IL State math competion question?

Question

A container is designed in the shape of an open rectangular solid ( bottom but no top). The container is to hold 1248cc. The base of the container is required to be a sqeare. Find the number of centimeters in the height of the container if its total surface area is to be the minimum possible. Assume no material needs to be wasted in construction. Express your answer as a decimal rounded to the nearest hundredth!!!

Show me your steps, I know the answer I just dont know how to get there!!!
Only 10 out of 100 got this one right on last years IL State math competion!!!!

Answer 1

It's an optimization problem

so V = 1248 cc
a^2.h = 1248 ( a is one of the side of the bottom square part)

Surface Area = a^2 + 4a.h
(perimeter times height = surface area of the sides)

h= 1248/a^2

S= a^2 + (4992/a)

dS/da = 2a - 4992/(a^2)

= (2a^3 - 4992)/a^2

for mininum dS/da = 0

so

2a^3 - 4992 = 0

a = 13.56

so h = Vol/a^2
= 6.78

so that's the height...

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you need to know calculus for this!


hmm.. anyways, seems like that was just a little mistake..

but see, if u are here in canada.. and u goin to any sciences.. and u have taken math course, Calculus, that is necessary for almost anything..u far sure would get this question...even if u aren't smart..

lol, it's a little strange to see that only 10!!

Answer 2

s = side of base

condition that volume is 1248:
s^2*h = 1248
h = 1248/s^2

Function for area of surface (4 sides and bottom)
f(s) = s^2 + 4*sh = s^2 + 4*s*(1248/s^2) = s^2 + 4992/s

f'(s) = 2s - 4992/s^2 = 0

2s = 4992/s^2, s^3 = 2496, s = 4*39^(1/3)

Use this value to solve for h (from the first equation, and then if you really want a numeric approximation rather than an exact answer use a calculator.

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"no man" is not quite right. He made the mistake of approximating the solution for the side (his a) and using that to solve for h. It is very possible this will give a solution for h that is off by a hundredth or more in either direction. He should have used the mathematically correct actual value for a to compute h, and then approximate that.

I tried this to check in this case. His answer, using a = 13.56, gives h = 6.787271255906231, rounds to 6.79. Keeping the cube root in the answer until the final result gives h = 6.782422886028335, rounds to 6.78. If the test is strict about the answer then his answer is incorrect, despite his work all being right.

Alternatively, using a better approximation to the intermediate result is OK. I think if he uses r to 4 decimal places then his result will be correct to 2 places.

Answer 3

You're VERY Observant. -And usually Right !! :o The Problem IS; for the beyond few Decades on this Country- Our Society & its Leaders, have inspired the Educational System to advertise extra Business & Service similar Subjects- cuz "THAT's in which the $$$ IS ( or so the Thinking WAS...). This "encouragement" has siphoned AWAY many Students that could OTHERWISE have long past into the Math & Science similar fields... This progress does NOT bode good for America's long run impact within the World... And if we do not readjust Our Educational priorities lovely Soon, many of the Technological Professionals on this Country- are NOT going to be FROM this Country !!! :(

Answer 4

The least square area of the solid will be formed by a cube.
Take the cube root of 1248
Approx. 11 cm.

Answer 5

I concur with the first answer