Monthly Archives: August 2016

Time has stopped !!!

My clock has stopped !! ūüė¶ ūüė¶ ūüė¶

My only time piece is a wall clock. One day I forgot to wind it and it stopped. I went to visit a friend whose watch is always correct, stayed awhile, and returned home. There I made a simple calculation and set the clock right.

How did I do this when I had no watch on me to tell how long it took me to return from my friend’s home?

-Nalin Pithwa 

ūüôā ūüôā ūüôā

Tea break please !!

Mathematics is a fantastic escape from the humdrum and mundane monotonous affairs of life !!

Express 100 with five 1’s. Express 100 three ways with five 5’s. You can use brackets, parentheses, and these signs: +, -, x and \div.

Have fun !

Nalin Pithwa

A Russian Duel !!

(From a translated Russian literature)

The Mathematics Circle in our school had this custom: Each applicant was given a simple problem to solve — a little mathematical nut to crack, so to speak. You became a full member only if you solved the problem.

An applicant named Vitia was given this array:

1 \hspace{0.1in} 1 \hspace{0.1in} 1 \hspace{0.1in}

3 \hspace{0.1in} 3 \hspace{0.1in} 3 \hspace{0.1in}

5 \hspace{0.1in} 5 \hspace{0.1in } 5 \hspace{0.1in}

7 \hspace{0.1in} 7 \hspace{0.1in} 7 \hspace{0.1in}

9 \hspace{0.1in} 9 \hspace{0.1in} 9 \hspace{0.1in}

He was asked to replace 12 digits with zeros so that the sum would be 20. Vitia thought a little, then wrote rapidly:

0 \hspace{0.1in} 1 \hspace{0.1in} 1 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 9 \hspace{0.1in}

----

X \hspace{0.1in} 2 \hspace{0.1in} 0 \hspace{0.1in}

and

0 \hspace{0.1in} 1 \hspace{0.1in} 0 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 3 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 7 \hspace{0.1in}

0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}

----

X \hspace{0.1in} 2 \hspace{0.1in} 0 \hspace{0.1in}.

He smiled and said, “If you substitute just ten zeros for digits, the sum will be 1111. Try it!”

The Circle’s president was taken aback briefly. But, he not only solved Vitia’s problem, but improved upon it:

“Why not replace only nine digits with 0’s —- and still get 1111?”

As the debate continued, ways of getting 1111 by replacing 8, 7, 6, and 5 digits with zeros were found.

Solve the six forms of this problem.

More duels for you to prove your mettle are coming !!

Nalin Pithwa

From Russia with love of numbers !!

Problem:

There are three ways to add four odd numbers and get 10:

1+1+3+5=10

1+1+1+7=10

1+3+3+3=10

Changes in the order of numbers do not count as new solutions.

Now add eight odd numbers to get 20. To find all eleven solutions, you will need to be systematic.

-Nalin Pithwa

PS: I found this gem in a Russian book (translated).

Applications of mathematics in sports — careers in mathematics

In India, as is well-known, the most popular out-door sport is cricket. There are various statistics associated with cricket, as with any other sport. Especially, in cricket, we find the following parameters discussed about in every match:

  • General statistics (matches, catches, stumpings)
  • Batting statistics (innings, Not Outs, runs, highest score, batting average, centuries, half-centuries, balls faced, strike rate, run rate, Net run rate).
  • Bowling Statistics: Overs. Bowls, Maiden Overs, Runs, Bowling Analysis, Wickets, No balls, Wides, Bowling Averages, Strike Rate, Economy Rate, Best Bowling In Innings, Best Bowling in Match, Five Wickets in an Inning, Ten Wickets in a match,
  • Dynamic and graphical statistics: The advent of saturation television coverage of professional cricket has provided an impetus to develop new and interesting forms of presenting statistical data to viewers. Television networks have thus invented several new ways of presenting statistics.
  • The T-20’s and ODI’s might have different statistics because their purposes are different. The county matches have a yet different purpose.

There are numerous books that deal with the applications of mathematics in sports. Three very good ones are:

  • de Mestre, N (1990): The Mathematics of Projectiles in Sport. Cambridge, UK: Cambridge University Press.
  • Hart, D. and T. Croft (1988). Modelling with Projectiles, Chicester, UK: Ellis Horwood.
  • Townend, M. S (1984), Mathematics in Sport, Chicester, UK: Ellis Horwood.

More later,

Nalin Pithwa

Apples from Moscow !!

Here is a puzzle, which I found in a Russian book of puzzles!!

Five Apples

Five apples are in a basket. How do you divide them among five girls so that each girl gets an apple, but one apple remains in the basket?

Well, an apple a day keeps the doctor away; a puzzle a day keeps the brain fresh !!

Please share your solutions!

-Nalin Pithwa

Applications of sinc function

The occurrence of the function \frac{\sin {x}}{x} in calculus is an isolated event. The function arises in diverse fields such as quantum physics (where it appears in solutions of the wave equation) and electrical engineering(in signal analysis and DSP filter design) as well as in the mathematical fields of differential equations and probability theory.

-Nalin Pithwa.

Sunday Coffee Time Puzzles !

  1. A train, an hour after starting, meets with an accident which detains it an hour, after which it proceeds at three-fifths of its former rate and arrives 3 hours after time; but, had the accident happened 50 miles further on the line, it would have arrived 1.5 hours sooner: find the length of the journey.
  2. A body of men were formed into a hollow square, three deep, when it was observed, that with the addition of 25 to  their number a solid square might be formed, of which the number of men in each side would be greater by 22 than the square root of the number of men in each aide of the hollow square: required the number of men.
  3. A set out to walk at the rate of 4 miles an hour; after he had been walking 2\frac{3}{4}, B set out to overtake him and went 4\frac{1}{2} miles the first hour, 4\frac{3}{4} miles the second, 5 the third, and so gaining a quarter of a mile every hour. In how many hours would he overtake A?
  4. A man wishing his two daughters to receive equal portions when they came of age bequeathed to the elder the accumulated interest of a certain sum of money invested at the time of his death in 4 per cent stock at 88; and to ¬†the younger he bequeathed the accumulated interest of a sum less than the former by \pounds 3500 invested at the same time in the 3 per cents, at 63. Supposing their ages at the time of their father’s death to have been 17 and 14, what was the sum invested in each case, and what was each daughter’s fortune?
  5. A and B travelled on the same road and at the same rate from Huntington to London. At the 50^{th} milestone from London, A overtook a drove of geese which were proceeding at the rate of 3 miles in 2 hours; and two hours afterwards met a waggon, which was moving at the 46^{th} milestone, and met the waggon exactly 40 minutes before he came to the 31^{st} milestone. Where was B when A reached London?
  6. To complete a certain work, a workman A alone would take m times as many days as B and C working together; B alone would take a times as many days as A and C together; C alone would take p times as many days as A and B together; show that the numbers of days in which each would do it alone are as m+1:n+1:p+1.
  7. A traveller set out from a certain place, and went 1 mile the first day, 3 the second, 5 the next, and so on, going every day 2 miles more than he had gone the preceding day. After he had been gone three days, a second sets out, and travels 12 miles the first day, 13 the second and so on. In how  many days, will the second overtake the first? Explain the double answer.
  8. A number of persons were engaged to do a piece of work which would have occupied them 24 hours if they had commenced at the same time; but instead of doing so, they commenced at equal intervals and then continued to work till the whole was finished, the payment being proportional to the work done by each: the first comer received eleven times as much as the last; find the time occupied.
  9. There are three towns A, B and C; a person by walking from A to B, driving from B to C, and riding from C to A makes the journey in 15.5 hours; by driving from A to B, riding from B to C, and walking from C to A, he could make the journey in 12 hours. On foot he could make the journey in 22 hours, oh horseback in 8.25 hours, and driving in 11 hours. To walk a mile, ride a mile, and drive a mile he takes altogether half an hour; find the rates at which he travels, and the distance between the towns.
  10. In a mixed company consisting of Poles, Turks, Greeks, Germans and Italians, the Poles are one less than the one-third of the number of Germans, and three less than half the number of Italians. The Turks and Germans outnumber the Greeks and Italians by 3; the Greeks and Germans form one less than the half the company; while the  Italians and Greeks form seven-sixteenths of the company; determine the number of each nation.

More fun later,

Nalin Pithwa

Some Applications of Derivatives — Part II

Derivatives in Economics.

Engineers use the terms velocity and acceleration to refer to the derivatives of functions describing motion. Economists, too, have a specialized vocabulary for rates of change and derivatives. They call them marginals.

In a manufacturing operation, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost (c) with respect to a level of production (x), so it is dc/dx.

For example, let c(x) represent the dollars needed needed to produce x tons of steel in one week. It costs more to produce x+h units, and the cost difference, divided by h, is the average increase in cost per ton per week:

\frac{c(x+h)-c(x)}{h}= average increase in cost/ton/wk to produce the next h tons of steel

The limit of this ratio as h \rightarrow 0 is the marginal cost of producing more steel when the current production level is x tons.

\frac{dc}{dx}=\lim_{h \rightarrow 0} \frac{c(x+h)-c(x)}{h}= marginal cost of production

Sometimes, the marginal cost of production is loosely defined to be the extra cost of producing one unit:

\frac{\triangle {c}}{\triangle {x}}=\frac{c(x+1)-c(x)}{1}

which is approximately the value of dc/dx at x. To see why this is an acceptable approximation, observe that if the slope  of c does not change quickly near x, then the difference quotient will be close to its limit, the derivative dc/dx, even if \triangle {x}=1. In practice, the approximation works best for large values of x.

Example: Marginal Cost

Suppose it costs c(x)=x^{3}-6x^{2}+15x  dollars to produce x radiators when 8 to 30 radiators are produced. Your shop currently produces 10 radiators a day. About how much extra cost will it cost to produce one more radiator a day?

Example : Marginal tax rate

To get some feel for the language of marginal rates, consider marginal tax rates. If your marginal income tax rate is 28% and your income increases by USD 1000, you can expect to have to pay an extra USD 280 in income taxes. This does not mean that you pay 28 percent of your entire income in taxes. It just means that at your current income level I, the rate of increase of taxes I with respect to income is dT/dI = 0.28. You will pay USD 0.28 out of every extra dollar you earn in taxes. Of course, if you earn a lot more, you may land in a higher tax bracket and your marginal rate will increase.

Example: Marginal revenue:

If r(x) = x^{3}-3x^{2}+12x gives the dollar revenue from selling x thousand candy bars, 5<= x<=20, the marginal revenue when x thousand are sold is

r^{'}(x) = \frac{d}{dx}(x^{3}-3x^{2}+12x)=3x^{2}-6x+12.

As with marginal cost, the marginal revenue function estimates the increase in revenue that will result from selling one additional unit. If you currently sell 10 thousand candy bars a week, you can expect your revenue to increase by about r^{'}(10) = 3(100) -6(10) +12=252 USD, if you increase sales to 11 thousand bars a week.

Choosing functions to illustrate economics.

In case, you are wondering why economists use polynomials of low degree to illustrate complicated phenomena like cost and revenue, here is the rationale: while formulae for real phenomena are rarely available in any given instance, the theory of  economics can still provide valuable guidance. the functions about which theory speaks can often be illustrated with low degree polynomials on relevant intervals. Cubic polynomials provide a good balance between being easy to work with and being complicated enough to illustrate important points.

Ref: Calculus and Analytic Geometry by G B Thomas.

More later,

Nalin Pithwa

 

Some Applications of Derivatives — Part I

Sensitivity to Change.

When a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x. The derivative f^{'}(x) is a measure of the senstivity to change at x.

Example:

The Austrian monk Gregor Johann Mendel (1822-1884), working with garden peas and other plants, provided the first scientific explanation of hybridization. His careful records show that if p (a number between 0 and 1) is the frequency of the gene for smooth skin in peas (dominant) and (1-p) is the frequency of the gene for wrinkled skin in peas, then the proportion of smooth-skinned peas in the population at large is

y = 2p(1-p)+p^{2}=2p-p^{2}

The graph of y versus p is an inverted parabola. If you plot and see, it suggests that the value of y is more sensitive to a change in p, when p is small than when p is large. Instead, this is borne out by the derivative graph which shows that (\frac{dy}{dp}) is close to 2, when p is  near zero, and close to zero, when p is near 1.

More later,

Nalin Pithwa

PS: why peas wrinkle

British geneticists had discovered that the wrinkling trail comes from an extra piece of DNA that prevents the ¬†gene that directs starch synthesis from functioning properly. With the plant’s starch conversion impaired, sucrose and water build up in the young seeds. As the seeds mature, they lose much of this water, and the shrinkage leaves them wrinkled.