Consider the following problem: A box with an open top is to be constructed from

Question

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cuttin ut a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. (b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base. (c) Write an expression for the volume V in terms of both x and y. (d) Use the given information to write an equation that relates the variables x and y (e) Use part (d) to write the volume as a function of only x. (f) Finish solving the problem by finding the largest volume that such a box can have. ft3 Talk to Submit Answer Save Progress

Explanation / Answer

c)
x is what we cut out

Length of base = y

So, V = x*y*y

V = x * y^2 ---> ANS

---------------------------------------------------------------------
d)

y = 3 - 2x ---> ANS

---------------------------------------------------------------------
e)

Say length of side of the sqr is x feet

So, when we fold it up and form a box,
we get these dimensions :

length of box = 3 - 2x
width of box = 3 - 2x
ht of box = x

Now, Volume is :
V = L*W*H

V = x(3 - 2x)(3 - 2x)

V = x(4x^2 - 12x + 9)

V = 4x^3 - 12x^2 + 9x ------> ANS

---------------------------------------------------------------

Now, deriving :
dV/dx = 12x^2 - 24x + 9 = 0

Divide all over by 3 :
4x^2 - 8x + 3 = 0

x = (8 +/- sqrt16) / 8

x = (8+4)/8 and (8-4)/8

x = 1.5 or 0.5

Gotta be 0.5

Now, largest volume is :
V = 4x^3 - 12x^2 + 9x

V = 4*0.5^3 - 12*0.5^2 + 9*0.5

V = 2 ----> ANS

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---