6)A rectangular box with an open top is to have a volume of 10 cubic meters.?

Question

Problem 8. A rectangular box with an open top is to have a volume of 10 cubic meters. The length of the base is twice its width. Material for the base costs $3 per square meter and for the sides costs $5 per square meter. (a) (15 points) Find the dimensions that minimize the cost of the box. (b) (5 points) How much does the cheapest box cost? (5 points) Which method described in the lecture are you using (endpoint method or first derivative test) and why?

Answer 1

l = 2w
lwh = 10
2w^2h = 10
C = 3lw + 5h(2)(l + w) = 6w^2 + 30hw
h = 5/w^2
C = 6w^2 + 150/w
for max or min cost
12w = 150/w^2
w^3 = 150/12 = 12.5
a)
w = 2.320794 m
l = 4.641589 m
h = 0.92832 m
b)
C = 6(2.320794)^2 + 150/2.320794
C = $96.95
c)
first derivative--missed the lecture

Answer 2

side=8^(1/3)=2 6x^2=6*4=24 [with top] 5x^2=5*4=20 since top part is missing.

Answer 3

2*A^2*B=10
B=5A^-2
dB/dA=-10A^-3

6A^2+30AB=F(A,B)
minimize
dF(A,B)/dA=12A+30B+30A(-10A^-3)=0
=12A+90A^-2-300A^-2
12A^3 =210
4A^3=70
A=(70/3)^1/3
Plug that into F along with A=0 and inf

A=(70/3)^1/3
Plug it into F to obtain B

First derifitive test.
The constraint wasn't linear