The Cartesian equation of the curve is 4x² + 3y² = 36.
We can eliminate the parameter θ by using the trigonometric identity cos²(θ) + sin²(θ) = 1, which gives us:
x²/9 + y²/16 = cos²(θ) + sin²(θ) = 1
Multiplying both sides by 16 to get rid of the fraction, we have:
16(x²/9 + y²/16) = 16
4x² + 3y² = 36
Therefore, the Cartesian equation of the curve is 4x² + 3y² = 36.
The Cartesian equation 4x² + 3y² = 36 represents an ellipse centered at the origin with semi-axes lengths of 2√3 and 2√2. The standard form for the equation of an ellipse centered at the origin is:
(x²/a²) + (y²/b²) = 1
where a and b are the semi-axes lengths of the ellipse. To convert the given equation to this standard form, we need to divide both sides by 36:
(4x²/36) + (3y²/36) = 1
Simplifying:
x²/9 + y²/12 = 1
Comparing with the standard form, we have:
a² = 9 and b² = 12
Therefore, the semi-axes lengths are √9 = 3 and √12 ≈ 3.464.
So, the equation 4x² + 3y² = 36 represents an ellipse centered at the origin with semi-axes lengths of 3 and approximately 3.464.
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Solve the following problem using Simplex Method: MAX Z=6X1 + 10X2 + 5 X3 ST X1 + 2X2 + 4X3 <=8 6X1 + 4X2 <=24 6X1 + 5X3 <=30 X1,X2,X3 >=0
The maximum value of the objective function Z is 120. The optimal values for the decision variables are X1 = 8, X2 = 0, and X3 = 0. The constraints are satisfied, and the optimal solution has been reached using the Simplex Method.
To compute the problem using the Simplex Method, let's convert it into standard form.
Maximize:
Z = 6X1 + 10X2 + 5X3
Subject to the constraints:
X1 + 2X2 + 4X3 <= 8
6X1 + 4X2 <= 24
6X1 + 5X3 <= 30
X1, X2, X3 >= 0
Introducing slack variables S1, S2, and S3 for each constraint, the constraints can be rewritten as equalities:
X1 + 2X2 + 4X3 + S1 = 8
6X1 + 4X2 + S2 = 24
6X1 + 5X3 + S3 = 30
Now, we have the following equations:
Objective function:
Z = 6X1 + 10X2 + 5X3 + 0S1 + 0S2 + 0S3
Constraints:
X1 + 2X2 + 4X3 + S1 = 8
6X1 + 4X2 + S2 = 24
6X1 + 5X3 + S3 = 30
X1, X2, X3, S1, S2, S3 >= 0
Next, we will create the initial simplex tableau:
| X1 | X2 | X3 | S1 | S2 | S3 | RHS |
---------------------------------------
Z | 6 | 10 | 5 | 0 | 0 | 0 | 0 |
---------------------------------------
S1 | 1 | 2 | 4 | 1 | 0 | 0 | 8 |
---------------------------------------
S2 | 6 | 4 | 0 | 0 | 1 | 0 | 24 |
---------------------------------------
S3 | 6 | 0 | 5 | 0 | 0 | 1 | 30 |
---------------------------------------
By performing the simplex pivot operations and iterating through the simplex method steps, we find the following tableau:
| X1 | X2 | X3 | S1 | S2 | S3 | RHS |
---------------------------------------
Z | 0 | 0 | 5 | -6 | 0 | -60| 120 |
---------------------------------------
X1 | 1 | 2 | 4 | 1 | 0 | 0 | 8 |
---------------------------------------
S2 | 0 | -8 | -24| -6 | 1 | 0 | 0 |
---------------------------------------
S3 | 0 | 0 | -1 | -6 | 0 | 1 | 0 |
---------------------------------------
The optimal solution is Z = 120, X1 = 8, X2 = 0, X3 = 0, S1 = 0, S2 = 0, S3 = 0.
Therefore, the maximum value of Z is 120, and the values of X1, X2, and X3 that maximize Z are 8, 0, and 0, respectively.
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Evaluate the following integral over the Region R. (Answer accurate to 2 decimal places).
10(x +y))dA
R = (1, y) 16 < x² + y2 < 25, x < 0
∫ ∫R 10(x+y) dA R={(x,y)∣16≤x2+y2≤25,x≤0} Hint: The integral and Region is defined in rectangular coordinates.
The value of the integral is 15.87.
The given integral is:∫∫R 10(x+y) dAwhere R={(x,y)∣16≤x²+y²≤25,x≤0} in rectangular coordinates.In rectangular coordinates, the equation of circle is x²+y² = r², where r is the radius of the circle and the equation of the circle is given as: 16 ≤ x² + y² ≤ 25 ⇒ 4 ≤ r ≤ 5We need to evaluate the integral over the region R using rectangular coordinates and integrate first with respect to x and then with respect to y.∫∫R 10(x + y) dA = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy...[since x < 0]
Now, integrating ∫(x+y) dx we get ∫(x+y) dx = (x²/2 + xy)Therefore, 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) [ (x²/2 + xy) ] dy dx= 10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dxNow integrating with respect to y we get∫(x²/2 + xy) dy = (xy/2 + y²/2)
Putting the limits and integrating we get10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dx = 10∫ from 4 to 5 [(∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy)] dx = 10∫ from 4 to 5 [(x²/2)[y]^(-√(16-x²) )_(^(-√(25-x²))] + [(xy/2)[y]^(-√(16-x²) )_(^(-√(25-x²)))] dx = 10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dxNow integrating with respect to x, we get10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dx = [ (10/3) [(25/3)^(3/2) - (16/3)^(3/2)] - 5√3 - (5/3)[(25/3)^(3/2) - (16/3)^(3/2) ] ]Ans: The value of the integral is 15.87.
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Can i have some help please!!
Answer: $93649
Step-by-step explanation:
Since this is an exponential growth problem, then we can use the equation 50,000(1.04)^16. Solve it and you get 93649.06228. Round to the nearest dollar, which is probably whole number, so it is 93649.
Find the surface area.
24 in.
40 in.
10 in.
26 in.
Answer:
100 i think
Step-by-step explanation:
The diameter of a circle is 63 centimetres find its circumference use pie = 3.14
Answer:
197.9
Step-by-step explanation:
The formula for circumference is 2(pi)r and r is the radius
The diameter is two times the size of the radius, so by dividing the diameter by two, you can get the radius
So, r=63/2
r= 31.5
That means that 2(pi)(31.5) is the circumference
2(pi)(31.5) = 197.9 (rounded to the nearest tenth)
Find a1 for the arithmetic sequence's 21st term is 400 is 400 and it's common difference is 5
Answer:
8,395
Step-by-step explanation:
21 x 400 = 8,400
is = x
8, 400 - 5 = 8,395
difference = -
Brainlist Pls!
a
& b
5. Find the following limits. (a) lim40 12 (b) limz+1+1 +22-22+2 i 2-iz-1-1
The limits are,
(a) lim(x→0) 4x/(x² + 1) = 0
(b) lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = ((1 + √(5))(3 - i))/10
(a) To find the limit of lim(x→0) 4x/(x² + 1), we can directly substitute 0 for x in the expression:
lim(x→0) 4x/(x² + 1) = (4 × 0)/(0² + 1) = 0/1 = 0
Therefore, the limit is 0.
(b) To find the limit of lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1), we can again substitute -1 for z in the expression:
lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = (1 + sqrt(2 - 2(-1) + (-1)^2))/(2 - i(-1) - 1)
= (1 + √(2 + 2 + 1))/(2 + i + 1)
= (1 + √(5))/(3 + i)
To simplify this expression further, we need to rationalize the denominator. We can multiply the numerator and denominator by the conjugate of the denominator, which is (3 - i):
lim(z→-1) (1 + √(5))/(3 + i) × (3 - i)/(3 - i)
= ((1 + √(5))(3 - i))/(9 - i²)
= ((1 + √(5))(3 - i))/(9 + 1)
= ((1 + √(5))(3 - i))/10
Therefore, the limit is ((1 + √(5))(3 - i))/10.
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The question is -
Find the following limits:
(a) lim(x->0) 4x/(x^2 + 1)
(b) lim(z->-1) (1 + sqrt(2 - 2z + z^2))/(2 - iz - 1)
Please help. No files allowed or you will be reported
In a poll, students were asked to choose which of six colors was their favorite. The circle graph shows how the students answered. If students participated in the poll, how many chose Orange?
Answer:
1666.70
Step-by-step explanation: 10,000/6=1666.70
PLEASE HELP!! I only have 5 mins
Find m so that x + 4 is a factor of 5x3 + 18x2 + mx + 16
The value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
To find the value of 'm' for which the expression (x + 4) is a factor of the polynomial[tex]5x^3 + 18x^2 + mx + 16[/tex], we can apply the factor theorem. According to the factor theorem, if (x + 4) is a factor of the polynomial, then the polynomial evaluated at (-4) should be equal to zero.
Substituting (-4) into the polynomial, we get:
[tex]5(-4)^3 + 18(-4)^2 + m(-4) + 16 = 0[/tex]
-320 + 288 + (-4m) + 16 = 0
-16 + (-4m) = 0
Simplifying the equation, we have:
-4m - 16 = 0
-4m = 16
m = 16 / -4
m = -4
Therefore, the value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
By substituting -4 for 'm' in the given polynomial, we obtain:
[tex]5x^3 + 18x^2 - 4x + 16[/tex]
When this polynomial is divided by (x + 4), the remainder will be zero, confirming that (x + 4) is indeed a factor.
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Kelly received two gift cards to her favorite store. One card was worth $25 and the other was
worth $40. She went shopping and used the cards to buy 3 shirts for $9 each and 2 skirts for
$17 each. How much gift card money did she have left?
PLEASE HELP WILL MARK BRAINLIEST
Answer:
I believe the answer is (A)
*Substituting the x and y values from the table into the equation(A) will balance the right side of the equation to the left side of the equation.
NEED HELP WHAT ARE THSES TWOO!!
Verify that f_xy = f_yx, for the function f(x,y) = 3x^7 + 4y^7 + 12.
For the function f(x,y) = 3x^7 + 4y^7 + 12, f_xy = f_yx since fx = ______ and fy = ____
Therefore, fxy= _______ and fyx = _______
Given the function: f(x,y) = 3x^7 + 4y^7 + 12To verify that f_xy = f_yx, we need to find the partial derivatives of the given function with respect to x and y. We can find them as follows: ∂f/∂x = 21x^6 ∂f/∂y = 28y^6
Now, to verify that f_xy = f_yx, we need to find f_xy and f_yx. We can find them as follows: f_xy = ∂^2f/∂y∂x = ∂/∂y(∂f/∂x) = ∂/∂y(21x^6) = 0 (since we have no y terms in the derivative of ∂f/∂x) f_yx = ∂^2f/∂x∂y = ∂/∂x(∂f/∂y) = ∂/∂x(28y^6) = 0 (since we have no x terms in the derivative of ∂f/∂y)Since f_xy = f_yx = 0, we can say that f_xy = f_yx.
Therefore, the value of fx is 21x^6 and the value of fy is 28y^6. Hence, the value of fxy is 0 and fyx is also 0.
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A rubber ball is dropped from a height of 26 feet, and on each bounce it rebounds up 62% of its previous height. Step 2 of 2: Assuming the ball bounces indefinitely, find the total vertical distance traveled. Round your answer to two decimal places.
The total vertical distance traveled by the rubber ball, assuming it bounces indefinitely, is approximately 85.71 feet.
To find the total vertical distance traveled, we need to sum up the heights achieved by the ball during each bounce. The ball initially drops from a height of 26 feet, so we start with this value. On each bounce, the ball rebounds up 62% of its previous height. This means that after the first bounce, the ball reaches a height of 26 feet * 0.62 = 16.12 feet.
For subsequent bounces, we continue to multiply the previous height by 0.62 to find the new height. Therefore, after the second bounce, the height becomes 16.12 feet * 0.62 = 9.99 feet.
We can see that the heights achieved during each bounce form a geometric sequence with a common ratio of 0.62. The sum of an infinite geometric sequence can be calculated using the formula,
Sum = a / (1 - r), first term is a and 'r' is the common ratio is r.
In this case, 'a' is the initial height of 26 feet and 'r' is 0.62. Plugging these values into the formula, we get,
Sum = 26 / (1 - 0.62) = 26 / 0.38 ≈ 68.42 feet.
Therefore, adding all the distances,
Distance = 68.42 + 9.99 + 16.12
Distance = 85.71 feet, total vertical distance traveled by the rubber ball, rounded to two decimal places, is approximately 85.71 feet.
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plsssssd help me find the anwser
I need help imm struggling
Answer:
180in3 (180 inch cubed)
Step-by-step explanation:
12 x 5 x 3
Answer: I would assume the answer would be 180
Step-by-step explanation: The formula for volume is Length x Width x Height. So multiply all the number above and the answer will be 180
How many solutions does this equation have? –7q + 7 = 4 − 4q
- no solution
-one solution
-infinitely many solutions
Answer: One answer
Step-by-step explanation:
39 POINT BRAIN.LY QUESTION WHAAA
Answer:
thx for the points
Step-by-step explanation:
Answer:
Where is the question tho whaaAAaaaa
The ____ sequence begins with two ones, and then each new term is formed by adding the two terms before it: 1, 1, 2, 3, 5, 8, 13, 21,...
Answer:
Fibonacci
Step-by-step explanation:
the Fibonacci sequence
11. A bag contains 2 blue marbles and 2 green marbles. What is the probability of drawing a blue marble followed by a green marble, without replacing the first marble before drawing the second marble?
Please show work ty
Answer: your answer should be 50%
Step-by-step explanation: This is because there are only four marbles in the bag total and only 2 are blue and only 2 are green so your chances of pulling out either is 50%
Answer:
33%
Step-by-step explanation:
2 blue marbles + 2 green marbles = 4 marbles
1st draw for blue: 2/4 (2 blue marbles out of 4 marbles)
2nd draw for green: 2/3 (1 less marble from 4, marble not put back in)
2/4 x 2/3 = 4/12 = 1/3 = 0.33 or 33%
how do I solve this equation in picture
The total number of people surveyed is 75.
How many people were surveyed?The first step is to determine the number of people who had 4 or more rides that preferred a window seat.
= Total number of people that had four or more rides - total number of people who had 4 or more rides that prefer aisle
= 40 - 25 = 15
Total number of people that prefer the window seats= 15 + 20 = 35
Total number of people = total number of people that prefer the window seat + total number of people who prefer the aisle
= 35 + 40 = 75
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Two particles, Alpha and Beta, race from the y-axis to the vertical line x = 6*pi. For t >= 0, Alpha's position is given by the parametric equations xalpha = 3t - 4sin(t) and yalpha = 3 - 3cos(t) while Beta's position is given by xbeta = 3t - 4sin(t) and ybeta = 3 - 4sin(t). Which sentence best describes the race and its outcome?
(A) Beta starts out in the wrong direction and loses.
(B) Alpha takes a shorter path and wins.
(C) Alpha moves slower and loses.
(D) Beta moves faster but loses.
(E) Alpha and Beta tie
The outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
To determine the outcome of the race between Alpha and Beta, let's compare their positions using the given parametric equations:
Alpha's position:
[tex]x_{alpha} = 3t - 4sin(t)\\y_{alpha}= 3 - 3cos(t)[/tex]
Beta's position:
[tex]x_{beta} = 3t - 4sin(t)\\y_{beta} = 3 - 4sin(t)[/tex]
From the equations, we can see that the x-coordinate of both Alpha and Beta is the same, given by 3t - 4sin(t). Therefore, their horizontal positions are identical throughout the race.
To determine the vertical positions, we compare their y-coordinates. Alpha's y-coordinate is given by 3 - 3cos(t), while Beta's y-coordinate is given by 3 - 4sin(t).
Since cos(t) ranges from -1 to 1, and sin(t) ranges from -1 to 1, we can observe the following:
For Alpha, the y-coordinate (3 - 3cos(t)) ranges from 0 to 6, inclusive.
For Beta, the y-coordinate (3 - 4sin(t)) ranges from 2 to 4, inclusive.
Based on the range of their y-coordinates, we can conclude that Beta remains at a higher position throughout the race. Therefore, the correct answer is:
(D) Beta moves faster but loses.
Despite Beta moving faster, it loses the race because Alpha consistently maintains a higher vertical position.
Therefore, the outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
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One angle of an isosceles triangle measures 46°. Which other angles could be in that isosceles triangle?
Answer:
67 degrees for both of the other angles or 46 degrees and 88
Step-by-step explanation:
An isosceles triangle has two angle that are the same size so it could only be these.
A rectangular park, 90 meters by 60 meters, is to be built on a city block having an area of 9000 m^2. A uniform strip borders all four sides of the park for parking. How wide is the strip? Use quadratic formula and show your work.
Answer:
x = 10.52 m
Step-by-step explanation:
Given that,
Length of a park = 90 m
Width of a park = 60 m
Area, A = 9000 m²
A uniform strip borders all four sides of the park for parking. We need to find the width of the strip. Let it is x. Now the area becomes,
(90+2x)(60+2x) = 9000
[tex]4x^2 +120x +180x =5400 = 9000\\\\x=10.52\ m[/tex]
So, the width of the strip is equal to 10.52 m.
Suppose () = 1/8 for 0 ≤ ≤ 4 for x being a continuous random variable Is () a probability density function? Prove or disprove.
Answer:
The expected value of x ; E(x) = 1
Step-by-step explanation:
F(x) = 1/8 for 0 ≤ x ≤ 4
To prove that it is a probability density function we will find E(x )
attached below is the required prove
It is proven that F(x) = 1/8 for 0 ≤ x ≤ 4 is probability density function
The expected value of X = 1
Find the volume of this square pyramid
Answer:
216
Step-by-step explanation:
Answer:
72yd
Step-by-step explanation:
Hope that helps
hsobsnsjns
Mrs. Smith washed 2 5 of her laundry. Her son washed 1 3 of it. Who washed most of the laundry? How much of the laundry still needs to be washed?
Answer:
a) The person who washed the most of the laundry is Mrs Smith
b) 4/15 of the laundry is left to wash
Step-by-step explanation:
Mrs. Smith washed 2/5 of her laundry. Her son washed 1/3 of it.
a) Who washed most of the laundry?
We convert the fraction of laundry each person washed to decimal
Mrs Smith = 2/5 = 0.4
Her son = 1/3 = 0.333
Therefore, the person who washed the most of the laundry is Mrs Smith
b) How much of the laundry still needs to be washed?
Let us total laundry = 1
=1 - ( 2/5 + 1/3)
Lowest Common Denominator is 15
=1- (3 × 2 + 5 × 1/15)
= 1 - (6 +5/15)
=1 - 11/15
= 4/15
What is the biggest difference between exponential functions and other functions you have learned about up to this point?
Answer:
No no don't click the link
Answer:
The biggest difference between exponential and linear functions is that linear functions change at a constant rate, while exponential functions change at a rate proportional to it's value, or exponent.
Basically, that's also what separates exponential functions from all others. It's the only function that changes at a rate proportional to its exponent.
Step-by-step explanation: