A maths problem

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thread starter yaminoodles Apr 29 2010 + PM | QUOTE | PERMALINK | REPORT

Can you guys help me with this please o.O The sum of 2 numbers is 20. Find the numbers and their product if their product is a maximum. It would be greatly appreciated if you could show me some working out as well :D

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ooAndu 04/29/10 + PM | QUOTE | PERMALINK | REPORT

a + b = 20, so b = 20 - a c = ab = a(20 - a) = 20a - a^2 Derive. = 20 - 2a Solve for a when c = 0 20 - 2a = 0 20 = 2a a = 10 sub in a b=10 therefore a=10 b=10 I hope I'm right...

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thisisbranny 04/29/10 + PM | QUOTE | PERMALINK | REPORT

What do they mean by maximum? I was guessing 10 and 10, but need to make sure what's maximum mean

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ingem 04/29/10 + PM | QUOTE | PERMALINK | REPORT

It's 10 and 10.

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SilentXynh + PM | QUOTE | PERMALINK | REPORT

Let the two numbers be x and y. x + y = 20 y = 20-x Let product of the two numbers be z. z = xy Substitute in y = 20-x so you get: z = x(20-x) z = 20x - x^2 Differentiate the function with respect to x to get: dz/dx = 20 - 2x To make sure that it's a maximum, take the second derivative to get d2z/dx^2 = -2 It's concave down, so we know now that it's for sure a maximum. To get the maximum, we find the turning point, which occurs when dz/dx = 0 20 - 2x = 0 x = 10 So y = 10 Product = 100 EDIT: ARG NINJA'D

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yaminoodles 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] ooAndu : a + b = 20, so b = 20 - a c = ab = a(20 - a) = 20a - a^2 Derive. = 20 - 2a Solve for a when c = 0 20 - 2a = 0 20 = 2a a = 10 sub in a b=10 therefore a=10 b=10 I hope I'm right... [/quote] Thanks :D but what does the c stand for o.o

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Jake305 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] ooAndu : Crap load of working out. a=10 b=10 [/quote] Yes, im pretty sure i got that answer by dividing 20 by 2... Just face it, mathematics is over-complicating the simple.

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HermanLi123 04/29/10 + PM | QUOTE | PERMALINK | REPORT

You guys are a bit overdoing it. You could just bring up the concept of limits, and that would work.

ooAndu + PM | QUOTE | PERMALINK | REPORT

@TS c is the product lol. And for those who don't know, a maximum product is the highest product you can get from 2 numbers which in this case add up to 20. 10x10=100, 11x9=99 etc. So 10 and 10 is the correct answer. @2above You'd think so in this case, but you'd need to work it out with more complicated questions. Better to learn the foundations with easy questions.

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thisisbranny + PM | QUOTE | PERMALINK | REPORT

[quote] SilentXynh : Differentiate the function with respect to x to get: dz/dx = 20 - 2x To make sure that it's a maximum, take the second derivative to get d2z/dx^2 = -2 It's concave down, so we know now that it's for sure a maximum. To get the maximum, we find the turning point, which occurs when dz/dx = 0 20 - 2x = 0 x = 10 So y = 10 Product = 100 [/quote] I understood everything til this, what's the d for? o.o

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SilentXynh + PM | QUOTE | PERMALINK | REPORT

[quote] ooAndu : @TS c is the product lol. And for those who don't know, a maximum product is the highest product you can get from 2 numbers which in this case add up to 20. 10x10=100, 11x9=99 etc. So 10 and 10 is the correct answer. @2above You'd think so in this case, but you'd need to work it out with more complicated questions. [/quote] I know it's a pretty obvious thing, but with your solution, you didn't formally show that it's a maximum :P EDIT: @ Above: Differentiating with respect to x. Differentiating z with respect to x gives you dz/dx

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yaminoodles 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] ooAndu : = 20a - a^2 Derive. = 20 - 2a [/quote] How did u get from there to there :P thanks.

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thisisbranny 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] SilentXynh : I know it's a pretty obvious thing, but with your solution, you didn't formally show that it's a maximum :P EDIT: @ Above: Differentiating with respect to x. Differentiating z with respect to x gives you dz/dx [/quote] I see..

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SilentXynh 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] yaminoodles : How did u get from there to there :P thanks. [/quote] He took the derivative of the function with respect to a.

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ooAndu 04/29/10 + PM | QUOTE | PERMALINK | REPORT

[quote] yaminoodles : How did u get from there to there :P thanks. [/quote] Oh come on, you don't even know how to derive? Easiest and simplest way to do it without needing to know anything: 20a - a^2 You move the 'power' of the a to the front and multiply, while subtracting the power by 1. Example: 20a is technically 20a^1. So you move it to become 1x20a^0 (1 is moved, and 1 is subtracted). Hence becoming 20 since 1x20=20 and a^0=1. a^2 turns into 2xa^1 (again, the power is moved to the front and the power on the a is subtracted by 1). So it turns out to be 2a. Yeah it technically doesn't explain anything at all but it's the easy way to know

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