(a) Show that the lines
r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s) = (1,0,0) + s(0,1,−2)
are skew.
(b) The two lines in (a) lie in parallel planes. Find equations for these two planes. Express your
answer in the form ax+by+cz +d = 0. [Hint: The two planes will share a normal vector n. How would one find n?]
would one find n?]

In an effort to replace the destruction of trees, five regions of the US have been tracking the annual rate of trees being planted (per thousand) in each region and the annual rate of trees being destroyed (per thousand). Additionally, the tree population (in thousands) by region has been recorded for five different years. The data is given in the tables below. Determine the number of trees planted and the number destroyed in each of the years listed.

Illustrate with a picture showing that the following are simply not true in Hyperbolic Geometry

a). The area of a triangle can be made arbitrarily large

b). The angle sum of all triangles is a constant.

Find Cartesian equation of a plane through the point a and parallel to the given vectors p and q.

a = (1, - 1, 3) , p= ‹2, -1, 0›, q= ‹2, 0, 6›.

(1 point) Let F=−3yi+4xjF=−3yi+4xj. Use the tangential vector form of Green's Theorem to compute the circulation integral ∫CF⋅dr∫CF⋅dr where C is the positively oriented circle x2+y2=25x2+y2=25.

ty'''+4y''=t^2cost

Can somebody explain to me Derivative of Hyperbolic Functions? May I know its importance and applications in real life? I need it for my essay. Thank you.

PS. I can't find it on the internet.

Write up a formal proof that the angle bisectors of a triangle are concurrent, and that the point of concurrency (the incenter) is equidistant from all three sides.

Problem: A bottle is shaped with a right cone placed on top of a right cylinder. The radius of the cone and the cylinder

is 1.5 inches. The height of the cone is 3 inches and the height of the cylinder is 5 inches.

1. Find the volume of the bottle.

2. About how much paper or plastic is needed to make a label for the cylindrical part of your bottle? Explain.

Laplace Question : y''-3y'+2y=4cos2t,y(0)=-2,y'(0)=0

A rectangular area adjacent to a river is to be fenced in, but no fencing is required on the side by the river. The total area to be enclosed is 114,996 square feet. Fencing for the side parallel to the river is $6 per linear foot, and fencing for the other two sides is $7 per linear foot. The four corner posts cost $25 apiece. Let xx be the length of the one the sides perpendicular to the river.

[A] Find a cost equation C(x)C(x):
C(x)=C(x)=    

[B] Find C'(x)C′(x):
C'(x)=C′(x)=    

[C] Find the appropriate critical value(s) for the appropriate domain in the context of the problem.
   

[D] Perform the second derivative test to determine if there is an absoulte minimum at the critical value found.
C''(x)=C′′(x)=   

[E] What is the best conclusion regarding an absolute maximum or minimum at this critical value. (MULTIPLE CHOICE)

a) At the critical value C''(x)>0C′′(x)>0 so I can conclude that there is a local/relative maximum there but I can't conculde anything about an absolute maximum for x>0x>0

b) Since C''(x)>0C′′(x)>0 for all x>0x>0 we can colude that the CC is concave up for all values of x>0x>0 and that we therefore have an absolute minimum at the critical value for x>0x>0

c) Since C''(x)>0C′′(x)>0 for all x>0x>0 we can colude that the CC is concave up for all values of x>0x>0 and that we therefore have an absolute maximum at the critical value for x>0x>0

d) The second derivative test is inclusive with regards to an absolute maximum or minimum and the first derivative test should be performed

e) At the critical value C''(x)>0C′′(x)>0 so I can conclude that there is a local/relative minimum there but I can't conculde anything about an absolute minimum for x>0x>0

[F] Find the minimum cost to build the enclosure: $

*Please show all work associated*

Use the Intermediate Value Theorem and the Mean Value Theorem to prove that the equation cos (x) = -10x has exactly one real root.

Not permitted to use words like "Nope", "Why?", or "aerkewmwrt".

Will be glad if you can help me with this question, will like to add some of your points to the one I have already summed up.. Thanks

Find an equation of the tangent plane to the surface x5+5z2ey−x=848 at point P=(3, 4, 11e√).

(Use symbolic notation and fractions where needed. Your answer should be in the form ax+by+cz=1.)