/sci/ - Science & Math

[Anonymous No.3844007]

Prepping for a test. Need help in a couple of problems.
I'll just list the problems out and please see if you can help me out, any help is appreciated. I just want a push in the right direction, not the answer.

1)Fine the area of the region common to the graphs of r= cosΘ and r=sinΘ

2)There are two spheres with radius 2 and tangent to 4x-y-z=2 at the point (1,1,1). Find their equations.

3)Find the parametric equation for the parabola y=x^2 using as parameter the slope m of the line joining the point (1,1) to the general point (x,y) on the parabola.

4) Prove that the distance between two parallel planes ax+by+cz=d1 and ax+by+cz=d2 is
|d2-d1|/sqrt(a^2+b^2+c^2)

I appreciate any help. I'm going to have to solve similar problems for the test. pic unrelated

[Anonymous No.3844012]

>>3844007
>1)Fine the area of the region common to the graphs of r= cosΘ and r=sinΘ

Find*

[Anonymous No.3844044]

1. Find the intervals where sin(x) and cos(x) coexist in the same quadrant of the unit circle. (I'm assuming this is only for [0,2pi] because if it isn't bounded, is it possible to find that area?)

2. really not sure... I've forgotten most of my multivariable calculus stuff

3. Find the equation for the distance from (1,1) to y(x^2)

4. Assume the distance isn't equal to that expression, then show that it is

[Anonymous No.3844155]

>>3844044
Got #4. I just got a vector PQ using the points (0,0,d1/c) and (0,0,d2/c) and used the theorem D=||proj PQ unto n||

For #3 I did dy/dx=2x=m
x=m/2
y=(m/2)^2=m^2/4
so the parametric is <t/2, t^2/4>

Still need help with 1 and 2.

For #1, cos (x) intersects sin (x) when x=pi/4 and pi/2.
And sin (x) instersects with cos (x) when x=pi/2 and 1.
Not sure what to do from there.
If it helps the area under a polar curve is 1/2 integral from Θ1 and Θ2 of r^2 dΘ.

Still lost with #2.

[Anonymous No.3844250]

>>3844155
with #2 go look back at the equation for spheres... it's something like (x+a)^3 + (y+b)^3 + (z+c)^3 = R^3, and I get the feeling if you use the definition of a sphere and the formula and whatnot the answer will come quite easily

again for #1 you're looking for where sin(x) and cos(x) coexist, so diagram out the four intervals of pi/2 from [0, 2pi] and do independent integrals for each of the four regions. Actually you may have to go more in depth, find where sin(x)=cos(x), and do your intervals from there... but same idea

[Anonymous No.3844527]

ok i've answered these problems. can't say how thankful i am.

i have 2 more problems i need help with, my test is in about 2 hours so again, any help is appreciated:

5.Prove that the lateral surface area of a cone of base radius r and height h is pi*r*sqrt(r^2 + h^2)

6.In R^3, find the equation of the plane tangent to the sphere (x-1)^2 + (y-2)^2 + (z-2)^2=9 at the point (0,1,2+sqrt(7)

Any ideas?

[Anonymous No.3844575]

>>3844527
OP again
for 6, i have the center of the circle Q(1,2,2) which makes a line to the point on the plane P(0,1,2+sqrt(7)). thus PQ=<1,1,-sqrt(7)is a line normal to the plane.

But i only know how to find the equation of a plane with a normal vector AND another point on the plane. Stuck :/

[Anonymous No.3844640]

>>3844575
bump for my grade

[Anonymous No.3844693]

>>3844640
fml. i guess i'll just wing the ones i don't know the best i can.

thanks for the help i actually received though.

[Anonymous No.3844733]

>>3844640
>>3844693
the best thing you can do for this test is to relax and go blow off some steam. Stressing yourself out before the test will make you receive a lower grade. Test anxiety is bad, mmmkay?