dansmath > ask dan > **questions and answers page****dan's q**uestions**and****a**nswers**page****I have expanded this section; check out the new questions, or ask your own!****Also you might want to look at some reader comments.**- Basic Skills Questions (or go to basic lessons)
- B1 - Writing a googolplex
- B2 - Fraction woes
- Algebra Questions (or go to algebra lessons)
- A1 - I need your help fast
- A2 - The 'two trains' problem
- A3 - Miminizing a weird function
- Trig Questions (or go to trig lessons)
- T1 - Sine, Cosine, Tangent
- Calculus Questions (or go to calculus lessons)
- C1 - Graphs of functions and derivatives
- Miscellaneous Questions (or go to 'other' lessons)
- M1 - Profit curve for econ?
- M2 - Too big for calculator!
- M3 - Scarecrow was not a wizard
- M4 - Good grapefruit prediction
- Basic Skills Questions
**Question B1 - Writing a googolplex**[ top of page ]**Dear Dan - Our math professor at U of Louisville posed this question:****If one million zeroes can be written on the front and back of a sheet of****paper, how many sheets of paper are necessary for a googol of zeros?**(which was not a problem - it's 10 to

We have to come up with answer- the 94th power sheets of
paper)
**and we have to be able to explain the** **problem and solution to a 12-year old. How do you explain such a****large number to someone - what can you relate it to? - Karen****Hi Karen**-- Thanks for writing -- interesting question. A googol is 10^100 and a 1 with- a googol zeroes after it is called a 'googolplex.'
- I thought maybe you could relate the
**thickness**of the paper (a ream of paper, 500 sheets, is

around 2" so each sheet is about .004" thick) and a large distance, say from the earth to the

moon (around 240,000 miles). I'll let you figure out how the number of sheets of paper

from here to the moon compares with 10^94 (still nowhere close?), so talk about how many

trillions of stacks of paper like that you'd need? - Another aspect is
**time**: if you wrote 1 number per second (or a printer spit out one sheet

per min), how many centuries would it take if all 6 billion people wrote (or had printers)?

By the way, 1 million seconds = 12 days and 1 billion sec = 31 years, approx! - Ask the 12-yr-old how far away tjey've
traveled (say New York, 3000 miles), then maybe

relate the earth-moon distance to that (80 times as far). -- Hope this helps! - Meanwhile don't waste paper writing out a googolplex in base 10. ;-}
- -- Dan the Math Man -- www.dansmath.com

**Question B2 - Fraction Woes**[ top of page ]**Hello Dave, I am a 7th grader and having a hard time with fractions.****Could you point me in the direction of a helpful site? Thanks, Will**

(Coming to you from Colorful Colorado, home of the Awesome Avalanche!!!)**Hi Will**... (My name's Dan but I'll answer to Dave...)- I was going to point you to my site, www.dansmath.com
, but you're right, I don't

have much currently there on fractions. I will give you some pointers but I would

suggest www.google.com -- type in 'Fraction math help' and I swear it will give

you some excellent active choices. Look for some on-line quizzes as well. - Here's one I just found: www.mathleague.com/help/fractions/fractions.htm
- You can also go to 'expert sites' such as www.askme.com and look for message

items or ask questions there. I'll be happy to answer specific questions or problems

for you too. - As for fractions themselves: here's a quick review: The top
is the '
**numerator**', and the - bottom is the '
**denominator**'; in 5/7 the 5 is the numerator and the 7 is the denominator. - Fractions with the same denominator can easily be added or
subtracted: keep the same

denominator and add or subtract the numerators. 5/7 + 3/7 = (5 + 3)/7 = 8/7 or 1 1/7. - Fractions can be converted or reduced: 4/6
= 2/3 (divide top & bottom by 2),

or 5/7 = 15/21 (multiply top & bottom by 3). - Fractions with different denoms have to be converted to a
common denom (LCM)

2/5 + 1/3 = 6/15 + 5/15 = 11/15 , etc. - Multiplying two fractions: multiply tops and bottoms separately:

2/3 * 5/8 = (2*5)/(3*8) = 10/24 = 5/12. - Dividing two fractions: iinvert the second fraction and then
multiply:

(4/9) / (3/7) = (4/9) * (7/3) = 28/27 or 1 1/27. - Then there's the whole connection with fractions and decimals and percents...
- --Dan the Math Man
**Algebra Questions****Question A1 - I need your help fast**[ top of page ]**Hi Dan, I live in Cleveland, Ohio, and I need your HELP.****Can you teach me algebra? I need to learn, and learn fast,****I am applying for an apprenticeship program and the****requirements are algebra. I really need this opportunity,****so can you please teach me? -- Brian****Hi Brian**- Algebra is a very important subject ; I call it the 'gateway to the

technical world'. If this apprentice program is asking you to know algebra, it- must mean it's for a job that uses it and depends on it. This means you can't
- learn it hastily or all at once. If I wanted to be fluent in speaking Russian, I
- wouldn't be able to do it in a few days or weeks.
- But I do have a lot of lessons on my
website, in all levels; arithmetic, prealgebra,

beginning alg. and on up. Check them out at www.dansmath.com --> - math lessons --> Basic Skills. There are also lots of links on my site or go to
- www.google.com and type 'algebra help.' It's a great way to find stuff.
- I also have a great textbook that could
be used for self-study, check it out at my

site at 'meet dan' then click textbook. If you want to order the book online, get - the details ready (Bach/Leitner Prealgebra, Houghton Mifflin) and then go to
- bookcenter.dvc.edu
and order with a credit card, they ship!
**-- Da**MathMa**n**

**Question A2 - The Two Trains Problem**[ top of page ]**Dear Dan - two trains leave stations 100 miles apart.****"A" train leaves at 12:00 and "B" train leaves at 1:00.****when, and at what mile, will they collide? there will be****no loss of life. > mark**

i would appreciate any help. big test on sat am. thank you.

**Hi Mark**- I mean mark.- If we knew the
**speed**of each train then we could set up distances: - t = hours past 12:00 ; and the trains
have gone dA and dB:

dA = (speed of A)*(t hours) ; dB = (speed of B)*(t - 1 hours)

because B starts at 1:00, 1 hr later, so 1 hr less time. (both are d = r t.) - Now our equation is dA + dB = 100 miles,
but we can't solve for t unless

we have some real numbers for those speeds. If you forgot to include those,

that's where you'd use them; if the speeds weren't given in the problem

then it wasn't a well-phrased question. - One thing we do know is if A is going more than 100 mph, the collision
- will occur at B's home station, maybe
with loss of life!
**- Dan the Math Man**

**Question A3 - Miminizing a Weird Function**[ top of page ]**Dear Dan, How would you find the minimum of****y = square root of (x^2 + 4) + square root of (x^2 - 6 +10).****Do you have to use calculus? - Tara****Hi Tara.**- You
**can**use calculus, to minimize y = sqrt(x^2 + 4) + sqrt(x^2 - 6x + 10)

by taking the derivative dy/dx and setting it equal to zero. That looked icky,

but do-able. But you'd prefer a calculus-free approach... - Or you can type the function into your
**graphing calculator**or some

computer algebra system like Mathematica (see my website), and try

to spot a min. This one looks like**x = 2**gives a smallest value of about

y = 4.25 or so. Then you could go back to the formula and see if you

could complete the square to see the min at x = 2:

y - sqrt[x^2 + 4] = sqrt[(x-3)^2 + 1] ; square both sides, simplify,

isolate the square root term, square both sides again, and the result should

have some sort of factor of (x - 2) if my intuition is right.- I'll let you have fun with the algebra. Hope this helps!
**- Dan the Man** - Trigonometry Questions
**Question T1 - Sine, Cosine, Tangent**[ top of page ]**Dan, please help!! I have been trying to figure out sine, cosine, and tangent****for about 2 months, and I can't quite get it. I just don't understand how to****get to the answer from: sin/cos/tan(angle)=(side length over side length)**

I also do not know how to do sin/cos/tan when you only have one side length.**Please, please, please help!!! ~ Michelle ~**- Hi Michelle - Have you heard of '
**SOHCAHTOA**' ? - Draw a right triangle with right angle at the lower right; the angle T at the left;
- the Opposite (vertical) and the Adjacent (base) and the Hypotenuse (rising diagonal).
- It means Sine = Opp / Hyp , Cosine = Adj / Hyp , Tangent
= Opp / Adj .

You might like sin(T) = y / r , cos(T) = x / r , tan(T) = y / x . - Along with the Pythagorean Theorem x^2 + y^2 = r^2 , you
can figure

out the whole story just by knowing any one side, and one other angle. - Please visit my (free!) web lessons at www.dansmath.com for more ; go to the
- Precalculus area and click Trig. There are pictures and even some animations
- there for you to enjoy / learn from. And write back any time with further or more
- specific questions.
**-- Dan the Math Man strikes again**

**Calculus Questions****Question C1 - Graphs of Functions and Derivatives**[ top of page ]**Dear Dan: Could you explain the principles on graphs with****relation to their derivative? - Shane****Hi Shane**- Happy to 'explane'; thanks for asking.- The
**derivative**of y = f(x) at a point (a, b) is the slope of the tangent line

to the graph at the point (a, b). This means that b = f(a) and that the - graph and the tangent line have the same slope at x = a.
- But the
**graph**of the derivative is different; it records the slope of the graph

at each point, so if the function is increasing, the derivative is positive, and

if the graph of f(x) is decr, then f '(x) is neg. - I have a
**picture**on my site of a graph, its deriv, and second deriv. The link is: - http://home.earthlink.net/matica.html ; go to the 'options for graphics' section.
- You've inspired me to make a movie of a moving tangent line
along a curve

y = f(x), and a second curve recording the slope f '(x) as it goes along. - Hope this helps a little; try graphing a few pairs {function
, derivative}

on your graphing calculator, like {y = x^2 , y = 2x} to see the relation.**-Dan** **Miscellaneous Questions**

**Question M1 - "Profit
Curve" **[
top of page ]

**Dear Dan, Can you tell me how to start and understand how to solve**- (simple)
**economicproblems using curves and diagrams?****If the volume** **goes up, I realize the price goes down, but how do I state these****circumstances on a graph? --Failed Economics****Dear Econ**-- I'll answer you with an example.- Let's call the volume or quantity '
**q**' and the unit price '**p**', where**p**is a function of**q**. **p**= f(**q**) in function notation.- The revenue
**R**is how much money you take in;**R**=**p*****q**= (price per item) * (number of items). **Fixed Cost:**For example if I sell**q**of my Dan's MathClinic CD's, and the price is $50 each,- then
**p**= 50 and the revenue is**R**=**pq**= 50**q**. - This makes a simple revenue curve (in blue in fig.1).
- If my costs
**C**(**q**) are 300 + 10**q**(see red curve in fig.1; $300 for the CD writer and $10 each in costs per CD), - then my profit
**P**(**q**) =**R**(**q**) -**C**(**q**) = revenue minus cost (see dotted line). - Here P(q) = 50q - (300 + 10q) = 40q - 300. Thus if 40q - 300 > 0 , I make a profit.
- Solving, 40q > 300 ; q > 300/40 = 7.5 ; I'd need to sell at least 8 CD's to make a profit.
- There's
**no maximum profit**here; the more I sell, the more I make. **Variable Cost:**Like you say, Econ, normally the price goes down as q goes up.- Let's say I lower my price by $2 each (for all the CD's), each 10 CD's I sell.
- This means if I sell 10, they're $48 each, sell 20, they're$46 each...
- Now p = f(q) = 50 - (2/10)q = 50 - q/5. Then R = pq = (50 - q/5)*q = 50q - (q^2)/5.
- This graph is shown in fig.2. This revenue curve has a max
at
**q = 125, p = $25**, - R = ($25)(125) =
**$3125 = max revenue**. - If we work in the cost C(q) = 300 + 10q, the profit will be
- P(q) = (50q - (q^2)/5) - (300 + 10q) = - (q^2)/5 + 40q - 300.
- The max profit is where the revenue curve and the cost curve are farthest apart,
- which is at q = 100 ; P(100) = R(100) - C(100) = 3000 - 1300
=
**$1700 max profit**. - Another thing is that the slopes of the curves are equal at the max profit point;
- that slope is called the "marginal"; the marginal revenue varies, while the
- marginal cost is always $10. This is usually called MR = MC (curves are parallel).
- What price
**max**imizes**revenue**? If q = 125 then p = 50 - q/5 =**$25**; - while with a $10 unit cost, if q = 100, then
**max profit**is p = 50 - 20 =**$30. -- Dan**

**Question M2 - Too Big for my Calculator!**[ top of page ]**Dear Dan,**wrote an AOL member,**Help me on this:****"Evaluate 514^623 to 5 significant digits.****Use scientific notation.****Show work." I tried this but my calculator overflowed.****Dear Digits,**- After you clean up your soggy calculator, you can use logarithms (logs) to solve your dilemma.
- The "inverse function" principle says that 10^(log(n)) = n , where log(n) = log base 10 of n.
- For example log(100) = 2 because 10^2 = 100 , so 10^(log(100)) = 10^2 = 100.
- This means that your base 514 = 10^(log(514)).
- Thus 514^623 = (10^(log(514))^623 = 10 ^ (log(514) * 623) ,
- by the exponent law (a^m)^n = a ^ (mn).
- Therefore since log(514) = 2.710963119 (plenty good for 5 sig figs), we get
- 514^623 = 10 ^ (2.710963119 * 623)
- = 10 ^ 1688.930023
- = 10^0.930023 * 10^1688
- = 8.5118337 * 10^1688
- = 8.5118 * 10^1688 , rounded to 5 sig figs.
- This means 514^623 = 851,183,...,..., ,,, ,544. (It has 1689 digits.)
- I worked out the last 3 digits for you as a bonus, keeping just the last
- three digits as I raised (514^7)^89 = (...504)^89 etc.
- Happy mathing! -- Dan the Man -- %;-}
- P.S. In case you want the whole number, I put it into Mathematica:
- 514^623 =
- 8511833779452615656486074261588685321885848269727007926473787021726299150613454731118227788880566249
- 9579341888420948898108481304117665427347156460108301497273089253879842812864182743351005163472035436
- 9091698147158225289509378576257427198682776595751872026868258927180938200163803994427647392715183966
- 8438066177414163107908914764766832312856288614285766846216688067422913987245591809525756132033707727
- 3261381164233095129638631892480829669588620858837489833863435391375254264716570918878595804922844974
- 2310418481180889249912455348674673062108084846174355138107409273786376749378544392942724724802378363
- 6951658353324022475138456232761933645669704603048983352602514049705723158039218408255041542765534406
- 4868815917217198525065278490345695277562166592816967071940449517006873597453967132291299704130560450
- 5723279949674682963922922526606392473508756303281655716042245944851970688413823936680306848990052843
- 4172747125372607254400181641278574025661557833730883587376180848061056695516518084197482071334661602
- 9788779118737691122722049733546225166341998238481750889164581840259239162814525821884767794193810740
- 1944924038500661882235340706891647456413345490150061465554646972653249409508235179697455196265802208
- 1857783642375526377369578939516800481364204375582244936236988756677536432530501400853958311462717242
- 5921231278348064084321169911247088080501465108150050001777239037687892455271721936588067472562372013
- 9226797016031858987695906723992518615071128448123951828909483023185215617848926913001735240233431571
- 8047094403631130175011120909154066915853750235015414384029541658961970066661248857253769694559849148
- 49683683205294629570227145647934661584612656822106000776999057985958189406303177926508544.

**Question M3 - Scarecrow was not a Wizard!**[ top of page ]- Adam, a student of mine, wrote:
**Hi Dan,** **I don't know if you've seen the movie The Wizard of OZ, but at the end of****the****movie when the Scarecrow receives the info that he's already had a brain,****he****says this, "The sum of the square roots of any two sides of an isosceles****triangle is equal to the square root of the remaining side. Oh joy, rapture!****I've got a brain!" But this is only true when you have the two sides and are****finding the hypotenuse. If you have the hypotenuse and are finding one of****the other legs then you subtract, so..if the writer of The Wizard of OZ knew****anything about math he woulda said,"The sum of the square roots of the two****legs not including the hypttenuse an isosceles triangle is equal to the square****root of the remaining side. Oh joy, rapture! I've got a brain!" I guess he****really didn't get a brain after all. Hehe... Adam****Hey, Adam.**Thanks for the reference. Yes I 've seen that scene many times, and it always- bugged me, not because of any subtraction you might need to do, but because of the incorrect
- use of the word "isosceles," which isn't "right." But you're right, he shouldn't have said "any
- two sides." But he also should have said "square"
of the sides, not "square root.
**-- Dan** - P.S. I sent this into the "Internet Movie Database Bloopers page."
- Ok, what about an "isosceles right triangle"? Are there any? Are there any with all three sides
- integers? (Pythag. triple) Anything close, within 1 degree? Think about these, ok?
- Also, you can look up the real Pythagorean Theorem on my Trigonometry page.

**Question M4 - Good Grapefruit Prediction**[ top of page ]**Dear Dan - Have a question, if you can help me.****It is estimated that 75% of the grapefruit crop is good, the other****25% have rotten centers which cannot be detected unless the****grapefruit are cut open. The grapefruit are sold in sacks of l0.****If you buy one sack, what is the probability that your sack will****contain at least 9 good grapefruit? What is the average number****of good grapefruit per sack? - Thanks; Wolfe****Hi Wolfe**- Thanks for writing.- This is a question about '
**binomial probability**', meaning there's a repeated experiment - with probability p, like n repeated coin flips with p = 0.5. Here the experiment is
- 'choosing a grapefruit', and the probability of success is p = 0.75, with q = 1 - p = 0.25.
- The
**average**number of 'good' grapefruits in a bag of n = 10 would be m = np = 10(0.75) - =
**7.5 good ones per bag**(75%). There'll never be 7.5 good fruits but that's the average in - the long run.
- The number of ways of '9 fruits being good' is 10: any of the 1st thru 10th g-fruit could
- be the bad one. And the prob of each case happening is p^9 * q^1 because there are 9
- good fruits and 1 bad. So we have P9 = (prob of 9 good) =
(10) * (0.75^9) * (0.25^1);

P9 = 10 * 0.07509 * 0.25 = 0.1877, or about an 18.8% chance. - The chance that all 10 are good is P10 = 1 * (0.75^10) * (0.25^0) = 0.0563. This must
- be added to the previous number: 0.1877 + 0.0563 = 0.2440
, or about
**24.4%.** - There are also tables that can give you this value directly. look for 'binomial probability
- tables' in the back of your book. Hope this helps!
**-- Dan the Stat Man**