Express cost as function of L and H, relate L and H, and...

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ptsoccerboy

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An open-top box is to be made so that its width is 4 ft and its volume is 40 ft^3. The base of the box costs $4/ft^2 and the sides cost $2/ft^2.

a. Express the cost of the box as a function of its length L and height H.

b. Find a relationship between L and H

c. Express the cost as a function of H only.

d. Give the domain of the cost function.

E. Use a graphing calculator or computer to approximate the dimensions of the bos having least cost.

Thank you!!!
 
Re: need help with functions

ptsoccerboy said:
An open-top box is to be made so that its width is 4 ft and its volume is 40 ft^3. The base of the box costs $4/ft^2 and the sides cost $2/ft^2.

a. Express the cost of the box as a function of its length L and height H.

b. Find a relationship between L and H

c. Express the cost as a function of H only.

d. Give the domain of the cost function.

E. Use a graphing calculator or computer to approximate the dimensions of the bos having least cost.

Thank you!!!
Click to expand...

Please share with us your work and your thoughts regarding this problem - so that we know where to start.
 
For a. i believe that the answer is
f(c) = 4LW + 2HW + 2HL
= 16L + 8H + 2LH

I am not sure if the relationship between L and H is that they are interchangeable?

I believe that the cost as a function of H only is: f(c) = 18H + 20

I do not know which cost function to take the domain of and how to graph it with the least amount of cost.
 
ptsoccerboy said:
For a. i believe that the answer is
f(c) = 4LW + 2HW + 2HL
= 16L + 8H + 2LH

not a bad start, but there are two sides of dimensions H by W and two sides with dimensions H by L ...
Cost = ($4)LW + ($2)(2HW) + ($2)(2HL)
substitute W = 4 ...
Cost = 16L + 16H + 4HL

I am not sure if the relationship between L and H is that they are interchangeable?

you know the volume is 40 ft<sup>3</sup>, so LWH = 40 ... since W=4, 4LH = 40 ... LH = 10.
finally, L = 10/H

I believe that the cost as a function of H only is: f(c) = 18H + 20

cost as a function of H is ...
f(H) = 16(10/H) + 16H + 40 = 160/H + 16H + 40

I do not know which cost function to take the domain of and how to graph it with the least amount of cost.

Graph f(H) in the window x:[0, 10] y:[0,200] ... the minimum cost will be the y-value of the lowest point on the graph ... min cost will be in the neighborhood of $140.
Click to expand...
 
ptsoccerboy said:
so what would the dimensions be??
Click to expand...
You've been provided with a complete explanation and almost all of the work, with instructions for what little bit is left. Please reply showing where you are stuck on that last bit.

Thank you.

Eliz.
 
i believe that the dimensions of the box w/ the least cost are the x and y values of the lowest point of the graph but im confused on wat the x represents. i think it represents height correct? so at the lowest point on the graph x, or height is 3 so length = 10/3
 
ptsoccerboy said:
i believe that the dimensions of the box w/ the least cost are the x and y values of the lowest point of the graph but im confused on wat the x represents. i think it represents height correct? so at the lowest point on the graph x, or height is 3 so length = 10/3
Click to expand...

Have you taken any calculus yet?

In that you wold have found that the height at minimum cost = sqrt(10)

then length = sqrt(10)

Since you are doing it by algebra, from graph height = 3.16 or 3.2

Then length = 3.16 or 3.13
 
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