Answer:
the dimensions that will minimize the cost of constructing the box is:
a = 5.8481 in ; b = 5.848 in ; c = 10.234 in
Step-by-step explanation:
From the information given :
Let a be the base if the rectangular box
b to be the height and c to be the other side of the rectangular box.
Then ;
the area of the base is ac
area for the front of the box is ab
area for the remaining other sides ab + 2cb
The base of the box is made from a material costing 8 ac
The front of the box must be decorated, and will cost 10 ab
The remainder of the sides will cost 4 (ab + 2cb)
Thus ; the total cost C is:
C = 8 ac + 10 ab + 4(ab + 2cb)
C = 8 ac + 10 ab + 4ab + 8cb
C = 8 ac + 14 ab + 8cb ---- (1)
However; the volume of the rectangular box is V = abc = 350 in³
If abc = 350
Then b = 
replacing the value for c in the above equation (1); we have :


Differentiating C with respect to a and c; we have:


--- (2)
---(3)
From (2)

----- (4)
From (3)

-----(5)
Replacing the value of a in 5 into equation (4)
![c = \dfrac{2800}{8*(\dfrac{4900}{8c^2})^2} \\ \\ \\ c = \dfrac{2800}{\dfrac{8*24010000}{64c^4}} \\ \\ \\ c = \dfrac{2800}{\dfrac{24010000}{8c^4}} \\ \\ \\ c = \dfrac{2800*8c^4}{24010000} \\ \\ c = 0.000933c^4 \\ \\ \dfrac{c}{c^4}= 0.000933 \\ \\ \dfrac{1}{c^3} = 0.000933 \\ \\ \dfrac{1}{0.000933} = c^3 \\ \\ 1071.81 = c^3\\ \\ c= \sqrt[3]{1071.81} \\ \\ c = 10.234](https://tex.z-dn.net/?f=c%20%3D%20%5Cdfrac%7B2800%7D%7B8%2A%28%5Cdfrac%7B4900%7D%7B8c%5E2%7D%29%5E2%7D%20%5C%5C%20%5C%5C%20%20%5C%5C%20%20c%20%3D%20%5Cdfrac%7B2800%7D%7B%5Cdfrac%7B8%2A24010000%7D%7B64c%5E4%7D%7D%20%5C%5C%20%5C%5C%20%20%5C%5C%20c%20%3D%20%5Cdfrac%7B2800%7D%7B%5Cdfrac%7B24010000%7D%7B8c%5E4%7D%7D%20%5C%5C%20%5C%5C%20%5C%5C%20c%20%3D%20%5Cdfrac%7B2800%2A8c%5E4%7D%7B24010000%7D%20%5C%5C%20%5C%5C%20%20c%20%3D%200.000933c%5E4%20%5C%5C%20%5C%5C%20%5Cdfrac%7Bc%7D%7Bc%5E4%7D%3D%200.000933%20%5C%5C%20%5C%5C%20%20%5Cdfrac%7B1%7D%7Bc%5E3%7D%20%3D%200.000933%20%5C%5C%20%5C%5C%20%5Cdfrac%7B1%7D%7B0.000933%7D%20%3D%20c%5E3%20%5C%5C%20%5C%5C%201071.81%20%3D%20c%5E3%5C%5C%20%5C%5C%20c%3D%20%5Csqrt%5B3%5D%7B1071.81%7D%20%5C%5C%20%5C%5C%20c%20%3D%2010.234)
From (5)
-----(5)

a = 5.8481
Recall that :
b = 
b = 
b =5.848
Therefore ; the dimensions that will minimize the cost of constructing the box is:
a = 5.8481 in ; b = 5.848 in ; c = 10.234 in