To evaluate the expression x² + y² - 1 in polar coordinates, we need to convert the Cartesian coordinates (x, y) to polar coordinates (r, θ).
In polar coordinates, x = rcos(θ) and y = rsin(θ). Substituting these values into the expression, we obtain r²cos²(θ) + r²sin²(θ) - 1. This expression can be simplified using trigonometric identities to obtain r²(cos²(θ) + sin²(θ)) - 1, which simplifies further to r² - 1.
When converting Cartesian coordinates (x, y) to polar coordinates (r, θ), we use the equations x = rcos(θ) and y = rsin(θ). Substituting these values into the expression x² + y² - 1, we have (rcos(θ))² + (rsin(θ))² - 1. Applying the trigonometric identity cos²(θ) + sin²(θ) = 1, we can simplify the expression to r²cos²(θ) + r²sin²(θ) - 1.
Since cos²(θ) + sin²(θ) = 1, the expression simplifies further to r²(1) - 1, which becomes r² - 1. Therefore, in polar coordinates, the expression x² + y² - 1 is equivalent to r² - 1. This means that when evaluating the expression in terms of polar coordinates, we only need to consider the square of the radial distance, r², and subtract 1.
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5 - y"+ 2y' = 2x+5-e-2x {undetermined coefficients)
The solution to the differential equation 5y'' - 2y' = 2x + 5 - e^(-2x) using the method of undetermined coefficients is given by y = C1 + C2e^(2x) - (5/2)x + B, where C1, C2, and B are constants determined by the initial or boundary conditions.
To solve the differential equation 5y'' - 2y' = 2x + 5 - e^(-2x) using the method of undetermined coefficients, we assume a particular solution of the form:
y_p = Ax + B + Ce^(-2x)
where A, B, and C are undetermined coefficients to be determined.
Taking the derivatives:
y_p' = A - 2Ce^(-2x)
y_p'' = 4Ce^(-2x)
Substituting these derivatives into the original differential equation, we have:
5(4Ce^(-2x)) - 2(A - 2Ce^(-2x)) = 2x + 5 - e^(-2x)
Simplifying the equation:
20Ce^(-2x) - 2A + 4Ce^(-2x) = 2x + 5 - e^(-2x)
(24C)e^(-2x) - 2A = 2x + 5 - e^(-2x)
For the equation to hold for all x, the coefficients on both sides of the equation must be equal.
Matching the coefficients:
24C = 0 -> C = 0
-2A = 5 -> A = -5/2
Therefore, the particular solution is:
y_p = (-5/2)x + B
To find the value of B, we substitute the particular solution back into the original differential equation:
5(-5/2) - 2(0) = 2x + 5 - e^(-2x)
-25/2 = 2x + 5 - e^(-2x)
Solving for x and e^(-2x) in terms of B:
2x = -25/2 - 5 + e^(-2x)
2x = -35/2 + e^(-2x)
As the left side is a linear function of x and the right side is a constant plus an exponential function, there is no value of x that satisfies this equation for all x. Hence, the equation is inconsistent, and there is no particular solution in the form y_p = Ax + B.
Therefore, the solution to the given differential equation using the method of undetermined coefficients is the complementary function (homogeneous solution) plus the particular solution, which is:
y = y_c + y_p = C1 + C2e^(2x) + (-5/2)x + B
where C1 and C2 are constants determined by the initial or boundary conditions, and B is an arbitrary constant.
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Question is in picture
Answer:
hypotenuse = 102.69
Step-by-step explanation:
7(13) + 4 = 95
3(13) = 39
hypotenuse² = 95² + 39² = 9025 + 1521 = 10546
hypotenuse = √10546 = 102.69
Answer:
It is 102.7Step-by-step explanation:
Let (h) is the hypotenuse so
[tex]h^{2} = {(7x + 4)}^{2} + {(3x)}^{2} \\ x = 13 \\ h^{2} = (95)^{2} + {(39)}^{2} \\ h = \sqrt{10546} \\ h = 102.7[/tex]
I hope that is useful for you :)
Each person is dealt 5 cards, show the total number of cards dealt for each players from 3 to 6 write the ratio of cards dealt
Answer:
Following are the responses to these question:
Step-by-step explanation:
Because every player has 5 cards handed
The card ratio is 5:1 per player.
So, if there are three teams
5 times 3=15 Cards
Four players=four times five=20 cards
Five players=5 times five=25 cards
6 cards=6 times 5=30 cards
The card ratio is 5:1 per player.
help me please :)
thank you :P
Answer: Should be 7
Step-by-step explanation:
14/2 is 7
What is the discriminant of the quadratic equation 0 = -x2 + 4x - 2? 4 8 012 O 24
Answer:
It's 8
Step-by-step explanation:
PLEASE HURRY 12 POINTS PLEASE HELPPPPPP
Step-by-step explanation:
to find the area of a circle, it is pi times radius squared. so 1 times pi squared is pi^2 the area is 4 plus 2pi^2
Two samples of sizes 25 and 35 are independently drawn from two normal populations has standard deviation of 0.9 and 0.8 respectively. Determine the variance sampling distribution for difference of two means.
A. 0.25
B. 0.51
C. 0.30
D. 0.051
Regardless of the shape of the population, the sampling distribution of the mean approaches a normal distribution as sample size increases
A. False
B. True
The variance of the sampling distribution for the difference of the two means is 0.0147142857.
The correct option is D. 0.051.
The variance of the sampling distribution for the difference of two means of sample populations can be calculated using the formula given below:
[tex]\Large\frac{{{\sigma }_{1}}^{2}}{n_{1}}+\frac{{{\sigma }_{2}}^{2}}{n_{2}}[/tex]
Where,[tex]{{\sigma }_{1}}$ and ${{\sigma }_{2}}[/tex] are the standard deviations of the two populations respectively, and [tex]{{n}_{1}} and ${{n}_{2}}[/tex] are the sample sizes of the first and second populations respectively.
Substituting the given values, we get
[tex]\Large\frac{0.9^2}{25}+\frac{0.8^2}{35}=0.009+0.0057142857[/tex]
=0.0147142857
Therefore, the variance of the sampling distribution for the difference of the two means is 0.0147142857.
Sampling distribution approaches normal distribution:
True. Regardless of the shape of the population, the sampling distribution of the mean approaches a normal distribution as the sample size increases.
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Consider the following recurrence: an= 8 n = 1 {2an-1 +8 n>1 it a. Give a closed-form expression for the recurrence. b. Prove, using proof by induction, that your answer from part a is equivalent to the recurrence an?
a.The closed-form expression for the given recurrence is: an = 8 * [tex]2^{(n-1)} + 6.[/tex]
b.In the proof by induction, we showed that the closed-form expression for the recurrence, an = 8 * [tex]2^{(n-1)}[/tex] + 6, holds true for both the base case and the inductive step. Thus, confirming its equivalence to the given recurrence.
a.What is the closed-form expression for the given recurrence?The closed-form expression for the given recurrence, an = [tex]2^n[/tex] * 8 - [tex]2^1[/tex] + 6, represents a direct formula to calculate the value of each term in the sequence. It involves exponentiation and arithmetic operations to determine the value based on the position (n) in the sequence.
b.How can we prove the equivalence between the closed-form expression and the recurrence using induction?In the proof by induction, we will first establish the base case, which is n = 1. From the recurrence, we have a1 = 8 * [tex]2^{(1-1)}[/tex] + 6 = 8. Substituting n = 1 into the closed-form expression, we also get a1 = 8 * [tex]2^{(1-1)}[/tex] + 6 = 8. The base case holds.
Next, we assume that the closed-form expression is true for an arbitrary positive integer k, i.e., ak = 8 * [tex]2^{(k-1)}[/tex] + 6.
Now, we will prove that it holds for k + 1, i.e., ak+1 = 8 * [tex]2^k[/tex] + 6.
Using the recurrence, we have ak+1 = 2 * ak-1 + 8 = 2 * (8 * [tex]2^{(k-1)}[/tex] + 6) + 8 = 16 * [tex]2^{(k-1)}[/tex] + 12 + 8 = 8 * [tex]2^k[/tex] + 6.
By comparing this with the closed-form expression, we see that ak+1 = 8 * [tex]2^k[/tex] + 6.
Therefore, the closed-form expression holds for k + 1.
By the principle of mathematical induction, we have proven that the closed-form expression, an = 8 * [tex]2^{(n-1)}[/tex] + 6, is equivalent to the given recurrence.
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The solution to the recurrence relation T(n) = T(n-1) + 2, with T(1) = 0, is T(n) = 2k.
To solve the given recurrence relation T(n) = T(n-1) + 2, for n > 0, with the initial condition T(1) = 0, we can use backward substitution.
1. Start with the base case T(1) = 0.
2. Substitute T(n-1) with T(n-2) + 2 in the original recurrence relation:
T(n) = T(n-1) + 2
= (T(n-2) + 2) + 2
= T(n-2) + 4
3. Repeat the substitution process until we reach the base case:
T(n) = T(n-1) + 2
= (T(n-2) + 2) + 2
= ((T(n-3) + 2) + 2) + 2
= T(n-3) + 6
4. Continue this process until n - k = 1, where k is a positive integer.
5. Finally, substitute n - k with 1:
T(n) = T(n-1) + 2
= (T(n-2) + 2) + 2
= ((T(n-3) + 2) + 2) + 2
= ...
= T(1) + 2k
6. Since T(1) = 0, we have:
T(n) = 0 + 2k
= 2k
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y=x+8
x+y=2
Solving Systems by Substitution
Answer:
{-3, 5}.
Step-by-step explanation:
y = x + 8
x + y = 2
Substitute y = x + 8 in the second equation:
x + x + 8 = 2
2x + 8 = 2
2x = -6
x = -3
Now plug this into the first equation
y = -3 + 8 = 5.
Suppose you expect to receive the following cashflows: $14,000 today followed by $6,000 each year for the next 11 years (so the last cash flow is at year 11). How much is this cashflow stream worth to you today if the appropriate discount rate is 7.1%? Round to the nearest dollar.
The cash flow stream consisting of $14,000 today followed by $6,000 each year for the next 11 years, discounted at a rate of 7.1%, is worth approximately $52,743 to you today.
To determine the present value of the cash flow stream, we need to discount each cash flow back to the present using the appropriate discount rate.
The present value (PV) of each cashflow can be calculated using the formula:
[tex]PV = CF / (1 + r)^n[/tex]
where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of periods.
Calculating the present value of each cash flow and summing them up, we get:
[tex]PV = \$14,000 + \$6,000 / (1 + 7.1\% / 100)^1 + \$6,000 / (1 + 7.1\% / 100)^2 + ... + \$6,000 / (1 + 7.1\% / 100)^11[/tex]
Evaluating the expression, we find that the present value of the cash flow stream is approximately $52,743 when rounded to the nearest dollar.
This means that if you discount the future cash flows at a rate of 7.1%, the combined value of all the cashflows today would be approximately $52,743.
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Given that z is a standard normal random variable, find z for each situation (to 2 decimals). Enter negative values as negative numbers. The area to the left of z is .2119. The area between -z and z is .9030. The area between -z and z is .2052. The area to the left of z is .9948. The area to the right of z is .6915.
After considering the given data we conclude that the z-score for each situation is
The area concerning left of z is .2119: z = -0.81
The area amongst -z and z is .9030: z = 1.44
The area amongst -z and z is .2052: z = 0.84
The area concerning left of z is .9948: z = 2.59
The area concerning right of z is .6915: z = 0.48
To evaluate the z-score for each situation, we can apply the z-table . Here are the steps to find the z-score for every situation:
The area concerning left of z is .2119:
Applying the z-table, we can evaluate that the z-score is -0.81.
The area amongst -z and z is .9030:
Applying the z-table, we can evaluate that the z-score is 1.44.
The area amongst -z and z is .2052:
Utilising the z-table, we can express that the z-score is 0.84.
The area concerning left of z is .9948:
Applying the z-table, we can calculate that the z-score is 2.59.
The area concerning right of z is .6915:
We need to evaluate the area concerning left of z first, which is
1 - 0.6915 = 0.3085.
Applying the z-table, we can compound that the z-score is 0.48.
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Which of the following systems of inequalities has point D as a solution?
Two linear functions f of x equals 3 times x plus 4 and g of x equals negative one half times x minus 5 intersecting at one point, forming an X on the page. A point above the intersection is labeled A. A point to the left of the intersection is labeled B. A point below the intersection is labeled C. A point to the right of the intersections is labeled D.
A. f(x) ≤ 3x + 4
g of x is less than or equal to negative one half times x minus 5
B. f(x) ≥ 3x + 4
g of x is less than or equal to negative one half times x minus 5
C. f(x) ≤ 3x + 4
g of x is greater than or equal to negative one half times x minus 5
D. f(x) ≥ 3x + 4
g of x is greater than or equal to negative one half times x minus 5
The point labeled D is to the right of the intersection of the two linear functions. This means that its x-coordinate is greater than the x-coordinate of the point of intersection.
We can find the point of intersection by setting the two functions equal to each other:
3x + 4 = (-1/2)x - 5
Solving for x, we get:
(7/2)x = -9
x = -18/7
So the point of intersection is (-18/7, -29/7).
Since the x-coordinate of point D is greater than -18/7, we can eliminate options A and C.
Now we need to check whether option B or option D includes point D as a solution. To do this, we can simply plug in the coordinates of D into the two inequalities and see which one holds true.
Option B:
f(x) ≥ 3x + 4
2 ≥ 3(D) + 4
2 ≥ 3D + 4
-2 ≥ 3D
D ≤ -2/3
g(x) ≤ (-1/2)x - 5
2 ≤ (-1/2)(D) - 5
7 ≤ -D
D ≥ -7
Since -2/3 is less than -7, option B does not include point D as a solution.
Option D:
f(x) ≥ 3x + 4
2 ≥ 3(D) + 42 ≥ 3D + 4
-2 ≥ 3D
D ≤ -2/3
g(x) ≥ (-1/2)x - 5
2 ≥ (-1/2)(D) - 5
7 ≥ -D
D ≤ -7
Since -2/3 is less than -7, option D does not include point D as a solution either.
Therefore, neither option B nor option D includes point D as a solution. The correct answer is that neither system of inequalities has point D as a solution.
Determine the Y intercept PLEASE HELP!!
X Y
-3 -3
0 -1
3 1
yo i need help please no links i just need the correct answer to pass this
Can anyone help me with this? Please and thank you!
Answer:
x=-6, y=-7
Step-by-step explanation:
Answer:
x = -6, y = -7
Step-by-step explanation:
One way to solve for x and y is using the substitution method
(1) 3x + 4y = -46
(2) 6x + y = -43
Solve for y in equation (2)
6x + y =-43, so y = -43 - 6x
Substitute y = -43 -6x into equation (1)
3x + 4(-43 -6x) = -46
3x -172 -24x = -46
-21x -172 = -46
-21x = 126
x = -6
Find y by substituting x = -6 into equation (2)
6(-6) + y = -43
-36 + y = -43
y = -7
HELP PLEASE! MARKING BRAINLIEST
WHT IS THE CIRCUMFERENCE OF THE CIRCLE SHOWN IN THE PICTURE? (Also show the process or tell me what the radius or diameter of the circle is)
Answer:
109.9 cm
Step-by-step explanation:
Circumference = (pi)(diameter)
They tell you diameter = 35
c = (3.14)(35)
c = 109.9 cm
Please lmk if you have questions.
Answer:
circumference : 109.96 cm
Step-by-step explanation:
The radius of a circle is half its diameter. The radius of a circle with a diameter of 35cm is 17.5cm.
The circumference of a circle is found by 2πr . So that would be 2π 17.5
which would be equal to 109.96cm (2 sig. fig.).
The area of a circle is found by πr2 . So that's π⋅17.5 which is equal to 962.11cm (2 sig.fig.).
Someone please help
Answer:
The first one
Step-by-step explanation:
Because
HELPPPPPO
In a school of 500 students, a random sample of 60
students are asked what
their favorite subject is. The
results are in the table. Based on this sample, how
many students in the school would we predict have
math as a favorite subject?
Answer:
150
Step-by-step explanation:
'x' = number of students out of 500 who selected math
18/60 = x/500
cross-multiply:
60x = 9000
x = 150
What is the least common multiple of 3,4 and 6?
Answer:
12
Step-by-step explanation:
Write down multiples of each number:
3, 6, 9, 12 . . .
4, 8, 12 . . .
6, 12 . . .
The first one they all have is the LCM.
Answer:
12
Step-by-step explanation:
Think of it this way. Simplifying 3, 4 and 6 into their simplest factors:
[tex]3=3[/tex]
[tex]4=2*2[/tex]
[tex]6=3*2[/tex]
6 is a multiple of both 3 and 2, which are both represented by the factors of 3 and 4. Thus, as it is doubled in these, it is not necessary to find the lowest common multiple of the numbers.
Now the LCM can be multiplied with the factors of the remaining numbers:
LCM[tex]=3*2*2[/tex]
Notice the first two numbers equal 6, the second and third equal 4, and the first only equals 3. This means the three numbers are represented in the LCM.
[tex]3*2*2=24[/tex]
And that is the LCM, so we are done. QED
Help ! I’m stuck. Any help would be gladly appreciated
Answer:
The answer is D
hope this helps
Someone help me please! With 3 and 4
Alan and Beth share $1190 in the ratio Alan : Beth = 5:2.
Work out how much Alan receives.
options:
$850
$1666
$34
$119
The share of money Alan receives is $850. Therefore, option C is correct answer.
Given that, the total amount is $1190 and the ratio Alan: Beth = 5:2.
We need to find the how much money Alan gets.
What is the ratio?The quantitative relation between two amounts shows the number of times one value contains or is contained within the other.
Now, 5+2=7
Money Alan receives=5/7×1190
=$850
The share of money Alan receives is $850. Therefore, option C is correct answer.
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PLEASE HELP! EASY MATH!!
Rosie measures the heights and arm spans of the girls on her basketball team. She plots the data and makes a scatterplot comparing heights and arm spans, in inches. Rosie finds that the trend line that best fits her results has the equation y = x + 2. If a girl on her team is 69 inches tall, what should Rosie expect her arm span to be?
A) 69 = x + 2
x = 67 inches
B) y = 69 - 2 = 67 inches
C) y = 69 inches
D) y = 69 + 2 = 71 inches
Answer:
Your answer would be D. I know this is kind of late, but maybe other people that come up here could get some help
Step-by-step explanation:
hey hottie !
Which of the following situations does Mrs. Ji Woo's hourly wage change by a constant percent?
A)Mrs. Ji Woo's starting hourly wage is $30.00 per hour the first year, and it increases by $2.50 each year.
B)Mrs. Ji Woo's hourly wage is $20 per hour in the first year, $22 per hour the second year, $24.20 per hour the third year, and so on.
C)Mrs. Ji Woo's starting hourly wage is $15.00 per hour. Her hourly wage is $15.75 after one year, $17.00 after two years, $18.75 after three years, and so on.
D)Mrs. Ji Woo's starting hourly wage is $28.00 per hour. She receives a $0.75 per hour raise after one year, a $1.00 per hour raise after the second year, a $1.25 raise after the third year, and so on.
Answer:
B I think
Step-by-step explanation:
I think it's B because in b it says so on which means his raise keeps going up each year
HELP PLEASE AND ASAP!!!!! look at the screen shot (10 pts)
Answer: 1/4
Step-by-step explanation:
3/12 simplified so divide the numerator and denominator by 3, you get 1/4
In a basketball game, Elena scores twice as many points as Tyler. Tyler scores four points fewer
than Noah, and Noah scores three times as many points as Mai. If Mai
scores 5 points, how many
points did Elena score? Explain your reasoning.
Answer:
22
Step-by-step explanation:
if mai scores 5 points and noah scores 3 times that then noah scored 15 points, and if tyler scores four less than noah than he scored 11 points, which you multiply by 2 to get elena's score which is 22
Wendy’s family lost the power at their house when there was a bad storm. The power was out for 3 days! Wendy’s neighbors lost power for 68 hours. Whose power was out for a greater amount of time?
Can someone help me on this one please?
Determine whether each quadrilateral is a parallelogram. Write yes or no. If yes, give a reason for your answer.
Answer:
yes
Step-by-step explanation:
All of the opposite sides are parallel to each other.
What is the answer to this question?
Answer:
C is the answer also can I have brian list
Step-by-step explanation:
The total number of cookies, y, contained in x packages can be represented by the equation y=24x. Which of the following graphs best represents this situation?
Answer: B)
Step-by-step explanation:
By checking which graph is satisfied, we choose points that the function has pass through.
First, we know that y = mx + b, where m is the slope, how the line change; and b is the y-intercept. In this equation, the slope is 24 which the line is increased. So we can eliminate the choice D, the line in D decreased.
Then we find where the first point and second point this graph will be.
When x = 0, y = 24x = 24(0) = 0, (0,24)
When x = 1, y = 24x = 24(1) = 24, (1,24)
1 package can have 24 cookies, only B have 24 cookies in 1 package.