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3 years ago

A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y=mx+b for this line

Mathematics

1 answer:

k0ka [10]3 years ago

8 0

Answer:

y=x+15

Step-by-step explanation:

1. find the slope

m=(24-19)/(9-4)

m=5/5

m=1

2. plug slope into slope intercept formula

y=1x+b

y=x+b

3. plug in for x and y from either coordinate point (just make sure they are from the same point!!)

24=9+b

15=b

4. plug m and b into formula

y=x+15

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Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) t

pashok25 [27]

Direct computation:

Parameterize the top part of the circle $x^2+y^2=1$ by

$\vec r(t)=(x(t),y(t))=(\cos t,\sin t)$

with $0\le t\le\pi$ , and the line segment by

$\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)$

with $0\le t\le1$ . Then

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)$

$=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt$

$=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}$

Using the fundamental theorem of calculus:

The integral can be written as

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}$

If there happens to be a scalar function $f$ such that $\vec F=\nabla f$ , then $\vec F$ is conservative and the integral is path-independent, so we only need to worry about the value of $f$ at the path's endpoints.

This requires

$\dfrac{\partial f}{\partial x}=\sin x\implies f(x,y)=-\cos x+g(y)$

$\dfrac{\partial f}{\partial y}=\cos y=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\sin y+C$

So we have

$f(x,y)=-\cos x+\sin y+C$

which means $\vec F$ is indeed conservative. By the fundamental theorem, we have

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=f(2,-\pi)-f(1,0)=-\cos2-(-\cos1)=\boxed{\cos1-\cos2}$

3 0

2 years ago

Eight times a number plus five another number is -13. The sum of the two numbers is 1. What are the numbers?

Pani-rosa [81]

Eight *(a number) plus 5*(another number) is -13.

translates to:

8(x) + 5(y) = -13

The sum of (the number) and (the other number) is 1.

translates to:

(x) + (y) = 1

We have a system of two equations involving two unknowns: x and y.

$\rm 8x+5y=-13\\
~~x+~~y=~~~~1$

We can easily solve the system using Substitution or Elimination. Let's use Elimination this time.

We'll multiply the second equation by -8 so that the x's match up.

$\rm ~~~8x+5y=-13\\ -8x-8y=-8$

When we add the equations together, the x's will fall out of the equation, summing to zero. The 5y and -8y will sum to -3y and the right hand side will sum to -21.

$\rm -3y=-21$

Divide by -3,

$\rm y=7$

Plug back into one of your original equations to find the value of x,

$\rm x+y=1\\
x+7=1$

Subtract 7,

$\rm x=-6$

8 0

2 years ago

Please I need help ASAP

jasenka [17]

Answer:

Step-by-step explanation:

3 0

2 years ago

Is my answer correct??

AlekseyPX

Yes I believe so. If you flip it and it matches up then yeah

3 0

2 years ago

Read 2 more answers

SOMEONE HELP!!! Can not get this wrong

pishuonlain [190]

Answer:

This is an geometric sequence with a common ratio r of 1/2

Step-by-step explanation:

When you find out r, you realize the ratio between the numbers to be dividing by 2, or 1/2. Since it is r and not common difference d, it is an geometric sequence.

7 0

3 years ago