dansmath.com - problem of the week - archives

 

 

Problem Archives page 28

with Answers and Winners. For Problems Only, click here.

 

1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110

111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190 . 191-200

201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261-270 . 271+ . index

 

 

271 - Dan's Directriconix

272-StockProfits2daMax

273 - Best Best-Fit Curve

problem #271 - posted wednesday, december 19, 2007

Dan's Directriconix (back to top)

The points equidistant from (0, 0) and (1, 1) form the perpendicular bisector

of the segment joining the given points, which has the equation x + y = 1.

The distance from a point to a line or curve is defined as the shortest distance.

Call the equidistant point (x, y) and the closest point on the curve (a, b).

a) The set (locus) of points equidistant from (0, 0) and the line x + y = 1 forms a familiar shape.

Name the shape, and find its equation.(as a relation between x and y or as parametric equations of a)

b) Determine the locus of points equidistant from the origin and the hyperbola x y = 1, x > 0.

(as a relation between x and y or as param. eqns of a)

c) Find the locus of points equidistant from the origin and the circle through (1, 1) that's

tangent to both axes. (in {x, y} or a form) These are possible with or without calculus; your choice!

Show your steps and reasoning; one point still available.

 

problem #272 - posted saturday, january 12, 2008

Stock Profits to the Max ! (back to top)

The price of dansmathcorp stock (DMC) is initially $100 per share. The stock can go up

or down each month, from $0-10, whole # $. Starting after the first month, you invest

$1000 per month in DMC.After six months you've put in $6000, the stock price is again

$100, and you sell off all the shares you have bought in the six months.

a) What sequence of stock prices will yield the maximum profit (if any)?

b) If the total price change (the sum of absolute values of monthly changes) is limited to

$40, then what sequence of prices gives the max profit? In each case find the maximum

profit and explain how you calculated it. Show your steps and reasoning; one point still available.

 

problem #273 - posted sunday, february 10, 2008

Best Best-Fitting Curve ? (back to top)

We at DVC feel that more experienced teachers can derive the quadratic formula faster!

Here is a table of years of experience (x) vs. minutes to formula (y), for a dozen teachers:

x years	1	2	4	7	9	10
y minutes	8	6	5	3	2	2

x years	1	3	6	8	13	15
y minutes	10	4	2	2	1	1

There are four types of curves we might try to fit to this data set :

Linear: y = a + bx Logarithmic: y = a + b ln(x) Exponential: y = a*b^x Power: y = a*x^b

i) Find the best "a" and "b" for each type of curve.

(measured by the least squares deviation method)

ii) Which is the best curve choice, for these data?

(measured by the largest abs value of correlation coeff rxy)

 

Use either the TI-8x, Excel, or the sample Pearson correlation formula

where sx and sy are the sample standard deviations for the x and y data values.

Show your steps, formulas, and reasoning; one point penalty for resubmissions.

 

 

THANKS to all of you who have entered, or even just clicked and looked. My site is in its 11th season - OVER 99,000 HITS so far! (Not factorial.) Help it grow by telling your friends, teachers, and family about it. YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2007 A.D.

 

Problem Archives Index

Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110

Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110

Probs & ans . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190 . 191-200

Probs only . . . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190 . 191-200

Problems & answers . . . . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261-270 . 271+

Problems only . . . . . . . . . . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261-270 . 271+

 

Browse the complete problem list, check out the weekly leader board,

or go back and work on this week's problem!

 

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