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How long should the rope be (in terms of the radius of the field) so the goat can reach and graze exactly half of the grass of the field?
Three college students are trapped in deep snow in northern Canada. To survive they must build an igloo using snow large.
Find a vector orthogonal to the plane of (subspace spanned by) the vectors u and v. Show work.
Write the vector equation of the line in R^3 which passes through the two points P: (1,-1,3) and Q: (2,-1,-1). Show work.
Prove that if u and v are given non-zero vectors in the arbitrary inner-product space V, and are such that =0, then {u,v} is a linearly independent subset.
Let Complex 3-space C^3 be equipped with the standard inner product and let W be the subspace of C^3 that is spanned by u_1= (1, 0, 1) .
Using only an unmarked straight edge and a compass construct the following: a rhombus given: one side and one angle
Find the equation of the tangent plane to the central conicoid x2 - 4y2 + 3z2 + 2 = 0 at the point (1,2,0).
In the taxi-cab plane show that ifA=(-5/2,2),B=(1/2,2), C=(2,2), P=(0,0), Q=(2,1) and R=(3,3/2)then A-B-C and P-Q-R.
Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... }
Reals with the "usual topology." Is there a way to prove this space is normal other than just saying it is normal because every metric space is normal?
For any quadrilateral one can define the so-called maltitudes. A maltitude on a side of a quadrilateral is defined as the line through the midpoint.
A rectangular building whose depth is twice its frontage is divided into two parts, a front portion and a rear portion.
James wanted a photo frame 3 in. longer than it was wide. The frame he chose extended 1.5 in. beyond the picture on each side.
If 2400 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/dt in terms of dx/dt.
(Extreme Value Theorem) prove if f:K->R is continuous on a compact set K subset or equal to R, then f attains a maximum and minimum value.
A closed rectangular box with volume 576 in^3 is to be made so its top (and bottom) is a rectangle whose length is twice its width.
The metal used to make the top and bottom of a cylindrical can costs 4 cents/in^2, while the metal used for the sides costs 2 cents/in^2.
The area of a circle which is inscribed in a square is 169pi. What is the area of the square?
Find a point x ? [0, 1) and a neighborhood N of x in [0, 1) such that f (N) is not a neighborhood of f (x) in C. Deduce that f is not a homeomorphism.
Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology on Y.
Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y.
Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space Rn.
Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T.