This may be simple, but I'm not getting it. An open box of maximum volume is to be made from a square piece of material, 24cm on a side, by cutting equal squares from the corners and turning up the sides. The table for it is:
Height(x) Volume(V)
1 484
2 800
3 972
4 1024
5 960
6 864
If volume,V, is a function of x write the function and its domain.
The table should be:
1, 484
2, 800
3, 972
4, 1024
5, 960
6, 864
Sorry!
volume= x^3 domain is 0<x<24cm
height x
length = 24 - 2x
width = 24 - 2x
volume = x(24-2x)(24-2x)
v = x(576 - 96 x + 4 x^2)
v = 4 x^3 -96 x^2 + 576 x
now x may not be negative
x may not be equal to or greater than 12 or there is no box bottom left
so
0 < x <12
To solve this problem, we need to understand the process of constructing the open box and how its volume is related to the variable x.
When we cut equal squares from the corners of the square piece of material, the side length of the resulting square base of the box will be (24 - 2x) cm, because we are removing x cm from each side.
To form the open box, we then fold up the sides of the square base, creating a height for the box. Let's call this height h, which is equal to x cm.
Therefore, the volume of the box, V, is given by the formula:
V = (24 - 2x)² * x
To simplify this equation, let's expand the square:
V = (576 - 48x + 4x²) * x
V = 4x³ - 48x² + 576x
So, the function V(x) represents the volume of the open box in terms of x:
V(x) = 4x³ - 48x² + 576x
The domain of the function V(x) depends on the context of the problem. In this case, we need to consider the restrictions caused by the physical dimensions.
Since we are cutting squares from the corners, x cannot be greater than half of the side length of the original square (24/2 = 12 cm). Therefore, the domain of the function is 0 ≤ x ≤ 12.