that after 7 practice sessions, the rate of change of the time it takes to perform the task with respect to the number of practice sessions is 36 minutes per practice session.
Therefore, T'(p) = 36.
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a pizza parlor offers five sizes of pizza and 14 different toppings. a customer may choose any number of toppings (or no topping at all). how many different pizzas does this parlor offer?
Answers
Therefore, there are 81,920 different pizzas that this parlor offers.
Since there are five different sizes of pizza, a customer can choose any one of the five sizes. For each size, the customer can choose to have any combination of the 14 toppings, or no toppings at all. This means that for each size of pizza, there are $2^{14}$ different possible topping combinations, including the option of having no toppings. So the total number of different pizzas that the parlor offers is:
=5*2¹⁴
=5*16,384
=81,920
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a solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. the total volume of the solid is 10 cubic centimeters. find the radius r (in cm) and height h (in cm) of the cylinder that produces the minimum surface area.
Answers
The radius r that produces the minimum surface area is 1.06 cm and the height h is 1.41 cm.
Let r be the radius of the cylinder and h be the height. You can find the equation for the volume of a solid by adding the volumes of the two hemispheres and the cylinder.
Volume = \((2/3)πr^3 + πr^2h\) = 10 cubic centimeters
Now let's find the values of r and h that minimize the surface area of the solid. This area consists of three parts:
Curved surfaces of two hemispheres and sides of a cylinder. This can be expressed as:
surface area = \(2πr^2 + 2πrh\)
To find the values of r and h that minimize this expression, we can use the Lagrangian multiplier method. Given that the volume of the solid is 10 cubic centimeters, we want to minimize the surface area. So it looks like this:
\(f(r, h) = 2πr^2 + 2πrh + λ[(2/3)πr^3 + πr^2h - 10]\)
Taking the partial derivatives for r, h, and λ and setting them to zero gives
\(4πr + 2πh = 2πr^2λ + (4/3)πr^3λ\)
\(2πr = πr^2λ + πrhλ\)
Solving these equations simultaneously gives:
h = 4r/3
λ = 2/(3r)
Substituting these values into the volume equation gives:
\(r^2h = 5/3\)
Substituting h = 4r/3 from above, we get:
\(r^3 = 15/8pi\)
Taking the cube root of both sides gives:
r=1.06cm
Substituting this value for r into the formula for h yields:
h=1.41cm
Therefore, the radius r that produces the minimum surface area is about 1.06 cm and the height h is about 1.41 cm.
learn more about the Lagrangian multiplier method
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