A man is on a game show with the top prize of 1,000,000. However, for every percent of money won he has to give that amount of money to taxes. (If he won 100,000, since it is 10% of 1,000,000 he would have to give away 10,000 and keep 90,000). What is the most amount of money he could possibly win if every money value is a possibility.
2006-07-12 16:58:25 · 9 answers · asked by Russel 1 in Science & Mathematics ➔ Mathematics
He can keep $250,000 if he wins $500,000 (50%). That's the max.
2006-07-12 17:13:27 · answer #1 · answered by Jim H 3 · 1⤊ 0⤋
Let X be the amount he won. Then he will give (X/1,000.000) % of X. i.e his net worth will be
f(x) =X - X(X/1,000,000) = X - (1/1,000,000)*X^2
This is the function that will determine how much money he'll get.
We can solve this two ways.
1) From Algebra:
The function is a parabola opening downwards. Hence the Vertex will be the maximum income.
i.e X = -b/2a = 1,000,000/2 = 500,000
Hence maximum income is f(500,000) = $250,000
OR
2) From Calculus:
Doing the first derivative test gives,
f'(x) = 1- x/500,000 = 0 --> = 500,000
Plugging this value of onto the original equation gives $250,000.
2006-07-13 09:46:35 · answer #2 · answered by haroun i 2 · 0⤊ 0⤋
This is a problem of finding the maximum of a function that describes the situation given. Assume that x is the gross prize money so that what goes to taxes is x/1000000 or x/10^6. The function for net prize money would be
P(x) = x - x(x/10^6) = x -x^2/10^6
The maximum of the function can be found be finding the derivative and setting it equal to zero. Thus,
P'(x) = 1 - (2/10^6) x = 1 - x/500000
If P'(x) = 0, then
1 - x/500000 = 0 => x/500000 = 1
x = 500000
The maximum net prize money is where the gross is 500000, so that the maximum net prize is 250000. If x is greater than 500000, the increasing taxes taken will grow faster than the increasing gross.
Hope that helps.
2006-07-12 17:16:03 · answer #3 · answered by Ѕємι~Мαđ ŠçїєŋŧιѕТ 6 · 0⤊ 0⤋
What you are realy asking is to maximize the following funtion:
x = amount of money won
x / 1000000 = percentage of money won
x * (1-percentage of money won) = what he gets to keep
Substitute In
x*(1-x/100000)
Expand
x - x^2/100000
Take first derivative
1 - 2x/1000000
Set first derivative = to 0
1 - 2x / 1000000 = 0
1 = 2x/ 1000000
1000000 = 2x
500000 = x
Substitute in x and the answer is 250,000
This is calculus in action
2006-07-12 17:18:37 · answer #4 · answered by lovingdaddyof2 4 · 0⤊ 0⤋
Assuming that each and every one 9 numbers are diverse and that you ought to repeat any decision, there are 9 to the fourth (4 available numbers) for a finished of 6,561 mixtures. Now shall we assume that once you employ a decision, you won't be able to repeat it. that ought to signify you've a decision of 9 numbers for the first slot, 8 for the 2d, 7 for the 0.33, and six for the most suitable for a finished of three,024 mixtures (9x8x7x6).
2016-10-14 10:12:12 · answer #5 · answered by Anonymous · 0⤊ 0⤋
10% is 10% regardles of the increments it's divided into. So the answer is 1,000,000 - 100,000. or 900,000
2006-07-12 17:05:18 · answer #6 · answered by tee_nong_noy 3 · 0⤊ 0⤋
The top prize is 1,000,000 so that is the answer.
2006-07-12 17:04:05 · answer #7 · answered by Anonymous · 0⤊ 0⤋
$ 999,999
so he have pay $90,000 he'll have $909,999
but if he have $1,000,000
he'll give out $ 100,000 and will havejust $900,000
so it's $999,999
2006-07-12 22:45:19 · answer #8 · answered by M. Abuhelwa 5 · 0⤊ 0⤋
$250,000
2006-07-12 17:07:57 · answer #9 · answered by Anonymous · 0⤊ 0⤋