SOLUTION: 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?

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Question 996106: 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?
Let x be the length of the base of the triangle. Write the area as a function of x. [First write the length of the equal-length sides in terms of the base, x, then write the height of the triangle in terms of the base.]
A(x) =

Length of base = ? miles
Length of the other two (equal-length) sides = ? miles each
Thank you
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Done part-way through but not including the derivative & maximization work:

Draw a triangle, base x, height h, the two equal sides each d. Cut x exactly in half forming two of x%2F2.

The drawing allows you to first form two equations.
system%282d%2Bx=3%2C%28x%2F2%29%5E2%2Bh%5E2=d%5E2%29

Starting with perimeter equation solve for d in terms of x.
This part will be d=%281%2F2%29%283-x%29.

Substitute this formula for d into the pythagorean relation ship equation and solve for h:

%28x%2F2%29%5E2%2Bh%5E2=%28%281%2F2%29%283-x%29%29%5E2
h%5E2=%281%2F2%29%5E2%2A%283-x%29%5E2-%28x%2F2%29%5E2
h%5E2=%281%2F4%29%28%283-x%29%5E2-x%5E2%29
h=%281%2F2%29sqrt%289-6x%2Bx%5E2-x%5E2%29
highlight_green%28h=%281%2F2%29sqrt%289-6x%29%29

A(x) will be the area function.
A%28x%29=%281%2F2%29x%2Ah, and now you have a formula for h.
highlight_green%28A%28x%29=%281%2F2%29x%2Asqrt%289-6x%29%29

Omitting the differentiation steps but starting with the product rule, I am finding dA%2Fdx=%289-9x%29%2F%284sqrt%289-6x%29%29; and you can continue the maximization process...

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

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?
Let x be the length of the base of the triangle. Write the area as a function of x. [First write the length of the equal-length sides in terms of the base, x, then write the height of the triangle in terms of the base.]
A(x) =

Length of base = ? miles
Length of the other two (equal-length) sides = ? miles each
Thank you
Length of base: x

Sum of equal sides: 3 – x

Length of each equal side: %283+-+x%29%2F2

Using the pythagorean formula: a%5E2+%2B+b%5E2+=+c%5E2, we get the height (H), as: H = sqrt%289+-+6x%29%2F2



Area, or A(x): %281%2F2%29bh, or %281%2F2%29+%2A+x+%2A+%28sqrt%289+-+6x%29%2F2%29

highlight_green%28A%28x%29+=+x+%2A+%28sqrt%289+-+6x%29%2F4%29%29