- 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.
- Dan's Directriconix (back to top)
- 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.
- Stock Profits to the Max ! (back to top)
- 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)
- Best Best-Fitting Curve ? (back to top)
- Show your steps, formulas, and reasoning.
- problem #274 - posted
wednesday, march 26, 2008
- Factorialus Sum Consecutivos. (back to top)
- Okay, I don't know Latin. But I do know that 5! = 120 = 39 + 40 + 41 . \/ (s.c.p.i.) \/
- a) How many ways can 5! be written as a sum of (two or more) consecutive positive integers?
- b) Describe a procedure to count the number of ways that any given natural number, m, can
- be expressed as an s.c.p.i. . c) How many ways can 74 factorial (74!) be written as an s.c.p.i.?
- Factorialus Sum Consecutivos. (back to top)
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- problem #277 - posted
wednesday, august 13, 2008
- Group the Numbers! (back to top)
- Use the 3 numbers in each group in an expression so that the 3 answers are equal.
- Each part has a unique solution. Example: group 1: 6, 18, 20; group 2: 21, 27, 30 ; group 3: 31, 45, 46:
- Solution: 6 / (20 - 18) = 27 / (30 - 21) = 45 / (46 - 31) = 3. Now do these:
- Group the Numbers! (back to top)
- a) group 1: 5, 14,
15; group 2: 17, 19, 34 ; group 3: 20, 34, 37 ;
- b) group 1: 5, 6, 23; group 2: 14, 22, 32 ; group 3: 32, 46, 47 .
- problem #278 - posted
monday, september 15, 2008
- Some Smallest Sums - Here are two cute number puzzles for you: (back to top)
- a) Given that the product of five consecutive positive integers divided by the sum of the
- five integers is divisible by 100, what is the smallest possible sum of the five integers?
- b) Four positive integers r, s, t, u have a product r s t u = 10!.
- What is the smallest possible sum r + s + t + u ? Show your steps, formulas, and reasoning.
- Note: 10! means 10 factorial, not just an emphatic ten.
- Some Smallest Sums - Here are two cute number puzzles for you: (back to top)
- problem #279 - posted
sunday, october 19, 2008
- last problem of season 11! (not factorial) . . . new season starts with Problem 280 ! (back to top)
- Triangles .: and Squares :: See if you can get all of these ( partial credit for partial answers )
- a) Where can you find or easily calculate the perfect squares in Pascal's Triangle?
- b) Notice 1 is a perfect square and also a triangular number. Find the next three
- . . triangular squares. (Discuss a method besides a brute force Excel search.)
- c) Prove that any triangular number > 1 is the sum of a square and two equal
- . . triangular numbers. Show steps, formulas & reasons.
- last problem of season 11! (not factorial) . . . new season starts with Problem 280 ! (back to top)
