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1. If a box with a square base and open top is to have a volume of 4ft (cubed), find the dimensions that require the least material.
2. If the perimeter of an isoceles triangle is 18cm, find the maximum area of the triangle.
3. Find the smallest product possible if the difference between two numbers is 27.
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I just helped someone with some optimization...let's see what I can do...
1) Equations first (well, technically I usually draw first, then equations, but it's kinda hard to do here...) - h(eight), s(ide), V(olume), S(urface) A(rea)
V = hs^2
SA = s^2 + 4hs
Known: V = 4
4 = hs^2 --> 4/s^2 = h
SA --> SA = s^2 + 16/s
SA' = 2s - 16/s^2
SA' = 0 at min and max
0 = 2s - 16/s^2 --> s^3 = 8 --> s = 2
Plug back into V equation to find h
2) Equations - a(length of equal sides), b(third side), P(erimeter), A(rea)
P = 2a + b
A = (1/2)b*sqrt(a^2 - b^2)
Known: P = 18
Solve P equation for a or b, plug into the A equation, find derivative of A equation, solve derivative for A' = 0, find variable, solve for other in P equation
3)Equations - d(ifference), f(irst number), s(econd number), p(product)
f - s = d
fs = p
Known: d = 27
Hint: One number will be negative.
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The answer to the third question is 0 because 27-0=27 and 27*0=0. You can't use negatives because subracting a negative is the same as adding and subtracting from a negative makes a greater negative number.
Actually, you could do 1-(-26) or something like that