a) The derivative of g(x) at x=2 is g'(2) = 2.
b) The second derivative of k(x) at x=1 is k"(1) = -30.
c) The derivative of n(x) at x=0 is n'(0) = -18.
a) To find g'(x), we need to take the derivative of g(x) with respect to x. Let's differentiate each term separately:
g(x) = 3x³ - 8x² - 2x + 35
The derivative of 3x³ is obtained by applying the power rule, which states that if we have a term of the form [tex]ax^n[/tex], the derivative is given by [tex]nax^{(n-1)[/tex]. In this case, the derivative of 3x³ is 3 * 3x², which simplifies to 9x².
The derivative of -8x² is obtained in a similar manner, resulting in -16x.
The derivative of -2x is -2.
Since 35 is a constant term, its derivative is zero.
Now, let's combine these derivatives to find g'(x):
g'(x) = 9x² - 16x - 2
To find g'(2), we substitute x = 2 into the derivative:
g'(2) = 9(2)² - 16(2) - 2
= 9(4) - 32 - 2
= 36 - 32 - 2
= 2
Therefore, g'(2) = 2.
b) To find k"(x), we need to take the second derivative of k(x) with respect to x. Let's differentiate each term:
k(x) = 1 / [tex]x^5[/tex]
The derivative of 1/[tex]x^5[/tex] can be found using the power rule and the chain rule. The power rule states that the derivative of [tex]x^n[/tex] is n[tex]x^{(n-1)[/tex], and the chain rule applies when we have a function within another function. In this case, the derivative of 1/[tex]x^5[/tex] is -5/[tex]x^6[/tex].
Taking the derivative of -5/[tex]x^6[/tex], we apply the power rule again, resulting in 30/[tex]x^7[/tex].
Now, let's find k"(x) by differentiating -5/[tex]x^6[/tex] again:
k"(x) = -30/[tex]x^7[/tex]
To find k"(1), we substitute x = 1 into the second derivative:
k"(1) = -30/([tex]1^7[/tex])
= -30/1
= -30
Therefore, k"(1) = -30.
c) To find n'(x), we need to take the derivative of n(x) with respect to x. We can apply the product rule to differentiate the two factors of n(x):
n(x) = (-4x + 2)(3x² - 5x + 2)
Using the product rule, the derivative of n(x) is given by:
n'(x) = (-4x + 2)(d/dx)(3x² - 5x + 2) + (3x² - 5x + 2)(d/dx)(-4x + 2)
To differentiate each term, we use the power rule:
(d/dx)(3x² - 5x + 2) = 6x - 5
(d/dx)(-4x + 2) = -4
Substituting these derivatives back into n'(x), we get:
n'(x) = (-4x + 2)(6x - 5) + (3x² - 5x + 2)(-4)
Now, let's find n'(0) by substituting x = 0 into the derivative:
n'(0) = (-4(0) + 2)(6(0) - 5) + (3(0)² - 5(0) + 2)(-4)
= (2)(0 - 5) + (0 - 0 + 2)(-4)
= (2)(-5) + (2)(-4)
= -10 - 8
= -18
Therefore, n'(0) = -18.
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Please answer the following
please explain how you got your answer
Answer:
1/4 or 25%
Step-by-step explanation:
Out of 12 students, 3 liked Hamlet 3/12 is 1/4 simio lidies and 1/4 can be 25%
Answer:
1/4 would be the answer
Step-by-step explanation:
There are 12 total students in the class, and 3 of them liked Hamlet best. So the chances of randomly picking a Shakespeare student that likes Hamlet best is 3/12. If we simplify this answer, we would get 1/4.
Use a scientific calculator to find the Tangent of 180 degrees.
A real estate major collected information on some recent local home sales. The first 6 lines of the database appear in the accompanying table. The columns correspond to the house identification number, the community name, the zip code, the number of acres of the property, the year the house was built, the market value, and the size of the living area (in square feet). Complete parts a and b below.
Yr Built FullMarket Value SFLA 12859 Neighborhood Mail_Zip Acre:s 413400536 Greenfield Manor 412800344 Dublin 412800352 Arcady 12859 12801 12309 10598 10562 12859 0.09 1.69 0.33 2.29 9.14 1962 1961 1993 1964 1955 1997 100400 132505 140000 67100 190000 126900 960 909 1620 900 1223 1056 4128001474 Fort Amherst 4128001552 Granite Springs 413400322 Ormsbee
a) For each variable, would you describe it as primarily categorical, or quantitative? If quantitative, what are the units? If categorical, is it ordinal or simply nominal?
Describe the variable House_ID. Choose the correct answer below.
A. The variable House_ID is categorical and ordinal.
B. The variable House_ID is categorical and nominal.
C. The variable House_ID is an identifier variable.
D. The variable House_ID is quantitative, with units house number.
Describe the variable Neighborhood. Choose the correct answer below.
A. The variable Neighborhood is categorical and ordinal.
B. The variable Neighborhood is quantitative, with units neighborhood name.
C. The variable Neighborhood is categorical and nominal.
D. The variable Neighborhood is quantitative, with units number of neighborhoods
The collected information is:
a) 1. House_ID and 2. Neighborhood: Categorical (Nominal) 3. Mail_Zip: Categorical (Ordinal) 4. Acres, 5. Yr_Built, 6. Full_Market_Value, 7. SFLA: Quantitative (Continuous)
b) House_ID: Categorical and nominal, serves as an identifier variable.
c) Neighborhood: Categorical and nominal, represents different neighborhood categories.
a) For each variable:
1. House_ID: Categorical and nominal. House_ID is a unique identifier for each house, and there is no inherent order to the values.
2. Neighborhood: Categorical and nominal. Neighborhood is a categorical variable that can be divided into different categories, such as "Greenfield Manor", "Dublin", and "Arcady". There is no inherent order to the values.
3. Mail_Zip: Categorical and ordinal. Mail_Zip is a categorical variable that can be divided into different categories, such as "12859", "12801", and "12309". The values are ordered in ascending order, with 12859 being the smallest value and 12309 being the largest value.
4. Acres: Quantitative and continuous. Acres is a quantitative variable that can take on any value between 0 and infinity. The units of measurement are acres.
5. Yr_Built: Quantitative and discrete. Yr_Built is a quantitative variable that can take on any value between 1955 and 1997. The units of measurement are years.
6. Full_Market_Value: Quantitative and continuous. Full_Market_Value is a quantitative variable that can take on any value between $67,100 and $190,000. The units of measurement are dollars.
7. SFLA: Quantitative and continuous. SFLA is a quantitative variable that can take on any value between 900 and 1620 square feet. The units of measurement are square feet.
b) Describe the variable House_ID. Choose the correct answer below.
A. The variable House_ID is categorical and ordinal. Incorrect
B. The variable House_ID is categorical and nominal. Correct
C. The variable House_ID is an identifier variable. Correct
D. The variable House_ID is quantitative, with units house number. Incorrect
c) Describe the variable Neighborhood. Choose the correct answer below.
A. The variable Neighborhood is categorical and ordinal. Incorrect
B. The variable Neighborhood is quantitative, with units neighborhood name. Incorrect
C. The variable Neighborhood is categorical and nominal. Correct
D. The variable Neighborhood is quantitative, with units number of neighborhoods. Incorrect
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Race) The longest racial grouping of respondents to the 2012 GSS was______, with ______%. The second-largest grouping was _____, with ______%.
The longest racial grouping of respondents to the 2012 GSS was non-Hispanic white, with 78.7%. The second-largest grouping was Black or African American, with 15.6%.
The General Social Survey (GSS) is a nationally representative survey of American adults that has been conducted annually since 1972. The GSS collects data on a wide range of topics, including race and ethnicity. In 2012, the GSS asked respondents to identify their race and ethnicity. The results showed that the largest racial grouping in the United States was non-Hispanic white, followed by Black or African American. in the 2012 GSS or any other related information, it is recommended to refer to the official documentation or reports from the General Social Survey (GSS).
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HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
Answer:
The answer is C
Step-by-step explanation:
The mean of math is 84, and the Mean of science is 85.
so, science is clearly one point higher
1. Find the following limits. 2x2 - 8 a) lim * 4x2 + 2 b) lim -0 c) lim 2x +5x+3 2x+3
The limit of the given expression as x approaches infinity is negative infinity. The limit of (4[tex]x^2[/tex] + 2) as x approaches negative infinity is positive infinity. The limit of (2x + 5x + 3) divided by (2x + 3) as x approaches negative infinity is 7/4.
In the first limit, as x approaches infinity, the dominant term in the expression is 2x^2. Since x^2 grows without bound as x becomes larger, the value of 2x^2 will also increase without bound, resulting in a limit of negative infinity.
In the second limit, as x approaches negative infinity, the dominant term in the expression is 4x^2. Similar to the first limit, since x^2 grows without bound as x becomes more negative, the value of 4x^2 will increase without bound, leading to a limit of positive infinity.
In the third limit, as x approaches negative infinity, both the numerator and denominator have the dominant term of 5x. Dividing the numerator and denominator by 5x, we get (2 + 5/x) divided by (2 + 3/x). As x approaches negative infinity, the terms with x in the denominator become negligible, resulting in the limit simplifying to 2/2, which equals 1. Therefore, the limit of (2x + 5x + 3)/(2x + 3) as x approaches negative infinity is 7/4.
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2 8/100 turned into decimal
Answer:
2.08
Step-by-step explanation:
We know 2 is a whole number.
8/100 would be converted to the hundredths place.
0.000
/\
|
This is the hundredths place.
So the decimal is 2.08
---
hope it helps
HELP PLEASE I WILL MARK BRAINST
Answer:
F=45
Step-by-step explanation:
G is 90 degrees and FC angles combined is 90 and divided is 45 each
which of the following functions are solutions of the differential equation y′′−9y′ 18y=0? a. y(x)=e6x b. y(x)=e−x c. y(x)=e3x d. y(x)=0 e. y(x)=6x f. y(x)=3x g. y(x)=ex
Only one of the following functions is a solution of the differential equation y′′−9y′+18y=0.
The second-order homogeneous linear differential equation is given as:y'' - 9y' + 18y = 0This differential equation is a linear homogeneous equation. We will have two roots of the characteristic equation: r1 = 3, r2 = 6So, the general solution to the differential equation is given as:y = c1e3x + c2e6xwhere c1 and c2 are arbitrary constants.a. y(x) = e6x is a solution because it is a part of the general solution of the differential equation.y(x) = e−x, y(x) = 0, y(x) = 6x, y(x) = 3x, y(x) = ex are not solutions because they don't satisfy the differential equation. Hence, the correct options are:a. y(x) = e6xTherefore, only one of the following functions is a solution of the differential equation y′′−9y′+18y=0.
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5
7
9
4
In percents?
Answer:
Convert the decimal to a percentage by multiplying the decimal by
100 .
500%
700%
900%
400%
Consider the function f(x) whose second derivative is f''(x)=9x+5sin(x). If f(0)=3 and f'(0)=2, what is f(3)?
Please show all your steps and explain why.
Evaluating this expression will give us the value of f(3).
To find the value of f(3), we need to integrate the second derivative of f(x) twice and use the given initial conditions to determine the constants of integration.
Step 1: Integrate the second derivative f''(x) with respect to x to find the first derivative f'(x):
∫(f''(x)) dx = ∫(9x + 5sin(x)) dx
f'(x) = (9/2)x^2 - 5cos(x) + C1
Step 2: Use the given initial condition f'(0) = 2 to find the constant C1:
f'(0) = (9/2)(0)^2 - 5cos(0) + C1
2 = 0 - 5 + C1
C1 = 7
Step 3: Integrate f'(x) with respect to x to find the function f(x):
∫(f'(x)) dx = ∫[(9/2)x^2 - 5cos(x) + 7] dx
f(x) = (9/6)x^3 - 5sin(x) + 7x + C2
Step 4: Use the given initial condition f(0) = 3 to find the constant C2:
f(0) = (9/6)(0)^3 - 5sin(0) + 7(0) + C2
3 = 0 - 0 + 0 + C2
C2 = 3
Now we have the function f(x):
f(x) = (9/6)x^3 - 5sin(x) + 7x + 3
To find f(3), substitute x = 3 into the function:
f(3) = (9/6)(3)^3 - 5sin(3) + 7(3) + 3
Therefore, Evaluating this expression will give us the value of f(3).
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Which of the following indicate that the result from a simple linear regression model could be potentially misleading? a. The error terms follow a normal distribution b. The error terms exhibit homoscedasticity c. Then n th error term (e_n) can be predicted with e_n = 0.91 * e_n - 1 d. The dependent and the independent variable show a linear pattern
The correct answer is: c. The n-th error term (e_n) can be predicted with e_n = 0.91 * e_n - 1.
This statement indicates that there is a correlation or relationship between consecutive error terms, where the n-th error term can be predicted based on the previous error term. In a simple linear regression model, the error terms are assumed to be independent and have no correlation with each other.
However, if there is a correlation between the error terms, it violates the assumption of independence, which can lead to biased and unreliable regression results. Therefore, this condition suggests that the result from the simple linear regression model could be potentially misleading.
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Find the area of the figure
Answer:
145
Step-by-step explanation:
Answer:
19 + 13 = 32 x 10 = 320 divided by 2 = 160yd2
Formula of a trapezoid:
B1 (base 1) + B2 (base2) x Height x 1/2 (basically dividing by 2).
Base 1 was 19
Base 2 was 13
and 10 was your height
And then divide by 2. ( x 1/2)
solve graphically this linear system of equations[tex]\left \{ {{x=3} \atop {y+1=0}} \right.[/tex]
Answer:
(3,-1)
Step-by-step explanation:
The given equations are :
x = 3 ..(1)
y+1 = 0 ....(2)
We need to solve the equations graphically. x = 3 means draw a line parallel to y axis.
For y+1=0
y = -1
Draw a line parallel to the negative x axis. The attached figure shows the graph for the given equations. The solution is (3,-1).
A school recieved 20 boxes of different stickers.12 boxes contain floral stickers, 14 boxes were with 3-D stickers. There were 6 boxes of sparkly alphabet stickers. How many of 3-D floral stickers were received? Fill in the chart.
Answer:
14
Step-by-step explanation:
Answer:
14
Step-by-step explanation:
Explanation on how to solve these problems (not answers to the problems).
Answer:
..
Step-by-step explanation:
First, you need to make sure that you know a few things.
1. A line is 180 degrees
2. Angles with the small boxes on them are right angles.
3. Angles that are across from each other are the same amount of degrees...
For example, look at angle 2, you can see that directly across from it is the angle 35 degrees, this makes angle 2 35 degrees as well.
In order to find angle 1, you need to subtract 35 from 180, to see how much is needed to fill the line, in this case it is 145.
So, angle 1 is 145 degrees, and angle two is 35.
One more thing, angles 1 and 7 are the same and so are 2 and 6 because they are on the same line, I can't remember why, but they are.
Let A = {10,20,30}. Find one non-empty relation on set A such that all the given conditions are met and explain why it works: Not Reflexive, Not Transitive, Antisymmetric. (Find one relation on A that satisfies all three at the same time - don't create three different relations).
A non-empty relation on set A which satisfies all the condition is {(10, 20), (20, 30), (30, 10)}.
To find a non-empty relation on set A = {10, 20, 30} that satisfies the given conditions (not reflexive, not transitive, and antisymmetric), we can define the following relation:
R = {(10, 20), (20, 30), (30, 10)}
Explanation:
Not Reflexive: A relation R on set A is reflexive if for every element x in A, (x, x) is in R. In this case, (10, 10), (20, 20), and (30, 30) are not present in the relation R, which makes it not reflexive.
Not Transitive: A relation R on set A is transitive if for every (x, y) and (y, z) in R, (x, z) is also in R. In this case, we have (10, 20) and (20, 30) in R, but (10, 30) is not present. Therefore, the relation R is not transitive.
Antisymmetric: A relation R on set A is antisymmetric if for every (x, y) and (y, x) in R, where x ≠ y, then x and y are not the same element. In this case, we have (10, 20) and (20, 10) in R, but 10 is not equal to 20. Similarly, we have (20, 30) and (30, 20) in R, but 20 is not equal to 30. Therefore, the relation R is antisymmetric.
By defining the relation R as {(10, 20), (20, 30), (30, 10)}, we satisfy all three conditions simultaneously: not reflexive, not transitive, and antisymmetric.
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A circular flower bed is 16m in diameter and has a circular sidewalk around it that is 3 m wide. Find the area of the sidewalk in square meters. Use 3.14 for pi .
Answer:
[tex]178.98\ \text{m}^2[/tex]
Step-by-step explanation:
d = Diameter of flower bed = 16 m
Thickness of sidewalk = 3 m
r = Radius of flower bed = [tex]\dfrac{d}{2}=\dfrac{16}{2}=8\ \text{m}[/tex]
R = Radius of flower bed with sidewalk = [tex]8+3=11\ \text{m}[/tex]
The required area is given by
[tex]A=\pi(R^2-r^2)\\\Rightarrow A=3.14\times (11^2-8^2)\\\Rightarrow A=178.98\ \text{m}^2[/tex]
The radius of the sidewalk is [tex]178.98\ \text{m}^2[/tex].
A group of students stood in a circle to play a game. The circle had a diameter of 22 meters. Which measurement is the closest to the circumference of the circle in meters?
Answer:
The closest measurement is 69 meters.
Step-by-step explanation:
We already know that 22 meters is the diameter and to find the circumference you need to multiply the diameter with π. We don't know π, so we are going with 3.14. When you multiply 22 and 3.14 together, you get 69.08 meters. You round the answer to the nearest meter, so it's 69 meters. So, the closest measurement is 69 meters.
A business student conducts an OLS regression analysis in excel (with usual defaults) with demand for strawberries (in 1000 units) as the dependent and price (in dollars) as an independent variable. The OLS regression line is given by y= 9 − 3x. If the pvalue of the intercept coefficient is 0 and the pvalue of the slope coefficient is 2% & If the standard error of the intercept coefficient is 6 and the standard error of the slope coefficient is 1; the true slope will be ______ to/from the estimated slope and the true intercept will be ________ to/from the estimated intercept.
Group of answer choices
equal, equal
different, equal
equal, different
different, different
The true intercept and slope are both different from the estimated values based on statistical significance.
The p-value of the intercept coefficient and that of the slope coefficients are 0 and 0.02(2%) respectively. This means that they are statistically significant. Thus we can infer that the true intercept and slope is not equal to 0.
The standard error of the intercept coefficient is 6, which means that the true intercept is likely to be within 6 units of the estimated intercept. The standard error of the slope coefficient is 1, which means that the true slope is likely to be within 1 unit of the estimated slope.
Therefore, the true slope and intercept will be different from the estimated slope and intercept.
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Please tell me if the transformation is a reflection or not!
Answer:
2, 3, & 5 are reflections, the others are rotations.
The function h defined by h(t)=(-4.9t + 29.4)(t+2) models the height in meters, of an object t seconds after it is launched. When will the object hit the ground?
Answer:
The time the object will hit the ground is 2 s.
Step-by-step explanation:
Given;
h(t) = (-4.9t + 29.4)(t + 2)
Open the bracket;
h(t) = -4.9t² + 29.4t -9.8t + 58.8
h(t) = -4.9t² + 19.6t + 58.8
When the object hit the ground, the final velocity will be zero;
[tex]v = \frac{dh}{dt} = 0 \\\\\frac{dh}{dt} = -9.8t + 19.6 = 0\\\\9.8t = 19.6\\\\t = \frac{19.6}{9.8} \\\\t = 2 \ s[/tex]
Therefore, the time the object will hit the ground is 2 s.
5x + 2 = x – 10 help
Answer:-3/4
Step-by-step explanation:
5x+2=x-1
5x-x=-1-2
4x=-3
X=-3/4
Josh is starting a round of golf the first hole is 130 yards long he needs advice which club to use for his first shot yes kept careful rack for records and how close is for shots came to the olive from the same distance his records include data from his onto golf clubs a ledge and an eight iron over the past year
Answer:
hello your question is incomplete attached below is the complete question
a) Histogram shows positive skewness and the box plot depicts the fact that the median of the 3-iron club is bigger while the variance of the 5-wood club is more than that of 3-iron club when compared to their distance
b)The Median is a better choice because it is not affected by skewness or outliners
c) Josh should use Club 3 iron. Reason : it has less variance
Step-by-step explanation:
A) Histogram and box plot is attached below
Histogram shows positive skewness and the box plot depicts the fact that the median of the 3-iron club is bigger while the variance of the 5-wood club is more than that of 3-iron club when compared to their distance
B) The Median is a better choice because it is not affected by skewness or outliners
Typical distance when Josh hits with a wedge = 4 , distance when Josh hits with an 8 iron = 10.5
c) Josh should use Club 3 iron. Reason : it has less variance
9-6x+x2 el dos va al cuadrado resuelvan lo porfA
Answer:
6
Step-by-step explanation:
do 9-6 then
*
1. Find the value of the discriminant.
3x2 - 6x + 3 = 0
O 29
0 -18
OO
O 23
0 -18 is the answer to your question
is it possible to have the ordered number pair 3;7 on the graph
Answer:
Yes it is possible. Start at (0,0) which is the orgin and go right on the x-axis 3 spaces, and go up on the y-axis 7 spaces to get to the ordered pair of (3,7).
Step-by-step explanation:
4. Solve the equation using the quadratic formula.
4x^2+3x-10 = 0
A.x= -2, x= 1.25
B.X= -2, x= 2
C.x= -1.25, x= 2
D.x= -1.25, x= 1.25
Answer:
A. x = -2, x = 1.25
Step-by-step explanation:
Use the quadratic formula
x = [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex]
Once in standard form, identify a, b, and c from the original equation and plug them into the quadratic formula.
4x² + 3x - 10 = 0
a = 4
b = 3
c = - 10
x = [tex]\frac{-3+\sqrt{3^{2}-4x4(-10) } }{2x4}[/tex]
Simplify
Evaluate the exponent
x = [tex]\frac{-3+\sqrt{9-4x4(-10)} }{2x4}[/tex]
Multiply the numbers
x = [tex]\frac{-3+\sqrt{9+160} }{2x4}[/tex]
Add the numbers
x = [tex]\frac{-3+\sqrt{169} }{2x4}[/tex]
Evaluate the square root
x = [tex]\frac{-3+13}{2x4}[/tex]
Multiply the numbers
x = [tex]\frac{-3+13}{8}[/tex]
Separate the equations
To solve for the unknown variable, separate into two equations: one with a plus and the other with a minus.
x = [tex]\frac{-3+13}{8}[/tex]
x = [tex]\frac{-3-13}{8}[/tex]
Solve
Rearrange and isolate the variable to find each solution
x = - 2
x = 1.25
Answer:
A. x = -2, x = 1.25
Step-by-step explanation:
Use the sum-product pattern
4x² + 3x - 10 = 0
4x² + 8x - 5x - 10 = 0
Common factor from the two pairs
(4x² + 8x) + (-5x - 10) = 0
4x (x + 2) - 5 (x + 2) = 0
Rewrite in factored form
4x (x + 2) - 5 (x + 2) = 0
(4x - 5)(x + 2) = 0
Create separate equations
(4x - 5)(x + 2) = 0
4x - 5 = 0
x + 2 = 0
Solve
Rearrange and isolate the variable to find each solution
x = 1.25
x = - 2
In your answers below, for the variable A type the word lambda, for y type the word gamma, otherwise treat these as you would any other variable We will solve the heat equation U-60<<6, 20 with boundary/initial conditions u(0, t) = 0, u(6, t) 0, and u(,0) - 0
The solution is the zero function for all values of x and t.
To solve the heat equation u_t = 60u_xx with the given boundary and initial conditions, we can use separation of variables. We assume that u(x, t) can be written as a product of two functions, X(x) and T(t), such that u(x, t) = X(x)T(t).
Substituting this into the heat equation, we have:
X(x)T'(t) = 60X''(x)T(t)
Dividing both sides by 60X(x)T(t), we get:
T'(t)/T(t) = 60X''(x)/X(x)
The left side of the equation only depends on t, while the right side only depends on x. Since both sides are equal to a constant, we can set them equal to -λ², where λ is the constant.
T'(t)/T(t) = -λ²
X''(x)/X(x) = -λ²
Now, let's solve these two equations separately:
T'(t)/T(t) = -λ²
This is a separable ordinary differential equation. Integrating both sides with respect to t, we get:
∫ T'(t)/T(t) dt = ∫ -λ² dt
ln|T(t)| = -λ²t + C₁
Taking the exponential of both sides, we have:
T(t) = e^(-λ²t + C₁)
T(t) = e^(C₁) * e^(-λ²t)
T(t) = A * e^(-λ²t), where A = e^(C₁)
Now, let's solve the second equation:
X''(x)/X(x) = -λ²
This is also a separable ordinary differential equation. Integrating both sides with respect to x, we get:
∫ X''(x)/X(x) dx = ∫ -λ² dx
∫ (X''(x)/X(x)) dx = -λ²x + C₂
Using the fact that X''(x) = d²X(x)/dx², we can rewrite the equation as:
∫ (d²X(x)/dx²)/X(x) dx = -λ²x + C₂
∫ (d²X(x)/dx²) / X(x) dx = ∫ -λ² dx
∫ (1/X(x)) d²X(x)/dx² dx = -λ²x + C₂
Integrating both sides again, we have:
ln|X(x)| = -λ²x + C₂x + C₃
Taking the exponential of both sides, we get:
X(x) = e^(-λ²x + C₂x + C₃)
X(x) = e^(-λ²x) * e^(C₂x) * e^(C₃)
X(x) = B * e^(-λ²x) * e^(C₂x), where B = e^(C₃) * e^(C₂x)
Putting the solutions for T(t) and X(x) together, we have:
u(x, t) = X(x)T(t)
u(x, t) = (B * e^(-λ²x) * e^(C₂x)) * (A * e^(-λ²t))
We can combine the constants A and B into a single constant C:
u(x, t) = C * e^(-λ²x) * e^(C₂x) * e^(-λ²t)
Applying the boundary condition u(0, t) = 0, we have:
u(0, t) = C * e^(-λ²0) * e^(C₂0) * e^(-λ²t) = 0
This implies that C * e^(-λ²t) = 0. Since e^(-λ²t) is never zero, we must have C = 0.
Therefore, the solution to the heat equation with the given boundary and initial conditions is: u(x, t) = 0
To learn more about heat equation
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6) You are shooting the puck along the path
seen in the picture of the air hockey table.
Find the mZxº and the mzmº.
Answer:
m = 58°
x = 64°
Step-by-step explanation:
When a puck touches one side of the air hockey table,
All the angles formed are located at a point on a straight line.
Therefore, sum of angles formed is 180°.
m° + m° + 64° = 180°
2m + 64 = 180
2m = 180 - 64
2m = 116
m = 58°
m∠y = m = 58° [Alternate interior angles]
Similarly, on the other side of the table,
58° + y° + x° = 180°
58° + 58° + x° = 180°
x + 116° = 180°
x = 180 - 116
x = 64°