Question 1000879
you can solve this graphically or you can solve it using calculus.
i don't know any other ways.
the graphical solution is shown below:
<img src = "http://theo.x10hosting.com/2015/110307.jpg" alt="$$$" </>
that 2.236 turns out to be sqrt(5).
the value of x is sqrt(5).
the value of y is 7.454


x is the length of a side of the square base.
y is the volume.


the formula for volume is derived as follows:


s = measure of one of the sides of the base.
h = measure of the height.
area of the base = s^2
volume = s^2*h


surface area is derived as follows:
sa = 2*s^2 + 4*h*s


cost of surface area is derived as follows:


cost of the base material is equal to 2 * the area of the base.
cost of the side material is equal to 3 * the area of the sides between the 2 bases.


the total cost is there equal to 2 * area of the bases plus 3 * area of the side faces.


you get total cost = 2 * (2s^2) + 3*(4hs) which becomes:


total cost = 4s^2 + 12hs


since total cost is 60, you get:


60 = 4s^2 + 12hs


in this equation, you can solve for h as follows:


subtract 4s^2 from both sides to get:
60 - 4s^2 = 12hs
divide both sides by 12s to get:
(60-4s^2) / 12s = h


the volume is equal to s^2*h
replace h with (60-4s^2)/12s to get:
volume = s^2 * (60-4s^2)/12s
factor out an s from the numerator and denominator and you get:
volume = s*(60-4s^2)/12


to graph this equation, make volume = y and make s = x.
formula becomes:
y = x*(60-4x^2)/12
this is the equation that was graphed above.


to solve this using calculus, get the derivative of the equation and set it equal to 0 and solve for x to find the maximum point on the graph.


the derivative turns out to be y' = 5-x^2
set it equal to 0 and you get 0 = 5-x^2
add x^2 to both sides to get x^2 = 5
take square root of both sides to get x = +/- sqrt(5).
it has to be sqrt(5) because negative values are not allowed.
the derivative is telling you that you need to evalute your equation at x = sqrt(5) to find the maximum volume.
that turns out to be same as what the graph is showing you.


the more detailed answer is:


x = s  = 2.236068
y volume = 7.4535599


round these to x = 2.236 and y = 7.454


x is equal to s which is the width
y is the volume


the volume is s^2*h
you can use this formula to solve for h.
from this formula, solve for h to get h = v/s^2.
that becomes h = 7.4535599/2.236068 = 1.49071195
round to 3 decimal place to get h = 1.491


the dimension of the box with the greatest volume that has a surface area that costs 60 dollars is therefore:


s = 2.236 
h = 1.491


if you recall, we made s = x when we graphed it.
when we found x, we automatically found s because they're equivalent to each other.


surface area = 2 * s^2 + 4hs


cost for surface area = 4s^2 + 12hs


when s = 2.236 and h = 1.491, we get cost for surface area = 59.999999999 which rounds to 60.


we got the maximum volume for a box that has a total cost for surface area of 60 dollars.


derivative was found using the following derivative calculator.


the calculator is very useful when you're not sure how to find the derivative.


<a href = "http://www.derivative-calculator.net/#" target = "_blank">http://www.derivative-calculator.net/#</a>