Circular pizza pies at Pat's Pizza's come in 12-inch and 14-inch diameters.
Which is closest to the difference in the areas of the pizza pies? Use 3.14 for pi.
The difference in the areas of the pizza pies will be 40.82 square inches.
What is the area of the circle?It is the close curve of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.
Let d be the diameter of the circle. Then the area of the circle will be
A = (π/4)d² square units
Then the difference in the areas of the pizza pies is given as,
A = (π/4) {14² - 12²}
A = (3.14 / 4) x (196 - 144)
A = 0.785 x 52
A = 40.82 square inches
The difference in the areas of the pizza pies will be 40.82 square inches.
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Please answer first CORRECT answer gets brainiest !!!
Let's call the number of minutes of calls "x".Plan A costs 15$ + 0.10x per month.
Plan B costs 22$ + 0.08x per month.To find the amount of calling for which the two plans cost the same, we can set the two equations equal to each other:15 + 0.10x = 22 + 0.08xSubtract 0.08x from both sides:15 + 0.02x = 22Subtract 15 from both sides:0.02x = 7Divide both sides by 0.02:x = 350So the two plans cost the same for 350 minutes of calling.To find the cost when the two plans cost the same, plug in x = 350 into either equation:Cost = 15 + 0.10x = 15 + 0.10 * 350 = 15 + 35 = 50.
When using fact-finding techniques, asking what is being done is the same as asking what could or should be done.- True- False
When using fact-finding techniques, asking what is being done is the same as asking what could or should be done is a false statement.
Fact-finding is a process that helps in
discovering facts, cues, or other valid details regarding a specific case or a situation. It refers to the gathering of information. It is often part of an initial mission, i.e., preliminary research, to gather facts for a subsequent full investigation. Accurate information can be collected with help of certain methods/ techniques. These specific methods for finding information of the system are termed as fact finding techniques. Interview, Questionnaire, Record View and Observations are the different fact finding techniques used by the analyst. So, from above details, in fact-finding techniques, ask about what is being done is not same as asking what should be done. Thus, it is false statement.
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Sixty percent of a city's population of 84,000 live
within 5 miles of the library. How many people live
within 5 miles of the library? Solve the problem
with either a math drawing or a percent table, or
both, explaining your reasoning.
Multiply 84,000 by 0.6 thus the answer is 50,400
Douglas will paint two walls in his living room. One wall is 15 3/4 feet long, and the other wall is 13 1/2 feet long. Each wall has a height of 10 feet. What is the area Douglas will paint in square feet?
Step-by-step explanation:
this is a college question ? oh, my ...
the area of a rectangle is
length × width
the 2 walls are 2 rectangles.
15 3/4 × 10
13 1/2 × 10
to be sure, let's convert both mixed numbers to full fractions :
15 3/4 = (15×4 + 3)/4 = 63/4
13 1/2 = (13×2 + 1)/2 = 27/2
so, the areas are
63/4 × 10 = 630/4 = 315/2 ft²
27/2 × 10 = 270/2 ft²
in total that is
315/2 + 270/2 = 585/2 = (584/2 + 1)/2 = 272 1/2 ft²
In the xy plane what is the y intercept of the graph of the equation y=6
The table shows the area of a square for specific side lengths.
side length (inches) 0.5 1 2
area (square inches)
0.25 1 4
3
*Be sure to explain how you know.*
9
The area A of a circle with radius r is given by the equation A = .1².
Is the area of a square with side length 2 inches greater than or less than the area of a
circle with radius 1.2 inches?
The area of a square with side length 2 inches greater than the area of a circle with radius 1.2 inches.
What is Area?Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. The area of a plane figure is the area that its perimeter encloses. The quantity of unit squares that cover a closed figure's surface is its area.
Given:
Area of Square = side x side
we have side = 2 inches
So, Area of Square = side x side
= 2 x 2
= 4 inch²
and, Area of Circle = πr²
We have radius = 1.2 inch
So, Area of Circle = 3.14 x (1.2)²
= 4.5216 inch²
So, Area of Square is greater than Area of Circle.
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There are 24 students in the class 3/6
of the students are boys how many students are boys
Answer:
12
Step-by-step explanation:3/6 is 1/2 when u simplify it. Then u divide 24 in half and get 12
Luke has a situation on his hands. A pipe in his house froze and cracked from cold weather. Now he has a water leak. The leak has been releasing 2/3 of a liter of water every 2 hours. If w represents the amount of water lost and h represents the number of hours, which equation represents this proportional relationship?
A. w=3h
B. w=1/3h
C. h=1/3w
D. 2/3w=2h
Answer: The equation that represents this proportional relationship is D: 2/3w = 2h.
To find the average rate of change of a function on a given interval, we can calculate the difference between the values of the function at the endpoints of the interval, and divide it by the difference between the values of the independent variable (in this case x) at the endpoints of the interval.
For the function y = 1/x on the interval [1, 3], we can use this method to find the average rate of change:
y(3) - y(1) = 1/3 - 1 = -2/3
x(3) - x(1) = 3 - 1 = 2
Average rate of change = (y(3) - y(1)) / (x(3) - x(1)) = (-2/3) / 2 = -1/3
So, the average rate of change of the function y = 1/x on the interval [1, 3] is -1/3.
Step-by-step explanation:
in the form, y=a(1+r)^t PLEASEEEEEE
Answer:
y
a= ——
(r + 1 )t
Step-by-step explanation:
divide both sides
The evolution of a population with constant migration rate M is described by the initial value problem dP/dt = kP + M. P(0) = Po. (a) Solve this initial value problem; assume k is constant. (b) Examine the solution P(t) and determine the relation between the constants k and M that will result in P() remaining constant in time and equal to Po- Explain, on physical grounds, why the two constants k and M must have opposite signs to achieve this constant equilibrium solution for P(t).
(a) Solving initial value problem, P(t) = (1/k) [(kPo + M)e^(kt) - M]
(b)To keep P(t) constant and equal to Po, k must be negative and M must be positive.
(a) To solve the initial value problem:
dP/dt = kP + M, P(0) = Po
We can rewrite the differential equation as:
dP/(kP + M) = dt
Integrating both sides, we get:
(1/k) ln|kP + M| = t + C
where C is a constant of integration. Solving for P, we get:
P(t) = (1/k) (e^(k(t+C)) - M/k)
To find the value of C, we use the initial condition P(0) = Po:
Po = (1/k) (e^(kC) - M/k)
Solving for C, we get:
C = (1/k) ln(kPo + M) - (1/k) ln|kPo|
Substituting this back into the expression for P(t), we get:
P(t) = (1/k) [(kPo + M)e^(kt) - M]
(b) To determine the conditions under which P(t) remains constant and equal to Po, we set P(t) = Po and solve for k and M:
Po = (1/k) [(kPo + M)e^(kt) - M]
Simplifying and rearranging, we get:
(k - 1)e^(kt) = M/Po
If k = 1, then we get 0 = M/Po, which is impossible unless M = 0 (i.e., there is no migration). Therefore, we assume k ≠ 1.
Taking the natural logarithm of both sides, we get:
kt + ln(k - 1) = ln(M/Po)
Solving for k, we get:
k = [ln(M/Po) - ln(k - 1)]/t
To keep P(t) constant and equal to Po, k must be negative (i.e., P(t) is decreasing) and M must be positive (i.e., there is net migration into the population). Physically, this makes sense because if the migration rate is positive, then more individuals are entering the population than leaving, which will tend to increase the population size. Conversely, if the growth rate k is negative, then the population is decreasing in the absence of migration, so the positive migration rate can counteract this decrease and maintain a constant population size.
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The following is the probability mass function for the number of times a certain computer program will malfunction:
x| 0 1 2 3 4 5 p(x)| 0.05 0.28 0.4 0.16 0.08 0.03 (a) What is the probability that the computer program will malfunction more than 3 times? (b) Compute E(X) = E(X^2)= V(x)=
For the probability mass function given for certain computer program answer of the following questions are:
a. Probability of the computer malfunction more than 3 times is 0.11.
b. Value of E(X) = 2.03 , E(X²) = 3.91 , V(X) = -0.2109
Probability of the computer program that malfunction more than 3 times
= p(4) + p(5)
= 0.08 + 0.03
= 0.11
E(X) = ∑x p(x)
= (0)(0.05) + (1) (0.28) + (2)(0.4)+(3)(0.16) + (4)(0.08) + (5)(0.03)
= 0 + 0.28 + 0.8 + 0.48 + 0.32 + 0.15
= 2.03
E(X²) = ∑x² p(x)
= (0)²(0.05) + (1)² (0.28) + (2)²(0.4)+(3)²(0.16) + (4)²(0.08) + (5)²(0.03)
= 0 + 0.28 + 0.16 + 1.44 + 1.28 + 0.75
=3.91
Var(X)
= V(x)
= E(X²) - [E(X) ]²
= 3.91 - (2.03)²
= 3.91 - 4.1209
= -0.2109
Therefore, the answer of the following as per given data are:
a. Required probability = 0.11
b. E(X) = 2.03 , E(X²) = 3.91 , V(X) = -0.2109
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which expression represents “3 times the quantity 1 plus p”?
1. 31+p
2. 3+1p
3. 3(1+p)
4. (3.1) + p
We can write the given statement as algebraic expression as -
3(1 + p).
What is function? What is expression?A function is a relationship between a dependent {y} and independent variable {x}. An expression is a combination of terms both constants and variables.Given is the mathematical statement as -
“ 3 times the quantity 1 plus p ”.
We can write the algebraic expression as -
3(1 + p)
Therefore, we can write the given statement as expression as -
3(1 + p).
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Find the 20 % trimmed mean of the following data. If necessary, round to one more decimal place than the largest number of decimal places given in the data. Lengths of Longest 3-Point Kick for NCAA Division 1-A Football 30 32 34 40 41 43 43 4445 45 461% 47 521 5153t 56t 56 57 60 Copy Data Answer How to enter your answer TablesKeypad
46.53 is the Trimmed mean after removing, 20% of the trimmed mean from all the given lengths of 3-Point Kick for NCAA Division 1-A Football.
The given data is:
Lengths of 3-point kicks = 30 32 34 40 41 43 43 44 45 45 46 47 52 51 53 56 56 57 60.
Trimmed mean = 20%
The trimmed mean is defined as the statistical measure of central tendency. That means calculating the mean value of the given tendency of observations.
Trimmed observations = 20 x (20/100) = 4
These, trimmed observations denote that 2 observations must be trimmed from both sides when they are arranged in ascending order.
Ascending order of lengths = 30 32 34 40 41 43 43 44 45 45 46 47 51 52 53 56 56 57 60.
The trimmed Mean = (34 40 41 43 43 44 45 45 46 47 51 52 53 56 56)/ 15
Trimmed mean = 698/15 = 46.53
Therefore, we can conclude that 46.53 is the Trimmed Mean after removing, 20% of the trimmed mean.
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4. Relaciona y encierra la respuesta correcta.
a) al, b2, c3, d4
b) a4, b3, c3, d1
c) a2, b3, c1, d4
d) a4, b1, c3, d2
Answer:
[tex]\boxed{\begin{minipage}{7cm}\\b) a4, b3, c3, d1 is the correct answer. why? B is only missing 50 percent and c only has 10 percent and finally only missing 20 percent a d\end{minipage}}\\\\\\\boxed{\begin{minipage}{7cm}\\Therfore, the correct answer is b) a4, b3, c3, d1. \end{minipage}}[/tex]
Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation).
Ordinary things that we encounter every day are electrically neutral. Therefore, since negatively charged electrons are a part of everything, positively charged particles must also exist in all matter.
(James E. Brady and Gerard E. Humiston, General Chemistry)
The argument in the given statement is a deductive argument. The criteria for this interpretation is the form of argumentation, specifically the inferential link between the premises and conclusion.
In a deductive argument, the conclusion logically follows from the premises and, if the premises are true, then the conclusion must be true as well. In other words, the conclusion is necessarily entailed by the premises. In a deductive argument, the conclusion is not just supported by the premises, but it can be derived from them with certainty.
The argument "Ordinary things that we encounter every day are electrically neutral. Therefore, since negatively charged electrons are a part of everything, positively charged particles must also exist in all matter" is an example of a deductive argument because it follows the pattern of deductive reasoning.
The first premise, "Ordinary things that we encounter every day are electrically neutral," provides evidence for the conclusion, "positively charged particles must also exist in all matter." The conclusion is logically entailed by the premises, and if the premises are true, then the conclusion must be true as well.
In this argument, the use of the word "therefore" signals the relationship between the premises and the conclusion. The word "therefore" indicates that the conclusion is a logical consequence of the premises, which supports the interpretation of this argument as a deductive argument.
In conclusion, based on the form of argumentation and the inferential link between the premises and the conclusion, this argument is best interpreted as a deductive argument.
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Find what is multiplied by to get ÷. Then, multiply to solve the problem. Do not reduce your answer. Enter your answers.
The Fraction 9/18 should be multiplied by 1/4 to get the value Equivalent to 9/18 ÷ 2.
What is Fraction?The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.
The number of parts into which the whole has been divided is shown by the denominator. It is positioned in the fraction's lower portion, below the fractional bar.How many sections of the fraction are displayed or chosen is shown in the numerator. It is positioned above the fractional bar in the upper portion of the fraction.We have to solve,
9/18 ÷ 2
As, Fraction does not support the division as usually number does.
So, we have to perform the multiplication by reciprocating the number,
9/18 x 1/2
= 9/ 36
= 1/4
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Which is the better deal?(Proportional Reasoning)
Deal A: % gallons of gas for %15.70 or Deal B. 11 gallons of gas for $42.00.Explain and Show work
Deal A is better if the cost of one gallon of gas under Deal A is less than $3.82.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Deal A offers % off the price of the gas. Let's say the regular price per gallon is $x. Then, the discounted price per gallon would be:
x - 0.5x = 0.5x
So, the cost of one gallon of gas under Deal A would be 0.5x.
We don't know the regular price per gallon, but we do know that the cost of % gallons under Deal A is $15.70. So, we can set up the following proportion:
% / 1 = $15.70 / 0.5x
% = 15.7 / 0.5x
Deal B offers 11 gallons of gas for $42.00. So, the cost of one gallon of gas under Deal B would be:
$42.00 / 11 = $3.82 (rounded to two decimal places)
Therefore, Deal A is better if the cost of one gallon of gas under Deal A is less than $3.82.
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Karina read a total of 20 2/4 pages in her science and social studies books combined. She read 12 3/4 pages in her science book. How many pages did she read in her social studies book?
______ pages
Answer:
10 1/4
Step-by-step explanation:
earrange the following lines of code to produce a method that computes the volume of a balloon with a given width and height Height o Width Note that the volume of an ellipsoid with radi a, b, cis V = nabe return volume) import java.util.Scanner public class Balloon double c = width/2; public static double balloon Volume(double width, double height) * Computer the volume of a balloon para wloth the horizontal dianter of the balloon para height the vertical diameter of the balloon ) 2 ( double = height/2; double volume. 4. Math.PI...cc/3; public public static void main(Stringt] args) Scanner in new Scanner
By rearranging the given lines of code, we have created a method that takes in the width and height of a balloon and calculates its volume using the formula for the volume of an ellipsoid.
To start, we need to import the Scanner class from the Java.util package, as we will be taking input from the user. Then, we define a public class called "Balloon" and initialize a variable "c" to be half the value of the width input.
Next, we create the method "balloonVolume" which takes in two parameters: width and height. We specify that width is the horizontal diameter of the balloon and height is the vertical diameter of the balloon. We then set two variables, "a" and "b" to be half the value of width and height respectively.
Using the formula for the volume of an ellipsoid, which is V = 4/3 * pi * a * b * c, we calculate the volume of the balloon and store it in the variable "volume." Finally, we return the volume.
In the main method, we create a Scanner object and prompt the user to input the width and height of the balloon. We then call the "balloonVolume" method and pass in the user's input as arguments. The method calculates the volume of the balloon and returns it, which we can then print out to the user.
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Factor xy-y+3x-3
Answer needs filled in
(Y+_) (x+_)
I got this answer off of
The probability of a certain flower having eight petals is 0.45. In a randomly selected batch of 240 of these flowers how many would be expected not to have eight petals? i need the answer for high school and i have no idea what the answer could be. ive asked my parents and they don't seem to know
We would expect approximately 132 flowers in the batch to not have eight petals.
What is the probability about?If the probability of a flower having eight petals is 0.45, then the probability of a flower not having eight petals is 1 - 0.45 = 0.55.
So, for we to find the expected number of flowers that do not have eight petals in a randomly selected batch of 240 flowers, we have to multiply the total number of flowers (240) by the probability of a flower not having eight petals (0.55):
Expected number of flowers without eight petals = 240 × 0.55
Expected number of flowers without eight petals = 132
Therefore, we would expect approximately 132 flowers in the batch to not have eight petals.
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Please help me with this question
The measure of angle ∠2 will be 111° by the supplementary property.
What are lines and angles?Lines are straight with little depth or width. Perpendicular lines, intersecting lines, transversal lines, and other types of lines will be covered. An angle is a shape formed by two rays emerging from a common point. In this field, you may also come across alternate and corresponding angles.
Supplementary angles are those that total 180 degrees. Angles 130° and 50°, for example, are supplementary angles because the sum of 130° and 50° equals 180°.
The value of the angle will be calculated as,
∠2 = 180 - 69
∠2 = 111°
Therefore, the value of the ∠2 is 111°.
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Mikhael wanted to rewrite the conversion factor “1 yard ≈ 0.914 meter” to create a conversion factor to convert meters to yards. He wrote “1 meter ≈ .”Tellhow Mikhael should finish his conversion, and explain how you know
1 meter = 1.094 yard create a conversion factor to convert meters to yards.
A meter example is what?Any instrument that detects and may store an electric or magnetic amount, such voltage or current, is referred to as a meter. Examples of meters include an ammeter and a voltmeter. One may refer to using this kind of device as "metering" or one could state that the volume being measured is now being "metered".
The dimensions of a yardThe yard (symbol: yd) is just an English unit of length that is equal to 3 feet or 36 inches according to the British imperial and American customary systems of measurement. According to an international agreement, it has been exactly regulated to 0.9144 metres since 1959. precise 0.9144 meters. One mile is 1,760 yards in length. One mile is 1,760 yards in length.
Given
[tex]1 yard=0.914 meter[/tex]
Create a convert meter to yard expression
[tex]1 yard=0.914 meter[/tex]
Divide both side by 0.914
[tex]\frac{1yard}{0.914} =\frac{0.914 meter}{0.914}[/tex]
[tex]\frac{1 yard}{0.914}= 1meter[/tex]
[tex]1.09409190372=1meter[/tex]
[tex]1meter=1.09409190372 yard[/tex]
[tex]1meter = 1.094 yard[/tex] Approx.
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Kara recently started a selling her baked goods at the local farmer's market. Her earnings at the end of the first day
were $13. At the end of the second day, her earnings were $20.
Which equation would you use to figure out by how much Kara's second day of earnings exceeded her first day of
earnings?
$13+$20=e
O $13-e = $20
O $20+e=$13
O $20-$13=e
k
Answer: 20-13=e
Step-by-step explanation: 20-13 is 7 so e=7 so you plug in 7 for e and which ever one makes sense, is the right answer so 20-13=e
9. IN A GROUP OF 60 STUDENTS, 31 SPEAK FRENCH, 23 SPEAK SPANISH AND 14 SPEAK NEITHER FRENCH NOR SPANISH DETERMINE THE NUMBER STUDENTS WHO SPEAK- (a) BOTH FRENCH AND SPANISH (FRENCH ONLY (SPANISH ONLY (FUS) = ?
The number of students that speak both French and Spanish are 7 students, 24 students speak French only, and 16 students speaks Spanish only.
How to determine the number of students that speaks the different languagesTo solve this problem, we can use the formula for the size of a union of two sets:
|A ∪ B| = |A| + |B| - |A ∩ B|,
where |A| represents the number of elements in set A.
Let F be the set of students who speak French, S be the set of students who speak Spanish, and N be the set of students who speak neither language. Then, we have:
|F| = 31,
|S| = 23,
|N| = 14.
We want to find the number of students who speak both French and Spanish, as well as the number who speak French only and the number who speak Spanish only. Let B be the set of students who speak both French and Spanish, F-only be the set of students who speak French only, and S-only be the set of students who speak Spanish only. Then:
|B| = (unknown),
|F-only| = (unknown),
|S-only| = (unknown).
We know that the total number of students is 60, so:
|F ∪ S ∪ N| = 60.
Using the formula for the size of a union, we get:
|F ∪ S ∪ N| = |F| + |S| + |N| - |F ∩ S| - |(F ∩ N) ∪ (S ∩ N)|.
We can simplify this expression using the fact that |N| = |(F ∩ N) ∪ (S ∩ N)|, since the sets (F ∩ N) and (S ∩ N) are disjoint:
60 = 31 + 23 + 14 - |B| - |N|.
|N| = 14.
Substituting |N| = 14, we get:
|B| = |F ∩ S| - 9.
So, to find |B|, we need to determine |F ∩ S|. We can use the formula for the size of an intersection:
|F ∩ S| = |F| + |S| - |F ∪ S|.
Substituting the known values, we get:
|F ∩ S| = 31 + 23 - |F ∪ S|.
|F ∪ S| = 31 + 23 - |F ∩ S|.
|F ∪ S| = 54 - |B|.
Substituting this into the expression for |F ∪ S ∪ N|, we get:
60 = 31 + 23 + 14 - |B| - 14 - |B|.
2|B| = 54 - 31 - 23 + 14 = 14.
|B| = 7.
So there are 7 students who speak both French and Spanish.
To find |F-only| and |S-only|, we can use the following formulas:
|F-only| = |F| - |B|,
|S-only| = |S| - |B|.
Substituting the known values, we get:
|F-only| = 31 - 7 = 24,
|S-only| = 23 - 7 = 16.
Therefore, there are 7 students who speak both French and Spanish, 24 students who speak French only, and 16 students who speak Spanish only.
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Given P(B) = 7/8 and P(AꓵB) = 3/8 find P(A | B) =
Step-by-step explanation:
P(A | B) = P(AꓵB)/P(B) = 3/8 / 7/8 = 3/8 × 8/7 = 3/7
Find the solution of the system x' = -4y, y' = -6x
where primes indicate derivatives with respect to t, that satisfies the initial condition x(0-2, y(0) =-2. y = ___
x = ___
Based on the general solution from which you obtained your particular solution, complete the following two statements: The critical point (0,0) is A. stable B. asymptotically stable C. unstable and is a A. center B. spiral C. node D. saddle point
Answer: The system of differential equations can be rewritten as a matrix equation:
[x', y']^T = [-4y, -6x]^T
The characteristic equation is obtained by setting the determinant of the matrix [A - λI] equal to 0, where A is the coefficient matrix, λ is an eigenvalue, and I is the identity matrix:
det([A - λI]) = det([-λ -4y; -6x, -λ]) = λ^2 + 24 = 0
Thus, λ = ±2i√6. The general solution to the system is then given by:
x(t) = c1cos(2√6t) + c2sin(2√6t)
y(t) = -2c1sin(2√6t) + 2c2cos(2√6t)
where c1 and c2 are constants determined by the initial conditions. Using the initial condition x(0) = -2, y(0) = -2, we can solve for c1 and c2:
-2 = c1cos(0) + c2sin(0)
-2 = -2c1sin(0) + 2c2cos(0)
c1 = -2
c2 = -2
So the particular solution to the system is:
x(t) = -2cos(2√6t) - 2sin(2√6t)
y(t) = 4sin(2√6t) - 4cos(2√6t)
The critical point (0,0) is a center, as the solution spirals towards the origin as t increases.
Step-by-step explanation:
Let R be a region in quadrant I with centroid (4,3) and area unit square. Find the volume of the solid generated when this region is rotated around y = -5 :) o 20 7 12 b) O 7 o 32 7 36 d) 16
The volume of the solid generated when the region R is rotated around y = -5 is 7π/4. so the correct answer is (b).
To find the volume , we can use the method of cylindrical shells.
Consider a vertical slice of R at x = a, where a is a point on the x-axis. Let h be the height of this slice and r be the distance from the line y = -5 to the point (a, h). Then the volume of the cylindrical shell generated by rotating this slice around the line y = -5 is given by:
V(a) = 2πrh Δa
where Δa is the thickness of the slice. Note that we multiply by 2 because the slice can generate a shell both above and below the line y = -5.
To find the values of h and r in terms of x, we can use the fact that the centroid of R is (4, 3), which means that the equation of the line passing through the centroid and the point (x, h) is given by:
(y - 3) = m(x - 4)
where m is the slope of the line. Since the line passes through the point (x, h), we have:
(h - 3) = m(x - 4)
Solving for h, we get:
h = mx - (4m - 3)
To find the slope m, we can use the fact that R has area 1. Since R lies in quadrant I, we know that m > 0. Then we have:
1 = ∫[0, 4] h dx = ∫[0, 4] (mx - (4m - 3)) dx
Simplifying, we get: 1 = (1/2) m(4^2) - (4m - 3)(4)
Solving for m: m = 1/8
Substituting this into the equation for h, h = (x/8) - 11/8
To find the distance r from the line y = -5 to the point (x, h), we have:
r = h + 5 = (x/8) - 3/8
Now we can integrate V(a) over the interval [0, 4] to get the total volume:
V = ∫[0, 4] V(a) da = ∫[0, 4] 2πr(h + 5) Δa
Substituting for r and h, we get:
V = ∫[0, 4] 2π[(x/8) - 3/8][(x/8) - 11/8 + 5] Δx
Simplifying, V = π/2 ∫[0, 4] (x^2/64) - (27/64)x + 2.25 Δx
Integrating, V = π/2 [(1/192)x^3 - (27/128)x^2 + 2.25x] [0, 4]
V = (π/2) [(1/12) - (27/32) + 9]
V = 7π/4
Therefore, the volume of the solid generated when the region R is rotated around y = -5 is 7π/4. Answer: (b)
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Prove by induction that for every n>=2
(2C2)+(3C2)+...+(NC2) = (N+1 C2)
The statement holds for all positive integers n ≥ 2 by mathematical induction.
How did we arrive at this assertion?We will use mathematical induction to prove that for all integers n ≥ 2,
(2C2) + (3C2) + ... + (nC2) = (n+1)C2.
Base Case:
For n=2, we have:
(2C2) = 1 and (2+1)C2 = 3C2 = 3/2
So the statement holds for the base case n=2.
Inductive Hypothesis:
Assume that the statement is true for some arbitrary positive integer k, i.e.,
(2C2) + (3C2) + ... + (kC2) = (k+1)C2
Inductive Step:
We must prove that the statement is also true for k+1, i.e.,
(2C2) + (3C2) + ... + ((k+1)C2) = (k+2)C2
Expanding the left-hand side of this equation, we have:
(2C2) + (3C2) + ... + ((k+1)C2) = [(2C2) + (3C2) + ... + (kC2)] + (k+1)C2
Using the inductive hypothesis, we know that:
(2C2) + (3C2) + ... + (kC2) = (k+1)C2
So we can substitute this expression to get:
(2C2) + (3C2) + ... + ((k+1)C2) = (k+1)C2 + (k+1)C2
Using the identity (n+1)C2 = nC2 + n + 1, we can simplify the right-hand side of this equation to get:
(2C2) + (3C2) + ... + ((k+1)C2) = 2(k+1)C2
Now, using the identity (n+1)C2 = nC2 + n + 1 again, we can simplify the right-hand side to get:
2(k+1)C2 = (k+1)(k+2)C2
So, we have:
(2C2) + (3C2) + ... + ((k+1)C2) = (k+1)(k+2)C2
This is the same as the right-hand side of the equation we wanted to prove. Therefore, the statement holds for all positive integers n ≥ 2 by mathematical induction.
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