[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]
Divergence is easier to compute:
[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]
[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]
Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then
[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]
[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]
[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]
To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.
Explanation:In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).
The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.
The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.
Learn more about Vector Field Analysis here:https://brainly.com/question/29828149
#SPJ3
What is a rational number between 9.5 and 9.7
Answer:
9.6 is one such number
Step-by-step explanation:
There are infinitely many.
If we are looking at decimals which I think that might be easier here, you are looking for ones that terminate or repeat (same pattern over and over).
So we are looking for a number between 9.5 and 9.7 that is a decimal that either terminates or repeats.
That number could be 9.6. That one is probably the easier one to see.
There are many more like these possibilities:
1) 9.6
2) 9.5001
3) 9.51
4) 9.54
5) 9.669
Answer:
9.57
Step-by-step explanation:
Because 9.56763865854637984..... (rounded 9.57) It is irrational because it has no pattern
The function h is defined below.
Find all values of x that are NOT in the domain of h.
If there is more than one value, separate them with commas.
h(x) = x^2 - 5x - 24 / x^2 + 6x +9
The answer is:
The only value of "x" that IS NOT in the domain of the function h, is -3.
Why?Since we are working with a quotient (or division), we must remember that the only restriction for this kind of functions are the values that make the denominator equal to 0, so, the domain of the function will include all the values of "x" that are different than the zeroes or roots of the denominator.
We have the function:
[tex]h(x)=\frac{x^2-5x-24}{x^2+6x+9}[/tex]
Where its denominator is :
[tex]x^2+6x+9[/tex]
Now, finding the roots or zeroes of the expression, by factoring, we have:
We need to find two numbers which product is equal to 9 and its addition is equal to 6, that number will be the same for both conditions (3):
[tex]3*3=9\\3+3=6[/tex]
So, the factorized form of the expression will be:
[tex](x+3)^2[/tex]
We have that the expression will be equal to 0 if "x" is equal to "-3", so, the only value of x that IS NOT in the domain of the function h, is -3.
Hence, the domain will be:
Domain: (-∞,-3)U(-3,∞)
Have a nice day!
Final answer:
The only value that is not in the domain of the function h(x) is x = -3, since it makes the denominator of the function equal to zero, thus making the function undefined.
Explanation:
To determine the values that are not in the domain of the function h(x) = x² - 5x - 24}{x² + 6x +9}, we need to identify any values of x that would make the denominator equal to zero, since division by zero is undefined. To do this, we will factor the denominator and find its roots.
First, let's factor the denominator:
x² +6x + 9 = (x + 3)(x + 3)
Now, we set the factored denominator equal to zero to find the values of x that are not in the domain:
(x + 3)(x + 3) = 0
This equation has one distinct root, x = -3.
Therefore, the only value that is not in the domain of h is x = -3. If you substitute x = -3 into the original function, the denominator becomes zero, and the function is undefined.
Prove or disprove:
a) Any positive integer can be written as the sum of the squares of two integers
b) For every integer n, the number 3(n2 + 2n + 3) ? 2n2 is a perfect square
Step-by-step explanation:
a). Let [tex]x^{2}+y^{2}=4[/tex]
[tex]y^{2}=4-x^{2}[/tex]
Now since [tex]y^{2}[/tex] is a square of an integer, its value is [tex]\geq[/tex] 0.
Therefore, [tex]4-x^{2}[/tex] ≥ 0, since x is an integer
where [tex]4-x^{2}[/tex] = 1,2,3,...
and x = 1
But as x = [tex]\sqrt{2}, \sqrt{3}[/tex],... cannot be integer
∴ [tex]y^{2}=4-x^{2}[/tex]
[tex]y^{2}=4-1[/tex]
[tex]y = \sqrt{3}[/tex]
= 1.732 which is not an integer
Thus, any positive integer cannot be written as the sum of the squares of the two integers.
b). Let n be an integer
∴ [tex]3(n^{2}+2n+3)-2n^{2}[/tex]
On solving we get,
[tex]3n^{2}+6n+9-2n^{2}[/tex]
[tex]n^{2}+6n+9[/tex]
[tex]n^{2}+3n+3n+9[/tex]
[tex]n(n+3)+3(n+3)[/tex]
[tex](n+3)(n+3)[/tex]
[tex](n+3)^{2}[/tex]
which is a perfect square
Hence proved.
Answer:
Step-by-step explanation:
a). Let
Now since is a square of an integer, its value is 0.
Therefore, ≥ 0, since x is an integer
where = 1,2,3,...
and x = 1
But as x = ,... cannot be integer
∴
= 1.732 which is not an integer
Thus, any positive integer cannot be written as the sum of the squares of the two integers.
b). Let n be an integer
∴
On solving we get,
which is a perfect square
Hence proved.
In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What's your favorite ice cream flavor? For people who buy ice cream, the all-time favorite is still vanilla. About 26% of ice cream sales are vanilla. Chocolate accounts for only 14% of ice cream sales. Suppose that 172 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.)
Answer:
44.72 Vanilla
24.08 Chocolate
Step-by-step explanation:
26% of 172 choose vanilla
14% of 172 choose chocolate
Find the scalar and vector projections of b onto a. a = 4, 7, −4 b = 3, −1, 1 scalar projection of b onto a vector projection of b onto a
Answer:
[tex]comp_{\vec{a}}\vec{b}=0.11 [/tex]
[tex]proj_{\vec{a}}\vec{b}=\left ( \frac{4}{81},\frac{7}{81},\frac{-4}{81} \right )[/tex]
Step-by-step explanation:
a=(4,7,-4) b=(3,-1,1)
Scalar projection of b onto a
[tex]comp_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|}[/tex]
[tex]a\cdot b=\left ( 4\times 3 \right )+\left ( 7\times -1 \right )+\left ( -4\times 1 \right )=1[/tex]
[tex]|a|=\sqrt{4^2+7^2+4^2}=9[/tex]
[tex]comp_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|}=\frac{1}{9}\\\Rightarrow comp_{\vec{a}}\vec{b}=0.11 [/tex]
Vector projection of b onto a
[tex]proj_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|^2}\cdot a[/tex]
[tex]\frac{a\cdot b}{|a|}=\frac{1}{9}[/tex]
[tex]\frac{a\cdot b}{|a|^2}\cdot a=\frac{1}{81}\left ( {4},{7},{-4} \right )[/tex]
[tex]proj_{\vec{a}}\vec{b}=\left ( \frac{4}{81},\frac{7}{81},\frac{-4}{81} \right )[/tex]
The scalar projection of vector b onto vector a is calculated as 1/9 and the vector projection is calculated as (4/81, 7/81, -4/81).
Explanation:The scalar projection of vector b onto a is calculated as the dot product of b and a divided by the magnitude of a.
So, first we calculate a.b = (4*3) + (7*-1) + (-4*1) = 12 - 7 - 4 = 1.
The magnitude of a is the square root of the sum of squares of its components, √(4^2 + 7^2 + -4^2) = √81= 9.
Therefore, the scalar projection is 1/9.
For vector projection, we multiply the scalar projection by the vector a divided by its magnitude, essentially scaling the a vector by the scalar projection.
This gives us (1/9) * (4/9, 7/9, -4/9) = (4/81, 7/81, -4/81).
Learn more about Projections here:https://brainly.com/question/34921881
#SPJ6
Listed below are the salaries, in $000, for a sample of 15 chief financial officers in the electronics industry. $ 516.0 $ 548.0 $ 566.0 $ 534.0 $ 586.0 $ 529.0 546.0 523.0 538.0 523.0 551.0 552.0 486.0 558.0 574.0 Click here for the Excel Data File Determine the mean, median, and the standard deviation.
Explained the mean, median, and standard deviation for a sample of CFOs' salaries in the electronics industry.
Mean: To find the mean, you add up all the salaries and divide by the number of salaries. For the CFOs, the mean salary is $544,773.
Median: To find the median, you arrange the salaries in order and find the middle value. For the CFOs, the median salary is $546,000.
Standard Deviation: The standard deviation measures the dispersion of data. For the CFOs, you can calculate it using the given formula to be approximately $29,534.
Seven friends want to play a game. They must be divided into two teams with three people in each team and one leader. In how many ways can they do it?
Answer:70
Step-by-step explanation:
Given
total of 7 friends need to be divided in two groups with 3 member each and 1 leader
leader can be chosen out of 7 person in [tex]^7C_1[/tex] ways
And 3 person out of remaining 6 persons in [tex]^6C_3[/tex] ways
thus a total of [tex] ^7C_1\times ^6C_3[/tex] ways is possible
If order is not matter then
[tex]\frac{140}{2} [/tex] ways are possible
A player scored 100100 points in a single professional basketball game. He made a total of 6262 baskets, consisting of field goals (worth two points) and foul shots (worth one point). Find the number of field goals and the number of foul shots that the player made.
Answer:
Step-by-step explanation:
I'm supposed to write the following intervals as sets indescriptive form:
a. (t, infinity), t a fixed real number
b. (0, 1/n), n a fixed natural number
---
I think it is:
a. (t, infinity) = {x: t < x < infinity}
b. (0,1/n) = {x: 0 < x < 1/n}
Is this correct?
Also, how do you indicate that t is a fixed realnumber and n a fixed natural number?
Answer:
Is correct.
Step-by-step explanation:
If you don't write the n and t in the left part of the set (before the : ) we assume that are fixed.
(t, infinity) = {[tex]x \in \mathbb{R} : t<x<\infty, t\in \mathbb{R}[/tex]}
(0,1/n) = {[tex]x \in \mathbb{R} : 0<x<\frac{1}{n}, n\in \mathbb{N}[/tex]}
Paul has just purchased a $1.250.000 home and made a $400,000 down payment The balance can be amortized at 3% for 30 years What are the monthly payments? (5 marks) Tay
Answer:
The monthly payment is $3583.63 ( approx )
Step-by-step explanation:
Given,
The total cost of the home = $1,250,000
There is a downpayment of $400,000,
Thus, the present value of the loan, PV = 1250000 - 400,000 = $ 850,000
Annual rate = 3 % = 0.03,
So, the monthly rate, r = [tex]\frac{0.03}{12}=\frac{1}{400}[/tex]
And, time ( in years ) = 30
So, the number of months, n = 12 × 30 = 360
Hence, the monthly payment of the loan,
[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]
[tex]=\frac{850000(\frac{1}{400})}{1-(1+\frac{1}{400})^{-360}}[/tex]
[tex]=3583.6342867[/tex]
[tex]\approx \$3583.63[/tex]
Exihibit a counterexample for the following. Every prime number is odd.
A counter example would be the number 2.
2 is a prime number, because it only has two factors ( itself and 1)
Yet 2 is also an even number, because it can be exactly divided by two.
So the number 2 proves that not every prime number is odd.
Convert 1000100 two to base sixteen
Answer:
(44)₁₆
Step-by-step explanation:
to convert it into hexa decimal we have select four pair of binary digit
(1000100)₂→(?)₁₆
100 0100
to solve this we have to no the decimal conversion of
0100 which is '4'
so,
conversion of (1000100)₂→(44)₁₆
Given the Homogeneous equation x^2ydy+xy^2dx=0, use y=ux, u=y/x and dy=udx+xdu to solve the differential equation. Solve for y.
Answer:
[tex] y=\frac{C}{x}[/tex].
Step-by-step explanation:
Given homogeneous equation
[tex] x^2ydy+xy^2dx=0[/tex]
[tex]\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{xy^2}{x^2y}[/tex]
Substitute y=ux , [tex]u=\frac{y}{x}[/tex]
[tex] \frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{y}{x}[/tex]
Now,
[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]
[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=-u[/tex]
[tex]\frac{\mathrm{d}u}{\mathrm{d}x}=-2u[/tex]
[tex]\frac{du}{u}=-\frac{dx}{x}[/tex]
Integrating both side we get
lnu=-2lnx+lnC
Where lnC= integration constant
[tex]lnu+ln{x}^2=lnC[/tex]
[tex]lnux^2=lnC[/tex]
Cancel ln on both side
[tex]ux^2=C[/tex]
Substitute [tex]u=\frac{y}{x}[/tex]
Then we get
xy=C
[tex]y=\frac{C}{x}[/tex].
Answer:[tex]y=\frac{C}{x}[/tex].
Write a function with a zero of 1/2. Explain how you know.
Answer:
Simplest function is f(x) = 2x - 1
Step-by-step explanation:
When a function has a zero, simply put, that point is where the graph goes through the x-axis. At this point, the y value is 0.
If a function has a 0 of 1/2, that translates to x = 1/2 and also (1/2, 0). That means that y = 0 when x = 1/2.
If x = 1/2 and is a zero, then x - 1/2 = 0 and 2x - 1 = 0 (notice that y = 0 here and solving for x would get you right back to x = 1/2).
The simplest function with a zero of 1/2 is
f(x) = 2x - 1
You know it is true that the zero is 1/2 for two reasons. First one is to set the right side equal to 0 and solve for x.
The second one is to graph the line y = 2x - 1 and see that it goes through the x axis at 1/2 where y = 0.
a. Assume that a 0.01 significance level will be used to test the claim that p 1less thanp 2. Which is better: A hypothesis test or a confidence interval? A hypothesis test is better. b. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; P-value method; critical value method? ▼ are equivalent, in that they will always lead to the same conclusion. Both of these methods use a standard deviation based on ▼ estimated values of the population proportions, the assumption that the two population proportions are equal, whereas the other method uses a standard deviation based on ▼ estimated values of the population proportions. the assumption that the two population proportions are equal.
Answer:
a. A hypothesis test is better
b. confidence interval method; and critical value method will lead to the same conclusion
Step-by-step explanation:
a. A hypothesis test is better
b. confidence interval method; and critical value method will lead to the same conclusion
when samples are taken from a population, the mean values of these samples are approximately normally distributed,that is, the mean values forming the sampling distribution of means is approximately normally distributed. It is also true that if the standard deviations of each of the samples is found, then the standard deviations of all the samples are approximately normally distributed, that is, the standard deviations of the sampling distribution of standard deviations are approximately normally distributed.
Parameters such as the mean or the standard deviation of a sampling distribution are called sampling statistics, S.
Final answer:
When comparing two population proportions, a hypothesis test is favored at a 0.01 significance level. The confidence interval and P-value methods are equivalent when assuming equal population proportions.
Explanation:
Hypothesis Test vs. Confidence Interval:
When dealing with inferences for two population proportions, a hypothesis test is better to test the claim that one proportion is less than the other at a 0.01 significance level.
Equivalence of Methods in Statistic Testing:
The confidence interval method and the P-value method are equivalent in that they both rely on assumed equal population proportions to draw conclusions.
Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x-values at which they occur. f left parenthesis x right parenthesisequalsx squared minus 6 x minus 9; left bracket 0 comma 7 right bracket
Answer:
The absolute minimum value of the function over the interval [0,7] is -18.
Step-by-step explanation:
The given function is
[tex]f(x)=x^2-6x-9[/tex]
Differentiate f(x) with respect to x.
[tex]f'(x)=2x-6[/tex]
Equate f'(x)=0 to find the critical points.
[tex]2x-6=0[/tex]
[tex]2x=6[/tex]
[tex]x=3[/tex]
The critical point is x=3.
Differentiate f'(x) with respect to x.
[tex]f''(x)=2[/tex]
Since f''(x)>0 for all values of x, therefore the critical point is the point of minima and the function has no absolute maximum value.
3 ∈ [0,7]
Substitute x=3 in the given function to find the absolute minimum value.
[tex]f(3)=(3)^2-6(3)-9[/tex]
[tex]f(3)=9-18-9[/tex]
[tex]f(3)=-18[/tex]
Therefore the absolute minimum value of the function over the interval [0,7] is -18.
The maximum value of the function f(x) = x² - 6x - 9 over the interval [0, 7] is 2, which occurs at x = 7, and the minimum value is -9, which occurs at both x = 0 and x = 3.
Explanation:The function given is f(x) = x² - 6x - 9. To find the maximum and minimum values of this function over the interval [0, 7], we first need to find the critical points of the function. These occur where the derivative of the function is zero or undefined. The derivative of f(x) is f'(x) = 2x - 6. Setting this equal to zero and solving for x gives x = 3. Since this value is within our interval, it is a critical point of the function.
Next, we evaluate the function at the endpoints of the interval and at our critical point. We have f(0) = -9, f(3) = -9, and f(7) = 2. This shows that the maximum value of the function over the interval is 2, which occurs at x = 7, and the minimum value is -9, which occurs at both x = 0 and x = 3.
Learn more about Maxima and Minima here:https://brainly.com/question/40354530
#SPJ3
A researcher studied the radioactivity of asbestos. She sampled 81 boards of asbestos, and found a sample mean of 193.2 bips, and a sample standard deviation of 49.5 bips. (a) Obtain the 94% confidence interval for the mean radioactivity. (b) (i) According the interval that you got, is 200 bips a plausible value for the true mean? (ii) What about 210 bips?
Answer:
a) Confidence interval is (182.86,203.54).
b) (i) Yes, 200 bips is a true mean as it lie in the interval.
(ii) No, 210 bips is not a true mean as it doesn't lie in the interval.
Step-by-step explanation:
Given : A researcher studied the radioactivity of asbestos. She sampled 81 boards of asbestos, and found a sample mean of 193.2 bips, and a sample standard deviation of 49.5 bips.
To find : (a) Obtain the 94% confidence interval for the mean radioactivity. (b) (i) According the interval that you got, is 200 bips a plausible value for the true mean? (ii) What about 210 bips?
Solution :
a) The confidence interval formula is given by,
[tex]\bar{x}-z\times \frac{\sigma}{\sqrt{n}} <C.I<\bar{x}+z\times \frac{\sigma}{\sqrt{n}}[/tex]
We have given,
The sample mean [tex]\bar{x}=193.2[/tex] bips
s is the standard deviation [tex]\sigma=49.5[/tex] bips
n is the number of sample n=81
z is the score value, at 94% z=1.88
Substitute all the values in the formula,
[tex]193.2-1.88\times \frac{49.5}{\sqrt{81}} <C.I<193.2+1.88\times \frac{49.5}{\sqrt{81}}[/tex]
[tex]193.2-1.88\times 5.5 <C.I<193.2+1.88\times 5.5[/tex]
[tex]193.2-10.34 <C.I<193.2+10.34[/tex]
[tex]182.86 <C.I<203.54[/tex]
Confidence interval is (182.86,203.54).
b) (i) According the interval [tex]182.86 <C.I<203.54[/tex]
200 bips a plausible value for the true mean as it lies in the interval.
(ii) 210 bips not lie in the confidence interval so it is not a true mean.
To obtain the 94% confidence interval for the mean radioactivity, use the formula: CI = X ± (Z * σ / √n). The 94% confidence interval for the mean radioactivity is (181.05, 205.35). To determine if a value is plausible, check if it falls within the confidence interval.
Explanation:To obtain the 94% confidence interval for the mean radioactivity, we'll use the formula:
CI = X ± (Z * σ / √n)
Where X is the sample mean, Z is the z-score corresponding to the desired confidence level, σ is the sample standard deviation, and n is the sample size.
For a 94% confidence level, the z-score is approximately 1.88. Plugging in the values:
CI = 193.2 ± (1.88 * 49.5 / √81) = 193.2 ± 12.15
The 94% confidence interval for the mean radioactivity is (181.05, 205.35).
(b) (i) To determine if 200 bips is a plausible value, we check if it falls within the confidence interval. Since 200 is within the interval (181.05, 205.35), it is a plausible value. (ii) Similarly, we check if 210 bips falls within the interval. Since 210 is not within the interval, it is not a plausible value.
Learn more about Confidence Interval here:https://brainly.com/question/34700241
#SPJ3
Find the cosine of the angle between the planes −1x+3y+1z=0 and the plane 5x+5y+4z=−4
Answer:
The he cosine of the angle between the planes is [tex]\frac{14}{11\sqrt{6}}[/tex].
Step-by-step explanation:
Using the definition of the dot product:
[tex]\cos\theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}[/tex]
The given planes are
[tex]-1x+3y+1z=0[/tex]
[tex]5x+5y+4z=-4[/tex]
The angle between two normal vectors of the planes is the same as one of
the angles between the planes. We can find a normal vector to each of the
planes by looking at the coefficients of x, y, z.
[tex]\overrightarrow{n_1}=<-1,3,1>[/tex]
[tex]\overrightarrow{n_2}=<5,5,4>[/tex]
[tex]\overrightarrow{n_1}\cdot \overrightarrow{n_2}=(-1)(5)+(3)(5)+(1)(4)=14[/tex]
[tex]|n_1|=\sqrt{(-1)^2+(3)^2+(1)^2}=\sqrt{11}[/tex]
[tex]|n_2|=\sqrt{(5)^2+(5)^2+(4)^2}=\sqrt{66}[/tex]
The cosine of the angle between the planes
[tex]\cos\theta =\frac{\overrightarrow{n_1}\cdot \overrightarrow{n_2}}{|\overrightarrow{n_1}||\overrightarrow{n_2}|}[/tex]
[tex]\cos\theta =\frac{14}{\sqrt{11}\sqrt{66}}[/tex]
[tex]\cos\theta =\frac{14}{11\sqrt{6}}[/tex]
Therefore the cosine of the angle between the planes is [tex]\frac{14}{11\sqrt{6}}[/tex].
To find the cosine of the angle between two planes, calculate the dot product of their normal vectors.
Explanation:To find the cosine of the angle between two planes, we need to determine the normal vectors of the planes and then calculate the dot product of the two normal vectors. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.
Given the planes -x+3y+z=0 and 5x+5y+4z=-4, the normal vectors are (-1,3,1) and (5,5,4) respectively.
Calculating the dot product of the two normal vectors, we get: (-1)(5) + (3)(5) + (1)(4) = 0. Therefore, the cosine of the angle between the planes is 0.
Prove:
For all sets A, B, and C, if A ⊆ B and B ⊆ Cc , then A ∩ C = ∅.
Step-by-step explanation:
Let's take "a" an element from A, a ⊆ A.
As A ⊆ B, a ⊆ A ⊆ B, so a ⊆ B.
Therefore, a ⊆ B ⊆ Cc, a ⊆ Cc.
Let's remember that Cc is exactly the opposite of C. That means that an element is in C or in Cc; it has to be in one of them but not in both.
As a ⊆ Cc, a ⊄ C.
As we can generalize this for every element of A, there is not element of A that is contained in C.
Therefore, the intersection (the elements that are in both A and C) is empty.
he volume of a cone of radius r and height h is one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius r and height h equal one-third the surface area of a cylinder with the same radius and height? If not, find the correct relationship. Exclude the bases of the cone and cylinder.
Answer:
The surface area of a cone of radius r and height h not equal to one-third the surface area of a cylinder with the same radius and height.
Relationship is [tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
Step-by-step explanation:
Given : The volume of a cone of radius r and height h is one-third the volume of a cylinder with the same radius and height.
To find : Does the surface area of a cone of radius r and height h equal one-third the surface area of a cylinder with the same radius and height?
If not, find the correct relationship. Exclude the bases of the cone and cylinder.
Solution :
Radius of cone and cylinder is 'r'.
Height of cone and cylinder is 'h'.
The volume of cone is [tex]V_c=\frac{1}{3}\pi r^2 h[/tex]
The volume of cylinder is [tex]V_C=\pi r^2 h[/tex]
[tex]\frac{V_c}{V_C}=\frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h}[/tex]
[tex]V_c=\frac{1}{3}V_C[/tex]
i.e. volume of cone is one-third of the volume of cylinder.
Now,
Surface area of the cone is [tex]S_c=\pi r\sqrt{(r+h)}[/tex]
Surface area of the cylinder is [tex]S_C=2\pi rh[/tex]
Dividing both the equations,
[tex]\frac{S_c}{S_C}=\frac{\pi r\sqrt{(r+h)}}{2\pi rh}[/tex]
[tex]\frac{S_c}{S_C}=\frac{\sqrt{(r+h)}}{2h}[/tex]
[tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
Which clearly means [tex]S_c\neq \frac{1}{3}S_C[/tex]
i.e. The surface area of a cone of radius r and height h not equal to one-third the surface area of a cylinder with the same radius and height.
The relationship between them is
[tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
How much money should you invest now in order to have $ 60 comma 000$60,000 in 2929 ?years? Assume that the money earns interest at 10.5 %10.5%?, compounded annually. Round to the nearest dollar amount.
Answer:
$3316 should be invested.
Step-by-step explanation:
Since, the amount formula in compound interest is,
[tex]A=P(1+r)^t[/tex]
Where, P is the principal amount,
r is the rate per period,
t is the time in years,
Here,
A = $ 60,000,
r = 10.5% = 0.105
t = 29 years,
By substituting value,
[tex]60000=P(1+0.105)^{29}[/tex]
[tex]P=\frac{60000}{1.105^{29}}=\$3316.23415377\approx \$3316[/tex]
Hence, $ 3316 should be invested.
Find dy/dx and d2y/dx2. x = cos(2t), y = cos(t), 0 < t < π dy dx = Correct: Your answer is correct. d2y dx2 = Correct: Your answer is correct. For which values of t is the curve concave upward? (Enter your answer using interval notation.) Incorrect: Your answer is incorrect.
The first and second derivatives of the given functions can be found using the chain rule of differentiation. The slope of the curve can be determined using these derivatives and to find out where the curve is concave upwards, the second derivative should be greater than zero.
Explanation:To find the first and second derivatives dy/dx and d2y/dx2, we first need to use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outside function multiplied by the derivative of the inside function.
For dy/dx = dy/dt * dt/dx. Since y = cos(t), dy/dt = -sin(t). And since x = cos(2t), dx/dt = -2sin(2t). Thus, dy/dx = [-sin(t)] / [-2sin(2t)].Now, let's find d2y/dx2 which is the derivative of dy/dx with respect to x. So, d2y/dx2 = d/dx (dy/dx). Here, please use the quotient rule and chain rule again for differentiation.
For a curve to be concave upward, the second derivative needs to be greater than zero. So, you need to set your second derivative function greater than zero and solve for t within the given interval 0 < t < π.
Learn more about Derivatives here:https://brainly.com/question/30365299
#SPJ6
The first derivative is [tex]\frac{dy}{dx} =\frac{1}{2cos(t)}[/tex] , and the second derivative is [tex]\frac{d^2y}{dx^2} = \frac{-1}{4cos^2tsin(2t)}[/tex]. The curve is concave upward in the interval (0, π/2).
We begin by finding the derivatives of x and y with respect to t.
Given x = cos(2t), differentiate to get:
[tex]\frac{dx}{dt} = -2sin(2t)[/tex]
Given y = cos(t), differentiate to get:
[tex]\frac{dy}{dt} = -sin(t)[/tex]
Using the chain rule,
[tex]\frac{dy}{dx} =\frac{dy}{dt} \frac{dt}{dx} = -sin(t) \cdot \frac{(-1)}{2sin(2t)} = \frac{1}{2cos(t)}[/tex]
Next, we need to find d²y/dx².
Start by finding the derivative of dy/dx with respect to t:
Given, [tex]\frac{dy}{dx} =\frac{1}{2cos(t)}[/tex] differentiate to get:
[tex]\frac{d(\frac{dy}{dx}) }{dt} = \frac{sin(t)}{2cos^2t}[/tex]
Using the chain rule again:
[tex]\frac{d^2y}{dx^2}= \frac{\frac{d(\frac{dy}{dx})}{dt} }{\frac{dx}{dt}} = \frac{-1}{4cos^2tsin(2t)}[/tex]
To determine where the curve is concave upward, we need d²y/dx² > 0.
Since sin(2t) is periodic, we look for values of t where sin(2t) is positive.
This is true for the interval (0, π/2). Thus, the curve is concave upward in the interval (0, π/2).
Determine the payment to amortize the debt. (Round your answer to the nearest cent.) Monthly payments on $170,000 at 4% for 25 years.
Answer:
The payment would be $ 897.32.
Step-by-step explanation:
Since, the monthly payment of a loan is,
[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]
Where, PV is the present value of the loan,
r is the monthly rate,
n is the total number of months,
Here,
PV = $170,000,
Annual rate = 4 % = 0.04
So, the monthly rate, r = [tex]\frac{0.04}{12}=\frac{1}{300}[/tex] ( 1 year = 12 months )
Time in years = 25,
So, the number of months, n = 12 × 25 = 300
Hence, the monthly payment of the debt would be,
[tex]P=\frac{170000(\frac{1}{300})}{1-(1+\frac{1}{300})^{-300}}[/tex]
[tex]=897.322628506[/tex]
[tex]\approx \$ 897.32[/tex]
The function g is defined by g(x)=2x^2-3.
Find g(2y)
Answer:
g(2y) = 8y^2 -3
Step-by-step explanation:
g(x)=2 x^2 -3
Let x=2y
g(2y) = 2 (2y)^2 -3
= 2 (4y^2) -3
= 8y^2 -3
(1 point) If the joint density function of X and Y is f(x,y)=c(x2−y2)e−2x, with 0≤x<∞ and −x≤y≤x, find each of the following. (a) The conditional probability density of X, given Y=y>0. Conditional density fX|Y(x,y)= 4(x^2-y^2)e^(-2x)/(1-2y^2) (Enter your answer as a function of x, with y as a parameter.) (b) The conditional probability distribution of Y, given X=x. Conditional distribution FY|X(y|x)= 3/4(x^2-y^2) (for −x≤y≤x). (Enter your answer as a function of y, with x as a parameter.)
Before you do anything, you have to find [tex]c[/tex] such that [tex]f_{X,Y}(x,y)[/tex] is a proper joint density function. Doing the math, you'll find that [tex]c=2[/tex].
Now, determine the marginal densities:
[tex]f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_{-x}^x2(x^2-y^2)e^{-2x}\,\mathrm dy[/tex]
[tex]\implies f_X(x)=\dfrac83x^3e^{-2x}[/tex]
[tex]f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\int_0^\infty2(x^2-y^2)e^{-2x}\,\mathrm dx[/tex]
[tex]\implies f_Y(y)=\dfrac12-y^2[/tex]
a. Then the density of [tex]X[/tex] conditioned on [tex]Y=y[/tex] is
[tex]f_{X\mid Y}(x\mid Y=y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}=\dfrac{4(x^2-y^2)e^{-2x}}{1-2y^2}[/tex]
b. The density of [tex]Y[/tex] conditioned on [tex]X=x[/tex] is
[tex]f_{Y\mid X}(y\mid X=x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}=\dfrac{3(x^2-y^2)}{4x^3}[/tex]
and so the distribution of [tex]Y[/tex] conditioned on [tex]X=x[/tex] is
[tex]F_{Y\mid X}(y\mid X=x)=\displaystyle\int_{-\infty}^uf_{Y\mid X}(y\mid X=x)\,\mathrm du[/tex]
[tex]F_{Y\mid X}(y\mid X=x)=\begin{cases}0&\text{for }y<-x\\\frac{2x^3+3x^2y-y^3}{4x^3}&\text{for }-x\le y\le x\\1&\text{for }y>x\end{cases}[/tex]
To find the conditional probability density of X, given Y=y>0, use Bayes' theorem. To find the conditional probability distribution of Y, given X=x, integrate the joint density over the range of y values.
Explanation:To find the conditional probability density of X, given Y=y>0, we need to calculate the conditional density fX|Y(x,y). This can be done using Bayes' theorem. First, find the marginal density of Y by integrating the joint density over the range of y values:
fY(y) = ∫f(x,y) dx = c(e^(-2y) - e^(-2y)/3)
Then, use Bayes' theorem to find the conditional density:
fX|Y(x,y) = f(x,y)/fY(y) = 4(x^2-y^2)e^(-2x)/(1-2y^2)
To find the conditional probability distribution of Y, given X=x, we need to calculate the conditional distribution FY|X(y|x). This can be found by integrating the joint density over the range of y values:
FY|X(y|x) = ∫f(x,y) dy = 3/4(x^2-y^2)
Learn more about Conditional Probability Density here:https://brainly.com/question/32732862
#SPJ11
Two accounts each began with a deposit of exist10,000. Both accounts have rates of 6.5%, but one account compounds interest once a year while the other account compounds interest continuously. Make a table that shows the amount in each account and the interest earned after one year, five, years, ten years, and 20 years. (Round ALL values to the nearest Dollar AND DO NOT include Dollar Signs)
Answer:
Step-by-step explanation:
Given data
principal P = $10000
rate (r) = 6.5%
to find out
amount in account and the interest earned after one year, five, years, ten years, and 20 years
solution
we know the formula for compounds interest continuously i.e.
amount = principal [tex]e^{rt}[/tex] ..............1
and
compounds interest annually i.e.
amount = principal [tex](1+r/100)^{t}[/tex] ..................2
here put value principal rate and time period 1, 5 10 and 20 years
and we get the amount for each time period
than for interest = amount - principal we get interest
as that we calculate all value
i put all value in table here
Final answer:
To calculate the future value of two accounts both starting with $10,000 at a 6.5% interest rate, one compounding annually and the other continuously, we use the formulas for annual and continuous compounding to compare the total amount and interest earned over 1, 5, 10, and 20 years. The continuous compounding results in higher amounts as shown in the provided table.
Explanation:
To compare the growth of two accounts with an initial deposit of $10,000 at an annual interest rate of 6.5%, with one account compounding annually and the other compounding continuously, we can use the following formulas:
For annual compounding: A = P(1 + r/n)^(nt)For continuous compounding: A = Pe^(rt)Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money).
r = the annual interest rate (in decimal form).
n = the number of times that interest is compounded per year.
t = the time in years.
e = the base of the natural logarithm, approximately equal to 2.71828.
To make the calculations for both accounts after 1, 5, 10 and 20 years, we will round all values to the nearest dollar and not include dollar signs as per the student's request.
Annual Compounding
Here, n is 1 since interest compounds once per year.
Continuous Compounding
Using the formula for continuous compounding, we do not need a value for n.
Let's look at the calculations:
TimeAnnual Compounded AmountContinuous Compounded AmountInterest Earned (Annual)Interest Earned (Continuous)1 year$10,695$10,709$695$7095 years$13,612$13,747$3,612$3,74710 years$18,504$18,907$8,504$8,90720 years$34,262$35,861$24,262$25,861
From the table, we can conclude that continuous compounding results in a higher total amount and interest earned over time compared to annual compounding for the same interest rate.
Use the future value formula to find the indicated value.
n=27; i=0.04; PMT=$109; FV=?
FV=
Answer:
The future value = $1780
Step-by-step explanation:
* Lets explain the formula of the future value
- Future Value of an annuity is used to determine the future value of a
stream of equal payments.
- The future value of an annuity formula can also be used to determine
the number of payments, the interest rate, and the amount of the
recurring payments
- The future value formula is [tex]FV=\frac{PMT}{i}(1-\frac{1}{(1+i)^{n}})[/tex]
where:
# FV = Future Value of the annuity
# PMT= Payment amount
# i = Annual interest rate
# n = Number of payments
* Lets solve the problem
- n = 27
- i = 0.04
- PMT = $109
- To find FV lets use the formula above
∵ n = 27 , i = 0.04 , PMT = 109
∴ [tex]FV=\frac{109}{0.04}(1-\frac{1}{(1+0.04)^{27}})[/tex]
∴ [tex]FV=2725(1-\frac{1}{(1.04)^{27}})=1779.9248[/tex]
∴ FV = 1779.92 ≅ 1780
∴ The future value = $1780
The future value is calculated using the formula FV = PMT * [(1+i)^n - 1] / i. With a regular payment of $109, an interest rate of 0.04, and 27 periods, you can compute the future value.
Explanation:The future value (FV) of an annuity can be calculated using the formula: FV = PMT * [(1+i)^n - 1] / i, where PMT is the regular payment, i is the interest rate, and n is the number of periods. In this case, the interest rate (i) is 0.04, the number of periods (n) is 27, and the regular payment (PMT) is $109. Plugging these values into the formula, we get FV = $109 * [(1+0.04)^27 -1] / 0.04. After performing these computations, you can find the future value (FV).
Learn more about Future Value here:https://brainly.com/question/32923864
#SPJ6
A 3 foot wide brick sidewalk is laid around a rectangular swimming pool. The outside edge of the sidewalk measures 30 feet by 40 feet. What is the perimeter, in feet, of the swimming pool?
Answer with Step-by-step explanation:
A 3 foot wide brick sidewalk is laid around a rectangular swimming pool.
The outside edge of the sidewalk measures 30 feet by 40 feet.
Length of swimming pool=(30-3-3) feet
=24 feet
Breath of swimming pool=(40-3-3) feet
= 34 feet
Perimeter of swimming pool=2(24+34) feet
(since perimeter of rectangle=2(l+b) where l is the length and b is the breath of the rectangle)
Perimeter of swimming pool=116 feet
Hence, Perimeter of swimming pool is:
116 feet
To find the perimeter of the swimming pool, subtract twice the width of the 3 feet sidewalk from the total length and width of the area, which is 40 and 30 feet respectively. This gives you the length and width of the pool. Then, add these dimensions together and multiply by 2 to find the perimeter, which is 116 feet.
Explanation:The subject of your question involves the concept of perimeter in math. To find the perimeter of the swimming pool itself, we need to subtract the width of the sidewalk from the total length and width measurements. Since the sidewalk is 3 feet wide and it is built around the pool, we need to account for this on both sides of the length and the width.
First, you subtract twice the width of the sidewalk from both the length and the width of the total area, which includes the pool and the sidewalk. So, the length of the pool would be 40 feet (total length) - 2*3 feet (two widths of the sidewalk) = 40 feet - 6 feet = 34 feet.
Similarly, the width of the pool would be 30 feet (total width) - 2*3 feet (two widths of the sidewalk) = 30 feet - 6 feet = 24 feet.
Then, you add the length and the width together and multiply by 2 to find the perimeter. The perimeter of the swimming pool would therefore be 2 * (34 feet + 24 feet) = 2 * 58 feet = 116 feet.
Learn more about Perimeter here:https://brainly.com/question/30252651
#SPJ3
The following problem refers to strings in A, B, ..., Z.
How many three-letter strings are there that begin with J ?
Answer: There are 676 three-letter strings that begin with J .
Step-by-step explanation:
The number of letters in English Alphabet = 26
If first letter is fixed as J , then the number of ways to make 3 letters strings [repetition is allowed ] is given by :-
[tex]1\times26\times26=676[/tex]
(The first place is occupied by J and the rest of the two place has 26 ways of getting any letter)
Now, there are 676 three-letter strings that begin with J .
Solve the congruence 169x 25 (mod 330)
First solve the congruence [tex]13y\equiv1\pmod{330}[/tex]. Euclid's algorithm shows
330 = 25 * 13 + 5
13 = 2 * 5 + 3
5 = 1 * 3 + 2
3 = 1 * 2 + 1
=> 1 = 127 * 13 - 5 * 330
=> 127 * 13 = 1 mod 330
so that [tex]y=127[/tex] is the inverse of 13 modulo 330. Then in the original congruence, multiplying both sides by 127 twice gives
[tex]127^2\cdot13^2x\equiv127^2\cdot5^2\pmod{330}\implies x\equiv127^2\cdot5^2\equiv403,225\equiv295\pmod{330}[/tex]
Then any integer of the form [tex]x=295+330n[/tex] is a solution to the congruence, where [tex]n[/tex] is any integer.
Final answer:
Using the Chinese Remainder Theorem, the final solution to the congruence is x = 9 + 330k, where k is an integer.
Explanation:
To solve the congruence 169x ≡ 25 (mod 330), we can use the Chinese Remainder Theorem. First, we factor 330 into its prime factors: 330 = 2 × 3 × 5 × 11. Next, we solve the congruences 169x ≡ 25 (mod 2), 169x ≡ 25 (mod 3), 169x ≡ 25 (mod 5), and 169x ≡ 25 (mod 11) separately.
For the congruence 169x ≡ 25 (mod 2), since 169 ≢ 1 (mod 2), we can ignore it. For the congruence 169x ≡ 25 (mod 3), we can rewrite it as x ≡ 1 (mod 3). For the congruence 169x ≡ 25 (mod 5), we can rewrite it as 4x ≡ 0 (mod 5), and since gcd(5, 4) = 1, we can divide both sides by 4 to get x ≡ 0 (mod 5). Lastly, for the congruence 169x ≡ 25 (mod 11), we can rewrite it as 5x ≡ 3 (mod 11), and solve for x which will give us x ≡ 9 (mod 11).
Using the Chinese Remainder Theorem, we can combine these solutions to get x ≡ 9 (mod 330). Therefore, the solution to the congruence is x = 9 + 330k, where k is an integer.