math question?

Question

a square based prism hasa volume of 8 meters^3, the price of the side is $4 per meters^2, top is $2 per m^2 and base is $10 per m^2, find minimum cost of box.

thanks for the help

Answer 1

I think it is 26 because you basically have to guess the length of the top base and sides. 8meters^3 represents the volume.The volume is all the three sides multiplied by each other. One side can be 1m so 1x10=10, 4x2=8, and 2x4=8.All added together is 26 dollars. I think 26 dollars is the right answer unless you can find the answer some other way.

Answer 2

6

Answer 3

6

Answer 4

top $2 x 8 x 8
sides $4 x 8 x 8 x 4 sides
bottom $10 x 8 x 8

enjoy your homework

Answer 5

let a = base
h = height

V = h a^2
base cost = 10 a^2
top cost = 2a^2
sides cost (4 sides) = 4(4ah) =16ah
Cost of the box = 12a^2 +16ah ----> call this C
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1st case: h is constant and a varies
V = h a^2 =8 ---> h = 8 /a^2

C = 12a^2 +128/a

dC/da = 24a -128/a^2 =0
24a^3 -128 =0
a^3 = 128/24 ---> a = 1.777 m
a = 1.777m ----> h = 8/a^2 = 2.533 m

Cost = 12a^2 +16ah = $109.91
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2nd case when a is constant and h varies
C = 12a^2 +128/a
h a^2 =8 ---> a = sqrt(8/h)
C = 12(8/h) +16h(sqrt(8/h))
C = 96/h + 16 sqrt(8h)
dC/dh = -96/h^2 +16sqrt(2/h) =0
-6/h^2 +sqrt(2/h) =0
-6/h^2 = -sqrt(2/h)
square both sides
36/h^4 = 2/h
36/h^3 = 2
h^3 = 36/2 =18 ---> h =2.621 m
a = 1.747

C = 12a^2 +16ah = 36.624 +73.262 = 109.886
*********************************
Summary:
Minimum cost = $109.886
a = 1.747m ; h = 2.621 m

Answer 6

side = 4 * 4 * 4 = $64
top = 4*2*2 = $16
base = 10 *2*4= $ 80

total = $160