Geometry

(See attached file for full problem description with proper symbols) The area A of an equilateral triangle varies directly as the square of the length of a side. If the area if the equilateral triangle whose sides are of length 1 cm is (  3 ) / 4 cm 2 , find the length s of an equilateral triangle whose area A is 3 cm 2 .

Finding the cubic yard

An engineer's plan shows a canal with a trapezoidal cross section that is 8 ft deep and 14 ft across at the bottom with walls sloping outward at an angle of 45 degrees. The canal is 620 ft long. A contractor bidding for the job estimates the cost to excavate the canal at $1.75 per cu yd. If the contractor adds 10% profit, what should the bid be?

Differential Geometry, imbedded submanifold.

Let phi : R^2 --> R be a function given by phi(x,y) = x^3 + xy + y^3 +1 For which points p =(0,0) , p=(1/3,1/3), p =(-1/3,-1/3) is the subset phi^-1 ( phi(p)) an imbedded sub-manifold of R^2. ( This question was given as part of Differential Geometry course, we use Do Carmo's book,Riemannian Geometry) Please justify all your steps and claims. And try to solve the problem in a manner similar to the book mentioned above.

Differential Geometry/Orientable Manifolds.

This problem is number 3 page 33 of handout chapter 0. I attached the handout for the chapter ( from Do Carmo's Book) and I hope you can solve it using the theorems and the same way Do Carmo does it, which shouldn't be very hard to follow. ( I believe the main idea is just to use the def of the orientable manifolds and show that the det > 0 ).

Proper affine function

I attached the problem and chapters 2 and 3 ( chapter 3 is about affine connections). Please solve it using the way that the book does it. ( Handouts are from Do Carmo's book).