Abox with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out 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 x and y.
v =
1(d) use the given information to write an equation that relates the variables x and y.
2(e) use part (d) to write the volume as a function of x.
v(x) =
3(f) finish solving the problem by finding the largest volume that such a box can have.
v = 4 ft3

a) See annex

b) See annex

x  =  0,5 ft

y =  2 ft   and

V = 2 ft³

Step-by-step explanation:  See annex

c) V = y*y*x

d-1) y = 3 - 2x

d-2) V = (3-2x)* ( 3-2x)* x   ⇒ V = (3-2x)²*x    

V(x) =( 9 + 4x² - 12x )*x    ⇒   V(x) = 9x + 4x³ - 12x²

Taking derivatives

V¨(x) = 9 + 12x² - 24x

V¨(x) = 0              ⇒   12x² -24x +9 = 0     ⇒  4x² - 8x + 3 = 0

Solving for x (second degree equation)

x =[ -b ± √b²- 4ac ] / 2a

we get    x₁  =  1,5       and    x₂ =  0,5

We look at y = 3 - 2x    and see that the value x₂ is the only valid root

then

x  =  0,5 ft

y =    2 ft   and

V = 0,5*2*2

V = 2 ft³

i believe the answer is d.

hope it you! : d

irrational, then, just means all the numbers that aren't rational. every integer is a rational number, since each integer n can be written in the form n/1. however, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

step-by-step explanation: