Math problem?? on Max and Min!?

Question

1) Two numbers hav a sum of 60. find the numbers if their product is a maxium.

2) A rectangular lot is bordered on one side by a stream and on the other 3 sides by 600m. of fencing, find the dimensions of the lot if its area is a maxium.


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GR 10 math! no calculus

GR 10 math! no calculus

Answer 1

Rewriten to take account of GR10 fact.

1) let the numbers be x and y. So x+y=60 or y = 60-x
We want to maximum P = xy = x(60-x) = 60x - x²
We can write this as
P = 900 - 900 +60x - x² ...... 900 picked as it is (0.5 * 60 / 1)²
P = 900 - (x²-60x+900)
P = 900 - (x-30)²
So P will be a maximum when x = 30
So y = 30
So both numbers are 30 and the product is 900.

Let the length of stream be y and the other side of the lot be x so the total length of fence is 2x + y = 600 (only one y as we don't fence the stream). So y = 600 - 2x
The area is A = xy = x(600 - 2x) = 600x - 2x²
This can be rewriten as:
A = 45000 - 45000 + 600x - 2x² ...... 45000 picked as it is (0.5 * 600 / 2)²
A = 45000 - 2(x² - 300x + 22500)
A = 45000 - 2(x-150)²
This is a maximum when x = 150m
So y = 300m
So Area is 45000m²

Answer 2

let x be one of the number then (60-x) is the other one

y = x(60-x)= -x^2 +60x
it is a parabola that opens downwards, so the vertex is the max
at -b/2a = -60/-2 = 30

Area = x(600-2x) = -2x^2 +600x
vertex = -600/-4 = 150

the lot is 150x300