Question

Suppose a landowner wishes to use 3 miles of fencing to enclose an isosceles triangular region of as large an area as possible. What should be the lengths of the sides of the triangle?

Answer 1

what's the area of an isosceles triangle?

Answer 2

it doesnt tell

Answer 3

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Answer 4

\[h = \sqrt{a ^{2} - \frac{ 1 }{ 4 }b ^{2}}\]
by pythagoras

Answer 5

yes, i got to that

Answer 6

area will be
\[\frac{ 1 }{ 2 }bh\]

Answer 7

you know what h is, so substitute that

Answer 8

also you're given from the ques that 2a + b = 3

Answer 9

from that you can find a in terms of b

Answer 10

hence area A =
\[\frac{ 1 }{ 2 }b \sqrt{\frac{ 3-b }{ 2 } - \frac{ 1 }{ 4 }b ^{2}}\]

Answer 11

differentiate A w.r.t b and set it equal to 0 (first derivative test)

Answer 12

is that clear?

Answer 13

I think finding the right equation was the main thing, after that you can apply the 1st and 2nd derivative rules to find the max or min

Answer 14

but the problem is i got a lot of trouble when differentiating it

Answer 15

because the equation is overly complex, and i can never get a resonable answer

Answer 16

Okay here's what I get after simplifying A
\[A = \frac{ 1 }{ 4 }b \sqrt{9-6b}\]

Answer 17

I think you can easily differentiate this?

Answer 18

thanks, let me try first

Answer 19

\[A' = \frac{ -6 }{ 8 }\frac{ b }{\sqrt{9-6b} } + \frac{ 1 }{ 4 }\sqrt{9-6b}\]

Answer 20

set A' = 0, and find the values of b

Answer 21

\[\frac{ 1 }{ 2 }b \sqrt{\frac{ 3-b }{ 2 }-\frac{ 1 }{ 4 }b ^{2}}\] are you sure about (3-b)/2 part?