Math Is Fun Forum

Suppose the order is for

of Platter A and

of Platter B. Then we have:

We want to minimize

subject to the above constraints. Let us then rewrite the above inequalities in terms of

and one of

and

, say

.

The 1st and 2nd inequalities give

, the 1st and 3rd inequalities give

, and the 2nd and 3rd inequalities give

. The minimum appears to be 855 – however

would imply

, which does not satisfy the 2nd inequality. So we must instead have

. Thus the minimum cost is $870 dollars for 10 of Platter A and 60 of Platter B.

In questions of this sort I tend look for a transformation to get rid of the

term:

When this is substituted into the original equation, the term in

is

We want this to vanish, so any

such that

and

will do. So we take

. Hence

Thus under the transformation the curve

becomes the hyperbola

. Furthermore as the transformation

represents a clockwise rotation of 45° about the origin followed by an enlargement of

at the origin, the conic section is preserved, i.e. the original curve is indeed a hyperbola.

NB: Be careful when using linear transformations on curves: only rotations, reflections and enlargements/contractions by a nonzero factor preserve conic sections. Any other transformation may distort the curve and alter its original nature.

#6. Rating: Medium

Let A and B be finite sets with m and n elements respectively, and suppose ƒ:A→B is a mapping. If ƒ is surjective (onto), show that m ≥ n.

Thanks bobbym. I've hidden my solution.

I see. Well, if

are differentiable, then linear independence implies

for all

. If they are not both differentiable, then it is possible that they are linearly independent yet

. See http://en.wikipedia.org/wiki/Wronskian# … dependence for an example.

According to the Old Testament, 31 kings were defeated by Joshua after the children of Israel had crossed the river Jordan (Joshua 12:9–24).

Is there a 0 (zero)?

#2. Rating: Medium

Do there exist integers n such that 3[sup]n[/sup]+7[sup]n[/sup] is a perfect square?