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tatiyna
3 years ago
12

Let c be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. find the exact length of c from t

he origin to the point (2, 2, 4/3).
Mathematics
1 answer:
Mandarinka [93]3 years ago
3 0
Parameterize the intersection by setting x(t)=t, so that

x^2=2y\iff y=\dfrac{x^2}2\implies y(t)=\dfrac{t^2}2
3z=xy\iff z=\dfrac{xy}3\implies z(t)=\dfrac{t^3}6

The length of the path C is then given by the line integral along C,

\displaystyle\int_C\mathrm dS

where \mathrm dS=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt. We have

\dfrac{\mathrm dx}{\mathrm dt}=1
\dfrac{\mathrm dy}{\mathrm dt}=t
\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{t^2}2

and so the line integral is

\displaystyle\int_{t=0}^{t=2}\sqrt{1^2+t^2+\dfrac{t^4}4}\,\mathrm dt

This result is fortuitous, since we can write

1+t^2+\dfrac{t^4}4=\dfrac14(t^4+4t^2+4)=\dfrac{(t^2+2)^2}4=\left(\dfrac{t^2+2}2\right)^2

and so the integral reduces to

\displaystyle\int_{t=0}^{t=2}\frac{t^2+2}2\,\mathrm dt=\dfrac{10}3
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Solve for x.<br> I was sure the answer was 2?<br> Please show me the breakdown. THX
Murljashka [212]

Answer:

x = 8 and y = 10

Step-by-step explanation:

Since the figures are congruent:

Angle PQR = EFG

6y + x = 68

Segment QR = FG

2x - 4 = 12

Solving that:

2x - 4 = 12

2x = 16

x = 8

Replacing x in 6y + x =68

6y + 8 = 68

6y = 60

y = 10

7 0
3 years ago
M(x)=−(x−5) 2 +25m, left parenthesis, x, right parenthesis, equals, minus, left parenthesis, x, minus, 5, right parenthesis, squ
Aleksandr [31]

Answer:

25

Step-by-step explanation:

Given the function that represents the amount of mosquitoes as;

M(x) = -(x-5)² + 25

To get the maximum mosquitoes possible, we need to first find x;

At the maximum dM/dx = 0

dM//dx = -2(x-5)

0 = -2(x-5)

-2(x-5) = 0

-2x + 10 = 0

-2x = -10

x = 10/2

x = 5

Substitute x = 5 into the function;

M(5) = −(5−5)² +25

M(5) = 0+25

M(5) = 25

Hence the maximum number of mosquitoes is 25

8 0
2 years ago
A certain radioactive material decays in such a way that the mass in kilograms remaining after t years is given by the function
Angelina_Jolie [31]

The mass of radioactive material remaining after 50 years would be 48.79 kilograms

<h3>How to determine the amount</h3>

It is important to note that half - life is the time it takes for the amount of a substance to reduce by half its original size.

Given the radioactive decay formula as

m(t)=120e−0.018t

Where

t= 50 years

m(t) is the remaining amount

Substitute the value of t

m(t) = 120e^-^0^.^0^1^8^(^5^0^)

m(t) = 120e^-^0^.^9

Find the exponential value

m(t) = 48.788399

m(t) = 48.79 kilograms to 2 decimal places

Thus, the mass of radioactive material remaining after 50 years would be 48.79 kilograms

Learn more about half-life here:

brainly.com/question/26148784

#SPJ1

7 0
1 year ago
The area of a rectangle is (27x5 + 9x4 - 18x3) square feet. The width of the rectangle is 9x2 feet. What is the length of the re
Yuliya22 [10]
If the width is 9x², then the length is 27x^5+9x^4-18x^3/9x², which equals 3x³+x²+2x. ☺☺☺☺
8 0
2 years ago
Read 2 more answers
Hector, Sam, Ted, and Gina are comparing their savings. Hector saved twice as much money as Sam. Ted saved $100 more than Hector
lutik1710 [3]

Answer:

2x+50

Step-by-step explanation:

Given: Hector saved twice as much money as Sam.

           Ted saved $100 more than Hector.

           Gina saved $50 less than Ted.

As given, lets assume the amount of money Sam has in saving be "x".

Now, finding saving of each person given.

Writing the algebric expression of each´s saving amount.

we know, Hector saved twice as much money as Sam.

∴ Hector saving= \$2x

Also, Ted saved $100 more than Hector.

∴ Ted´s saving= \$(2x+100)

Next finding Gina´s saving

As given, Gina saved $50 less than Ted.

∴ Gina´s saving= \$ (2x+100)-\$ 50

Opening parenthesis and solving it.

⇒ Gina´s saving= \$ 2x+\$50

Algebric expression of Gina´s saving= 2x+50

5 0
3 years ago
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