[tex]f(x,y)=x^4+y^4-4xy+1[/tex]
has critical points wherever the partial derivatives vanish:
[tex]f_x=4x^3-4y=0\implies x^3=y[/tex]
[tex]f_y=4y^3-4x=0\implies y^3=x[/tex]
Then
[tex]x^3=y\implies x^9=x\implies x(x^8-1)=0\implies x=0\text{ or }x=\pm1[/tex]
If [tex]x=0[/tex], then [tex]y=0[/tex]; critical point at (0, 0)If [tex]x=1[/tex], then [tex]y=1[/tex]; critical point at (1, 1)If [tex]x=-1[/tex], then [tex]y=-1[/tex]; critical point at (-1, -1)[tex]f(x,y)[/tex] has Hessian matrix
[tex]H(x,y)=\begin{bmatrix}12x^2&-4\\-4&12y^2\end{bmatrix}[/tex]
with determinant
[tex]\det H(x,y)=144x^2y^2-16[/tex]
At (0, 0), the Hessian determinant is -16, which indicates a saddle point.At (1, 1), the determinant is 128, and [tex]f_{xx}(1,1)=12[/tex], which indicates a local minimum.At (-1, -1), the determinant is again 128, and [tex]f_{xx}(-1,-1)=12[/tex], which indicates another local minimum.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.
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
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.
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
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).
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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.
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If a resistor follows Ohm’s Law, what does the slope represent when you plot V as a function of I (with V on the y-axis)? What about when you plot I as a function of V (with I on the y-axis)?
Answer:
Resistance
Conductance
Step-by-step explanation:
In any graph the slope represents the ratio of y axis to x axis.
So, in the first case the voltage (V) is on the y axis and current (I) on the x axis. Here the slope is V/I = R. Therefore, slope represents resistance.
In the second case current (I) is on the y axis and voltage (V) is on the x axis. Here the slope formed will be I/V = 1/R = G. Therefore, slope represents conductance.
Final answer:
In physics, particularly when discussing Ohm's Law, the slope of a V versus I plot represents the resistance of the resistor, and conversely, the slope of an I versus V plot signifies the conductance. These relationships illustrate the fundamental interplay between current, voltage, and resistance in electrical circuits.
Explanation:
If a resistor follows Ohm’s Law, when plotting V (voltage) as a function of I (current), with V on the y-axis, the slope of the graph represents the resistance (R) of the resistor. This is directly derived from Ohm's Law, V = IR, which shows that voltage (V) is equal to the current (I) multiplied by the resistance (R). The slope in this context is R, because the graph shows how much voltage is needed to achieve a certain current flow, exhibiting a linear relationship. This slope is a constant value for ohmic devices, showcasing that the resistance does not change with varying voltage or current in these cases.
Conversely, when plotting I as a function of V, with I on the y-axis, the slope represents the conductance (the reciprocal of resistance). From the equation I = V/R, we could also interpret that the slope in a graph of I versus V would represent 1/R, indicating how current changes in response to changes in voltage. This also demonstrates a linear relationship for ohmic devices, where the conductance remains constant.
In essence, these plots offer a visual representation of Ohm's Law in action, and the slope on these graphs provides valuable information about the resistor’s resistance or conductance, highlighting the fundamental relationship between voltage, current, and resistance in an electrical circuit.
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.
The slope of a graph representing the number of calories that Chad burns (y-axis) with respect to the number of times he works out (x-axis) is 230. Which point will lie on the graph? A. (19, 4,370) B. (18, 3,960) C. (18, 4,730) D. (19, 3,960)
Answer:
Option A. (19, 4,370)
Step-by-step explanation:
we know that
A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]
In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin
In this problem
The relationship between the number of calories that Chad burns (y-axis) with respect to the number of times he works out (x-axis) represent a proportional variation
so
[tex]y=kx[/tex]
we have that
[tex]k=230[/tex] ----> the constant is equal to the slope
substitute
[tex]y=230x[/tex] ----> linear equation that represent the situation
If a point lie on the graph, the the point must satisfy the linear equation
Verify each case
case A) (19,4,370)
For x=19, y=4,370
substitute in the equation
[tex]4,370=230(19)[/tex]
[tex]4,370=4,370[/tex] ----> is true
therefore
The point lie on the graph
case B) (18,3,960)
For x=18, y=3,960
substitute in the equation
[tex]3,960=230(18)[/tex]
[tex]3,960=4,140[/tex] ----> is not true
therefore
The point does not lie on the graph
case C) (18,4,730)
For x=18, y=4,730
substitute in the equation
[tex]4,730=230(18)[/tex]
[tex]4,730=4,140[/tex] ----> is not true
therefore
The point does not lie on the graph
case D) (19,3,960)
For x=19, y=3,960
substitute in the equation
[tex]3,960=230(19)[/tex]
[tex]3,960=4,370[/tex] ----> is not true
therefore
The point does not lie on the graph
Answer:
It would be A
Step-by-step explanation:
its not the same for everyone but the number is (19, 4,370)
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
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.
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.
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.
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]
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]
Suppose you just received a shipment of eleven televisions. Three of the televisions are defective. If two televisions are randomly selected, compute the probability that both televisions work. What is the probability at least one of the two televisions does not work?
Answer: The probability that both televisions work : 0.5329
The probability at least one of the two televisions does not work : 0.4671
Step-by-step explanation:
Given : The total number of television : 11
The number of defective television : 3
The probability that the television is defective : [tex]p=\dfrac{3}{11}\approx0.27[/tex]
Binomial distribution formula :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(X) is the probability of getting success in x trials, p is the probability of success and n is the total trials.
If two televisions are randomly selected, then the probability that both televisions work:
[tex]P(0)=^2C_0(0.27)^0(1-0.27)^{2-0}=(1)(0.73)^2=0.5329[/tex]
The probability at least one of the two televisions does not work :
[tex]P(X\geq1)=1-P(0)=1-0.5329=0.4671[/tex]
(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)
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Find the value of x.
A) x = 2
B) x = 3
C) x = 33
D) x = 52
Answer:
B) x = 3
Step-by-step explanation:
The given triangle has three angles:
∠R, ∠S and ∠T
The exterior angle theorem says that
∠S+∠T = ∠R
Putting the values of angles
(57+x) + 25x = 45x
57 + x + 25x = 45x
57+26x = 45x
57 = 45x - 26x
57 = 19x
x = 57/19
x = 3
Hence, the correct answer is:
B) x = 3 ..
Answer:
B
Step-by-step explanation:
X = 3
In this problem, y = c1 cos 4x + c2 sin 4x is a two-parameter family of solutions of the second-order DE y'' + 16y = 0.
If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. (If not possible, enter NOT.) y(0) = 0, y(π/8) = 4.
Answer:
y=4 sin(4x)
Step-by-step explanation:
So you are given y(0)=0. This means when x=0, y=0.
So plug this in:
0=c1 cos(4*0)+c2 sin(4*0)
0=c1 cos(0) +c2 sin(0)
0=c1 (1) +c2 (0)
0=c1 +0
0=c1
So our solution looks like this after applying the first boundary condition:
y=c2 sin(4x).
Now we also have y(pi/8)=4. This means when x=pi/8, y=4.
So plug this in:
4=c2 sin(4*pi/8)
4=c2 sin(pi/2)
4=c2 (1)
4=c2
So the solution with the given conditions applies is y=4 sin(4x) .
Testing:
y'=16 cos(4x)
y''=-64 sin(4x).
y''+16y=0
-64 sin(4x)+16(4 sin(4x))
-64 sin(4x)+64 sin(4x)
0
So the solution still works.
The solution of the differential equation y'' + 16 y = 0 is,
y = 4 sin4x
Here,
The second-order differential equation is, y'' + 16y = 0.
And, y = c₁ cos 4x + c₂ sin 4x is a two-parameter family of solutions of the second-order DE y'' + 16y = 0.
We have to find the solution of the differential equation that satisfies the given side conditions, y(0) = 0, y(π/8) = 4.
What is Differential equation?
A differential equation is a mathematical equation that relates some function with its derivatives.
Now,
y = c₁ cos 4x + c₂ sin 4x is a two-parameter family of solutions of the second-order DE y'' + 16y = 0.
Apply given conditions y(0) = 0, y(π/8) = 4 on y = c₁ cos 4x + c₂ sin 4x.
We have given y(0) = 0,
This means x = 0, y = 0
So put this in y = c₁ cos 4x + c₂ sin 4x.
We get,
0 = c₁ cos 4*0 + c₂ sin 4*0
0 = c₁
c₁ = 0
Hence equation become,
y = c₂ sin4x
And, y(π/8) = 4 , this means that x = π/8 , y = 4
So put in equation y = c₂ sin4x we get,
4 = c₂ sinπ/2
c₂ = 4
Hence, the solution become after putting the value of c₁ = 0 and c₂ = 4,
y = c₁ cos 4x + c₂ sin 4x
y= 4 sin4x
So, The solution of the differential equation y'' + 16 y = 0 is,
y = 4 sin4x
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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.
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For the month of January in a certain city, 42% of the days are cloudy. Also in the month of January in the same city, 41% of the days are cloudy and rainy. What is the probability that a randomly selected day in January will be rainy if it is cloudy?
Answer: 0.9762
Step-by-step explanation:
Let A be the event that days are cloudy and B be the event that days are rainy for January month .
Given : The probability that the days are cloudy = [tex]P(A)=0.42[/tex]
The probability that the days are cloudy and rainy = [tex]P(A\cap B)=0.41[/tex]
Now, the conditional probability that a randomly selected day in January will be rainy if it is cloudy is given by :-
[tex]P(B|A)=\dfrac{P(B\cap A)}{P(A)}\\\\\Rightarrow\ P(B|A)=\dfrac{0.41}{0.42}=0.97619047619\approx0.9762[/tex]
Hence, the probability that a randomly selected day in January will be rainy if it is cloudy = 0.9762
the probability that a randomly selected day in January will be rainy if it is cloudy is approximately 97.62%.
The question asks us to find the probability that a day will be rainy given that it is cloudy. From the information provided, we know that in this city, 42% of the days in January are cloudy, and 41% are both cloudy and rainy. To find the probability that it is rainy given that it is cloudy, we use the concept of conditional probability.
The formula for conditional probability is P(A|B) = P(A u2229 B) / P(B), where P(A|B) is the probability of A given B, P(A \\u2229 B) is the probability of both A and B occurring, and P(B) is the probability of B occurring.
Let A be the event that it is rainy and B be the event that it is cloudy. Therefore, P(A|B) = P(A u2229 B) / P(B) = 0.41 / 0.42 = 0.9762 or 97.62%.
Hence, the probability that a randomly selected day in January will be rainy if it is cloudy is approximately 97.62%.
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]}
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.
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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).
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.
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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).
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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.
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:
QUESTION 20 5.0 Calculate the standard deviation for the following set of numbers: 73, 76, 79, 82, 84, 84, 97
Answer:
7.16.
Step-by-step explanation:
The variance is ∑ (x - m)^2 / N and the standard deviation is the square root of this.
m is the mean of the data . Here it is 82.14.
Construct a table:
x (x - m) (x - m)^2
73 -9.14 83.54
76 -6.14 37.70
79 -3.14 9.86
82 -0.14 0.02
84 1.86 3.46
84 1.86 3.46
97 14.86 220.82
Total: 358.86
Variance = 358.86 / 7 = 51.27
Standard deviation = √51.27 = 7.16.
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]