The spring constant, k = 27 N/m. The system is critically damped. The position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).
(a) Given, m = 1 kg, c = 12 and the spring is stretched 0.5 meters by a force of 13.5 N. We need to find the spring constant k. Let's consider the force applied on the spring, F = -kx
Here, x = 0.5 m and F = 13.5 N. So, 13.5 = -k (0.5) => k = -27 N/m
Since, we cannot have negative spring constant. Therefore, k = 27 N/m.
(b) The given differential equation is x" + 12x' + 27x = 0.
On comparing with the general equation of a damped mass-spring system, we get m = 1 kg, c = 12, k = 27.
The damping factor is defined as β = c/2m = 6.
Using the damping factor we can determine the type of the system. c2 - 4mk > 0 - overdamped systemc2 - 4mk = 0 - critically damped systemc2 - 4mk < 0 - underdamped system In this case, c2 - 4mk = 12^2 - 4(1)(27) = 0
This indicates that the system is critically damped.
(c) The general equation for a damped mass-spring system is x(t) = e-βt(A cos ωdt + B sin ωdt)
Where, A and B are constants that are determined using initial conditions. x(0) = A = 1 and x'(0) = -βA + ωdB/dt = -2 => B = (1/ω) (-βA + 2)
Now, we need to determine ω.ω2 = k/m - β2/4m2 = 27/1 - (12/2)2/4(1)(27) = 3ω = √3
Substitute the value of A, B and ω in the general equation, we get x(t) = e-6t(cos √3t + √3/3 sin √3t)
Hence, the position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).
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Can somebody help me!!??
Answer:
D) 8
Step-by-step explanation:
We're looking for [tex]x[/tex], which means the 7x and 5x will have the SAME x.
7(8) = 56
For the C corner, it is cornered as a 90 degree angle, so that means the (7x) + 34 degrees NEEDS to equal 90.
7(8) = 56 + 34 = 90!!
Now we have to see if 5(8) is correct
5(8) = 40 + 50 [The degree mark] = WHICH EQUALS 90 TOO..
Therefore, the answer is D) 8
[tex](2x5y6)(4x - 3y - 3) [/tex]
for what positive values of k does the function y=sin(kt) satisfy the differential equation y′′ 64y=0?
The function y = sin(kt) satisfies the differential equation y'' - 64y = 0 for pospositiveypospositiveyitiveitive values of k that are multiples of 8.
To determine the values of k for which the function y = sin(kt) satisfies the given differential equation, we need to substitute y into the equation and solve for k. Let's start by finding the first and second derivatives of y with respect to t.
The first derivative of y with respect to t is y' = kcos(kt), and the second derivative is y'' = -k^2sin(kt). Substituting these derivatives into the differential equation gives us:
(-k^2sin(kt)) - 64sin(kt) = 0Simplifying the equation, we get:
sin(kt) = -64*sin(kt)/k^2
We can divide both sides of the equation by sin(kt) (assuming sin(kt) is not zero) to get:
1 = -64/k^2
Solving for k^2, we find k^2 = -64. Since k must be positive, there are no positive values of k that satisfy this equation. Therefore, there are no positive values of k for which the function y = sin(kt) satisfies the given differential equation y'' - 64y = 0.
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PLEASE HELP!!!! 69 points my guy!!!
Answer:
D. f(q) = 2q + 3Step-by-step explanation:
Given equation:
6q = 3s - 9q is independent variable, we need to solve it for s:
6q = 3s - 93s = 6q + 9s = 2q + 3Correct choice is
D. f(q) = 2q + 3Luis's car used 4/5
of a gallon to travel 29 miles. At what rate does the car use gas, in miles per gallon?
Answer:
36.25 miles per gallon
Step-by-step explanation:
4/5 = .8
29/.8 = x/1
cross-multiply:
.8x = 29
x = 36.25
An experiment consists of tossing 3 fair (not weighted) coins, except one of the 3 coins has a head on both sides. Compute the probability of obtaining exactly 3 heads The probability of obtaining exactly 3 heads is
The probability of obtaining exactly 3 heads is 5/16.
We can find the probability of obtaining exactly 3 heads by considering the different ways in which this can happen.
First, suppose we toss the two normal coins and the biased coin with the two heads.
There is a probability of getting heads on each toss of the biased coin: 1/2
A probability of getting heads on each toss of the normal coins: 1/2
Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/2) = 1/8
Now suppose we toss the two normal coins and the biased coin with the two heads, but we choose to use the biased coin twice. In this case, we need to get two heads in a row with the biased coin, and then a head with one of the normal coins.
The probability of getting two heads in a row with the biased coin is: 1/2, and the probability of getting a head with one of the normal coins is: 1/2. Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/2) = 1/8
Finally, suppose we use the biased coin and one of the normal coins twice each. In this case, we need to get two heads in a row with the biased coin, and then two tails in a row with the normal coin. The probability of getting two heads in a row with the biased coin is: 1/2,
and the probability of getting two tails in a row with the normal coin is :
(1/2) * (1/2) = 1/4.
Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/4) = 1/16
Adding up the probabilities from each case, we get:
1/8 + 1/8 + 1/16 = 5/16
Therefore, the probability of obtaining exactly 3 heads is 5/16.
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PARK is a parallelogram. Find the value of x.
Answer: 40
Step-by-step explanation:
Consider the following second order linear ODE y" - 5y + 6y = 0, where y' and y" are first and second order derivatives with respect to x. (a) Write this as a system of two first order ODEs and then write this system in matrix form. (b) Find the eigenvalues and eigenvectors of the system. (e) Write down the general solution to the second order ODE. (a) Using your result from part 3 (or otherwise) find the solution to the following equation. y' - 5y + y = 32
a. System in the matrix form is x' = Ax where A = [tex]\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex] and x = [y, u].
b. The eigenvalues of the system are λ₁ = 5 and λ₂ = 1 and eigenvector are v₁ and v₂ = v₁, and v₁ is any non-zero value.
c. The general solution is equal to y(x) = c₁ × [tex]e^{(5x)[/tex] × [v₁] + c₂× [tex]e^{(x)[/tex]× [v₂].
a. Solution to the equation. y' - 5y + y = 32 is y(x) = c₁ × [tex]e^{(5x)[/tex] + c₂ × [tex]e^{(x)[/tex].
(a) To write the second order linear ODE as a system of two first order ODEs,
Introduce a new variable u = y'.
Then, we have,
u' = y'' - 5y + 6y
= -5y + 6u
Now, write this as a system of two first order ODEs,
y' = u
u' = -5y + 6u
To express this system in matrix form,
Define the vector x = [y, u] and the matrix A = [tex]\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex]
The system can then be written as,
x' = Ax
(b) To find the eigenvalues and eigenvectors of matrix A, solve the characteristic equation,
|A - λI| = 0
where I is the identity matrix.
Substituting the values of A, we have,
[tex]|\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex] [tex]-\lambda\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]|[/tex] = 0
[tex]\left[\begin{array}{ccc}-\lambda&1\\-5&6-\lambda\end{array}\right][/tex] = 0
(-λ)(6-λ) - (-5)(1) = 0
λ²- 6λ + 5 = 0
Factoring the quadratic equation, we get,
(λ - 5)(λ - 1) = 0
So the eigenvalues are λ₁ = 5 and λ₂ = 1.
To find the corresponding eigenvectors,
solve the equation (A - λI)v = 0 for each eigenvalue.
Let us start with λ = 5
(A - 5I)v = 0
[tex]|\left[\begin{array}{ccc}1&1\\-5&6\end{array}\right]|[/tex] v = 0
v₁ + v₂ = 0
-5v₁ + v₂ = 0
From the first equation, we get v₂ = -v₁.
Substituting this into the second equation, we have -5v₁ - v₁ = 0,
which simplifies to -6v₁ = 0.
This implies v₁ = 0, and consequently, v₂ = 0.
So, for λ = 5, the eigenvector is v₁ = 0 and v₂ = 0.
Now, let us find the eigenvector for λ = 1.
(A - I)v = 0
[tex]|\left[\begin{array}{ccc}-1&1\\-5&5\end{array}\right][/tex] v = 0
-v₁ + v₂ = 0
-5v₁ + 5v₂ = 0
From the first equation, we get v₂ = v₁.
Substituting this into the second equation, we have -5v₁ + 5v₁ = 0,
which simplifies to 0 = 0.
This implies that v₁ can be any non-zero value.
So, for λ = 1, the eigenvector is v₁ and v₂ = v₁, where v₁ is any non-zero value.
(e) The general solution to the second order ODE can be expressed using the eigenvalues and eigenvectors as follows,
y(x) = c₁ ×[tex]e^{(\lambda_{1} x)[/tex] × v₁ + c₂ × [tex]e^{(\lambda_{2} x)[/tex]× v₂
Plugging in the values we found earlier, the general solution becomes,
y(x) = c₁ × [tex]e^{(5x)[/tex] × [v₁] + c₂× [tex]e^{(x)[/tex]× [v₂]
where [v₁] and [v₂] are the eigenvectors corresponding to the eigenvalues λ₁ = 5 and λ₂ = 1 respectively.
(a) To find the solution to the equation y' - 5y + y = 32,
Use the general solution obtained above.
Comparing the equation with the standard form y' - 5y + 6y = 0,
The equation corresponds to the case where λ₂ = 1.
Substitute λ = 1, v₁ = 1, and v₂ = 1 into the general solution.
y(x) = c₁ × [tex]e^{(5x)[/tex] × [1] + c₂ × [tex]e^{(x)[/tex] × [1]
Simplifying this expression, we have,
y(x) = c₁ × [tex]e^{(5x)[/tex] + c₂ × [tex]e^{(x)[/tex]
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The radius of a circle is 3 kilometers. What is the circle's area
Answer:
28.27
Step-by-step explanation:
A=πr2=π·32≈28.27433
Answer:
If you're using 3.14 for pi, it's 28.26
Step-by-step explanation:
hey guys could yall solve this problem for me? thanks
Answer:
Given ABCD ~ EFGH
FG = BC(EF/AB)
FG = 7(9/6)
FG = 63/6
FG = 10.5
GH = CD(EF/AB)
GH = 11(9/6)
GH = 99/6
GH = 16.5
EH = AD(EF/AB)
EH = 12(9/6)
EH = 108/6
EH = 18
answer the question true or false. the null distribution is the distribution of the test statistic assuming the null hypothesis is true; it is mound shaped and symmetric about the null mean .
False, the null distribution is the distribution of the test statistic assuming the null hypothesis is true; it is mound-shaped and symmetric about the null mean.
The null distribution is the distribution of the test statistic under the assumption that the null hypothesis is true. However, its shape and symmetry are not necessarily predetermined.
The null distribution can take various forms depending on the specific test and the underlying data. It may or may not be mound shaped or symmetric about the null mean. The shape and characteristics of the null distribution are determined by the specific hypothesis being tested, the sample size, and other factors.
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Is this statement true or false? You calculate a finance charge by subtracting the cost of the purchase from the total payment,
Answer:
True
Step-by-step explanation:
Brainliest?
Answer:
I Believe the answer is True
Step-by-step explanation:
A whale is at the surface of the ocean to breathe. What is the whale's elevation?
A certain statistic bˆ is being used to estimate a population parameter B. The expected value of bˆ is equal to B. What property does bˆ exhibit?
Answer:
Unbiased
Step-by-step explanation:
If b^ is equal to B this means that it is an unbiased estimator. When there is an absence of bias, we have an unbiased estimator. As an unbiased estimator it gives accurate information most of the time. The result it gives is not over estimated and also it is not underestimated.
Expected value = true value
Parameter estimates are correct on average
Thank you
Find the inverse of the following matrix. Write entries as integers or fractions in lowest terms. If the matrix is not invertible, type "N" for all entries. -5-1021 A = -2-5 9 1 2 -4
The inverse of matrix A is given by;
A^-1 = |5/139 -189/139 29/139 |
|-10/139 129/139 -27/139 |
|-5/139 19/139 -3/139 |
The given matrix is A =
| -5 -10 21 |
| -2 -5 9 |
| 1 2 -4 |
To find the inverse of a matrix, first find the determinant of that matrix. The determinant of matrix A is given as;
|A| = -5(-5(-4) - 2(9)) - (-10)(-2(-4) - 1(21)) + (21)(-2(2) - 1(-5))
|A| = -5(10) + 100 - 21(9)
|A| = -50 + 100 - 189
|A| = -139
Thus, the determinant of matrix A is -139. Now, we can use the formula of inverse of a 3x3 matrix;
A^-1 = 1/|A| * |(b22b33 - b23b32) (b13b32 - b12b33) (b12b23 - b13b22)|
| (b23b31 - b21b33) (b11b33 - b13b31) (b13b21 - b11b23)|
| (b21b32 - b22b31) (b12b31 - b11b32) (b11b22 - b12b21)|
where b is the cofactor of each element of matrix A.
The cofactor of element aij is denoted as Aij and given as Aij = (-1)i+j|Mij|.
Thus, the cofactors of matrix A are;
|-5 -10 21|
| -2 -5 9 |
| 1 2 -4 |
M11 = | -5 9 |
| 2 -4 |
M12 = | -2 9 |
| 2 -5 |
M13 = | -2 -5 |
M21 = | -10 21 |
| 2 -9 |
M22 = | -5 -21 |
| -2 5 |
M23 = | -2 -2 |
M31 = | -10 -5 |
| 2 9 |
M32 = | -5 -9 |
| 2 2 |
M33 = | -2 -2 |
Now we can find the inverse of matrix A as follows;
A^-1 = 1/-139 * |(5 189 -29)|
|(-10 -129 27)|
|(-5 19 -3) |
Hence, the inverse of matrix A is given by;
A^-1 = |5/139 -189/139 29/139 |
|-10/139 129/139 -27/139 |
|-5/139 19/139 -3/139 |
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calculate the exact distance between the points (8, -3) and (-2, 4). sophia calculus
The exact distance between the points (8, -3) and (-2, 4) can be calculated using the distance formula in mathematics.
The formula for finding the distance between two points (x1, y1) and (x2, y2) in a two-dimensional Cartesian coordinate system is given by: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). Using the coordinates (8, -3) and (-2, 4), we can substitute the values into the distance formula: Distance = sqrt((-2 - 8)^2 + (4 - (-3))^2) = sqrt((-10)^2 + (7)^2) = sqrt(100 + 49) = sqrt(149) ≈ 12.207
Therefore, the exact distance between the points (8, -3) and (-2, 4) is approximately 12.207 units.
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Question 8 of 9
Carlita has a swimming pool in her backyard that is rectangular with a length of 26 feet and a width of 16
feet. She wants to install a concrete walkway of width c around the pool. Surrounding the walkway, she
wants to have a wood deck that extends w feet on all sides. Find an expression for the perimeter of the wood
deck.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
We have given a rectangular swimming pool with a length of 26 feet and a width of 16 feet. We need to find the perimeter of the wood deck that surrounds the concrete walkway of width c around the pool and extends w feet on all sides.
Let's solve the given problem as follows:Firstly, let's calculate the dimensions of the concrete walkway. Let the width of the concrete walkway be 'c' feet.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
So, the dimensions of the pool and concrete walkway are (26 + 2c) ft. x (16 + 2c) ft.The dimensions of the wood deck that surrounds the concrete walkway by w feet on all sides will be (26 + 2c + 2w) ft. x (16 + 2c + 2w) ft.Now, let's write the expression for the perimeter of the wood deck.P = 2(Length + Width)P = 2[(26 + 2c + 2w) + (16 + 2c + 2w)]P = 2[42 + 4c + 4w]P = 84 + 8c + 8wThe expression for the perimeter of the wood deck is 84 + 8c + 8w. Hence, the answer is 84 + 8c + 8w.
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Find the general solution to the differential equation dy xoay + 3y = x2 dx b) Find the particular solution to the differential equation dy dx = (y + 1)(3x2 – 1) E subject to the condition that y = 0 at x = 0 c) Find the particular solution to the differential equation dy dx = y X- subject to the condition that y = 2 at x = 1
a) The differential equation is y = [tex]e^{(4x)/x}[/tex] + 3 b) The particular solution is y = [tex]e^{x^3 - x}[/tex] - 1 c) The particular solution to the differential equation is given by the equations is y = 2x or y = -2x.
a) To find the general solution to the differential equation:
x(dy/dx) + 3y = [tex](e^{4x})/{x^2}[/tex]
We can start by rearranging the equation:
dy/dx = [[tex](e^{4x})/{x^2}[/tex] - 3y]/x
This equation is linear, so we can use an integrating factor to solve it. The integrating factor is given by:
μ(x) = e^(∫(1/x) dx) = [tex]e^{ln|x|}[/tex] = |x|
Multiplying both sides of the equation by the integrating factor:
|x| * dy/dx - 3|xy| = [tex]e^{(4x)/x}[/tex]
Now, let's integrate both sides with respect to x:
∫(|x| * dy/dx - 3|xy|) dx = ∫([tex]e^{(4x)/x}[/tex]) dx
Using the properties of absolute values and integrating term by term:
∫(|x| * dy) - 3∫(|xy|) dx = ∫([tex]e^{(4x)/x}[/tex]) dx
Integrating each term separately:
∫(|x| * dy) = ∫([tex]e^{(4x)/x}[/tex]) dx + 3∫(|xy|) dx
To integrate ∫(|x| * dy), we need to know the form of y. Let's assume y = y(x). Integrating ∫[tex](e^{4x)/x}[/tex] dx gives us a natural logarithm term.
Integrating 3∫(|xy|) dx can be done using different cases for the absolute value of x.
By solving these integrals and rearranging the equation, you can find the general solution for y(x).
b) To find the particular solution to the differential equation:
dy/dx = (y + 1)(3x² - 1)
subject to the condition that y = 0 at x = 0.
We can solve this equation using separation of variables. Rearranging the equation:
dy/(y + 1) = (3x² - 1) dx
Now, let's integrate both sides:
∫(dy/(y + 1)) = ∫((3x² - 1) dx)
The left-hand side can be integrated using the natural logarithm function:
ln|y + 1| = x³ - x + C1
Solving for y, we have:
[tex]y + 1 = e^{x^3 - x + C1}\\y = e^{x^3 - x + C1} - 1[/tex]
Using the initial condition y = 0 at x = 0, we can find the particular solution. Substituting these values into the equation:
0 = [tex]e^{0 - 0 + C1}[/tex] - 1
1 = [tex]e^{C1}[/tex]
C1 = ln(1) = 0
Therefore, the particular solution is:
y = [tex]e^{x^3 - x}[/tex] - 1
c) To find the particular solution to the differential equation:
x(dy/dx) - y = y
subject to the condition that y = 2 at x = 1.
We can simplify the equation:
x(dy/dx) = 2y
Now, let's separate variables and integrate:
(1/y) dy = (1/x) dx
Integrating both sides:
ln|y| = ln|x| + C2
Simplifying further:
ln|y| = ln|x| + C2
ln|y| - ln|x| = C2
ln(|y/x|) = C2
|y/x| = [tex]e^{C2}[/tex]
Since we are given the initial condition y = 2 at x = 1, we can substitute these values into the equation:
|2/1| = [tex]e^{C2}[/tex]
2 = [tex]e^{C2}[/tex]
C2 = ln(2)
Therefore, the particular solution is:
|y/x| = [tex]e^{ln(2)}[/tex]
|y/x| = 2
Solving for y, we have two cases:
y/x = 2
y = 2x
y/x = -2
y = -2x
So, the particular solution to the differential equation is given by the equations:
y = 2x or y = -2x.
The complete question is:
a) Find the general solution to the differential equation
x dy/dx + 3y = (e⁴ˣ)/(x²)
b) Find the particular solution to the differential equation dy/dx = (y + 1)(3x² - 1)
subject to the condition that v = 0 at x = 0
c) Find the particular solution to the differential equation
x dy/dx (y) = y
subject to the condition that y = 2 at x = 1
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CAN SOMEONE HELP PLS :D will mark brainliest ;)
Answer: the second one/ \/36/6
Step-by-step explanation:
Can someone answer all of them? Tysm!
In a river bank
you can put the answers from the end
1.) 1/12
2.) 2/13
3.) 8/41
4.) 1/4
5.) 11/74
6.) 2/35
7.) 15/58
8.) 5/18
9.) 1/7
10.) 1/13
Which dot plot represents the data in this frequency table?
Number 3 4 5 7 8
Frequency 3 2 4 2 3
Answer:
Im so sorry im late! The answer is A i just took the quiz!
Step-by-step explanation:
The correct dot plot is given in option 1.
What is a dot plot?Any data that may be shown as dots or tiny circles is called a dot plot. Given that the height of the bar created by the dots indicates the numerical value of each variable, it is comparable to a bar graph or a simple histogram. Little amounts of data are shown using dot plots.
As per the given data:
There are 4 options, with each option represented by a diagram also the number and frequency table is given
Number: 3 4 5 7 8
Frequency: 3 2 4 2 3
We can find the correct diagram of the dot plot by observing the number of cross against each value of the number on the line and then matching the obtained value with the given number and frequency table.
Hence, the correct dot plot is given in option 1.
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What does the best fit line estimate for the y value when x is 100
The best fit line estimate for the y value when x is 100 in this case is 205.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
To determine what the best fit line estimate for the y value is when x is 100.
Assuming that you have a linear regression model with the equation y = mx + b, where "m" is the slope and "b" is the y-intercept, you would need to know the values of "m" and "b" to estimate the y value for a given x value.
If you have these values, you can substitute x = 100 into the equation and solve for y.
The resulting value will be the estimated y value for the given x value.
If the equation of the best fit line is y = 2x + 5, then the estimated y value when x is 100 would be:
y = 2(100) + 5
y = 200 + 5
y = 205
Hence, the best fit line estimate for the y value when x is 100 in this case is 205.
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what is 1/12 in simplest form
It cant be written any way else its already in its simplest form
complete the table... plz help
Answer:
3=20
4=15
5=12
Step-by-step explanation:
What is the probability of getting a number greater than or equal to 5 when rolling a number cube numbered 1 to 6?
Answer:
There is a 1/3 (or 0.33%) probability of rolling a number greater than or equal to 5.
Step-by-step explanation:
First, find what numbers are greater than or equal to 5:
5 and 6
Find what options you can get on a number cube:
1, 2, 3, 4, 5, and 6
Out of the 6 possible outcomes, there are only 2 that will get a number greater than or equal to 5. Write this as a fraction:
2/6
Simplify:
1/3
Knowledge and Understanding 14. Simplify (1112 - 6vw - 3wa)-(-702 + vw + 13w). 15. Which of the following is equivalent to the expression (5a + 26 - 4c)? a. 25a2 + 20ab - 40ac +482 - 16bc + 1602 b. 25a2 + 10ab - 20ac + 482 - 86C + 16c2 + c. 25a2 + 482 + 1602 d. 10a + 4b-8c 16. Expand and simplify. (b + b)(4 - 5)(25 - 8) 17. Simplify. P-2 3p + 3 X 9p +9 P + 2 3r2 - 18. Simplify. 63 62 po* + 5m3 - 15r + 12 2m2 + 2r - 40 19. Simplify. xi21 4 X + 2 3 x-1
14. (1112 - 6vw - 3wa)-(-702 + vw + 13w) = 1814 - 7vw - 3wa - 13w
15. The equivalent of the expression (5a + 26 - 4c) is 25a2 + 10ab - 20ac + 482 - 86c + 1602 + c.
16. (b + b)(4 - 5)(25 - 8) = -34
14. Simplify (1112 - 6vw - 3wa)-(-702 + vw + 13w).
Given expression is (1112 - 6vw - 3wa)-(-702 + vw + 13w)
⇒ 1112 - 6vw - 3wa + 702 - vw - 13w
⇒ 1814 - 7vw - 3wa - 13w
15. We are to find the equivalent of the expression (5a + 26 - 4c).
a. 25a2 + 20ab - 40ac +482 - 16bc + 1602
b. 25a2 + 10ab - 20ac + 482 - 86C + 1602
c. 25a2 + 482 + 1602
d. 10a + 4b-8c5a + 26 - 4c
= 5a - 4c + 26 = 25a2 - 20ac +482 - 4c2 + 52 - 8ac
= 25a2 - 20ac + 482 - 4c2 + 10a - 8c = Option (b)
⇒ 25a2 + 10ab - 20ac + 482 - 86c + 16c2 + c.
16. Expand and simplify. (b + b)(4 - 5)(25 - 8)
Given expression is (b + b)(4 - 5)(25 - 8) = 2b(-1)(17) = -34
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determine the inteerval on which the following function is convave upor concave down. identify any inflection points. f(x)= x^4-4x^3 3
To determine the intervals on which the function [tex]f(x) = x^4 - 4x^3 + 3[/tex] is concave up or concave down, we need to find the second derivative of the function and analyze its sign.
First, let's find the first derivative of f(x):
[tex]f'(x) = 4x^3 - 12x^2[/tex]
Next, let's find the second derivative of f(x) by taking the derivative of f'(x):
[tex]f''(x) = 12x^2 - 24x[/tex]
To determine the intervals of concavity, we need to find where f''(x) is positive (concave up) or negative (concave down). We can do this by analyzing the sign of f''(x).
Now, let's find the values of x for which f''(x) = 0, as these will give us potential inflection points:
[tex]12x^2 - 24x = 0[/tex]
12x(x - 2) = 0
x = 0 or x = 2
The potential inflection points are x = 0 and x = 2.
To analyze the concavity of the function, we'll use test points within the intervals between the potential inflection points and beyond.
For x < 0, let's choose x = -1 as a test point:
[tex]f''(-1) = 12(-1)^2 - 24(-1) = 12 + 24 = 36[/tex]
Since f''(-1) = 36 > 0, the function is concave up for x < 0.
For 0 < x < 2, let's choose x = 1 as a test point:
[tex]f''(1) = 12(1)^2 - 24(1) = 12 - 24 = -12[/tex]
Since f''(1) = -12 < 0, the function is concave down for 0 < x < 2.
For x > 2, let's choose x = 3 as a test point:
[tex]f''(3) = 12(3)^2 - 24(3) = 108 - 72 = 36[/tex]
Since f''(3) = 36 > 0, the function is concave up for x > 2.
In summary:
The function is concave up for x < 0 and x > 2.
The function is concave down for 0 < x < 2.
The inflection points are x = 0 and x = 2.
I hope this clarifies the concavity and inflection points of the given function. Let me know if you have any further questions!
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solve the equation 3a - 6 = -12.
Answer:
a=-2
Step-by-step explanation:
3a−6=−12
Step 1: Add 6 to both sides.
3a−6+6=−12+6
3a=−6
Step 2: Divide both sides by 3.
3a/ 3 = −6/ 3
a=−2
Your grandmother always has a jar of cookies on her counter. One day while you are visiting, you eat 5 cookies from the jar. In the equation below, c is the number of cookies remaining in the jar and b is the number of cookies in the jar before your visit.
Answer:
b - 5 = c
Step-by-step explanation:
if you take the number of cookies before the visit (B) and then you eat 5, that would be (B - 5). after you eat the cookies, C is the amount left. so it’s a subtraction problem without 2 numbers. so the equation would be
b - 5 = c
or
before - 5 = after
Consider the triple integral defined below: = f(x, y, z) dv 2y² 9 Find the correct order of integration and associated limits if R is the region defined by 0 ≤ ≤1-20≤x≤2- and 0 ≤ y. Remember that it is always a good idea to sketch the region of integration. You may find it helpful to sketch the slices of R in the zy-, zz- and yz-planes first. Hint: There are multiple correct ways to write dV for this integral. If you are stuck, try dV=dz dzdy s s s f(x, y, z) ddd I=
The correct order of integration and associated limits for the given triple integral I = ∫∫∫[f(x, y, z)] dx dy dz with limits: 0 ≤ x ≤ 1 - 2y, 0 ≤ y, 0 ≤ z ≤ 2y²
The correct order of integration and associated limits for the given triple integral, let's first examine the region of integration R and its slices in different planes.
Region R is defined by 0 ≤ z ≤ 2y² and 0 ≤ x ≤ 1 - 2y.
1.Slices in the zy-plane: In the zy-plane, z is restricted to 0 ≤ z ≤ 2y², and y is unrestricted. Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dz dy dx
2.Slices in the zx-plane: In the zx-plane, z is unrestricted, and x is restricted to 0 ≤ x ≤ 1 - 2y. Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dx dz dy
3.Slices in the yz-plane: In the yz-plane, y is unrestricted, and z is restricted to 0 ≤ z ≤ 2y². Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dy dz dx
Considering the given hint, we can choose any of the above orders of integration as all of them are correct ways to write the integral. However, for simplicity, let's choose the order: I = ∫∫∫ f(x, y, z) dz dy dx.
Now, let's determine the limits of integration for each variable in this order:
∫∫∫ f(x, y, z) dz dy dx = ∫∫ [∫[f(x, y, z) dz] from z=0 to z=2y²] dy dx
The innermost integral with respect to z is evaluated from 0 to 2y². The next integral with respect to y is evaluated from 0 to a certain limit determined by the region R. Finally, the outermost integral with respect to x is evaluated from 0 to 1 - 2y.
Therefore, the order of integration and the associated limits for the triple integral are:
I = ∫∫∫ f(x, y, z) dz dy dx
I = ∫∫ [∫[f(x, y, z) dz] from z=0 to z=2y²] dy dx
I = ∫∫∫[f(x, y, z)] dx dy dz with limits: 0 ≤ x ≤ 1 - 2y, 0 ≤ y, 0 ≤ z ≤ 2y²
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