Miguel is designing shipping boxes that are rectangular prisms. The shape of one box, with height h in feet, has
a volume defined by the function V (h) = hh - 5)(h - 6). Graph the function. What is the maximum volume for the domain 0 < h < 6? Round to the nearest cubic foot.
To graph the function V(h) = h^2(h - 5)(h - 6), we can plot points by selecting different values of h and calculating the corresponding values of V(h).
Let's choose a few values of h:
h = 0: V(0) = (0^2)(0 - 5)(0 - 6) = 0
h = 1: V(1) = (1^2)(1 - 5)(1 - 6) = 10
h = 2: V(2) = (2^2)(2 - 5)(2 - 6) = -24
h = 3: V(3) = (3^2)(3 - 5)(3 - 6) = -54
h = 4: V(4) = (4^2)(4 - 5)(4 - 6) = -48
h = 5: V(5) = (5^2)(5 - 5)(5 - 6) = 0
Now we can plot these points on a graph:
(0, 0), (1, 10), (2, -24), (3, -54), (4, -48), (5, 0)
The graph should show a shape similar to a parabola opening upwards. Since we are only interested in the domain 0 < h < 6, we can see that the maximum volume occurs between 1 and 5.
To find the maximum volume within this domain, we can analyze the sign changes of V(h). At h = 1, V(h) changes from negative to positive, and at h = 5, V(h) changes from positive to negative. This indicates that the maximum volume occurs somewhere between these two values.
To estimate the maximum volume, we can find the vertex of the parabola. The vertex of a quadratic function in the form ax^2 + bx + c can be found using the formula h = -b/(2a).
For V(h) = h^2(h - 5)(h - 6), a = 1, b = -11, and c = 0. Plugging these values into the formula, we get h = -(-11)/(2*1) = 11/2 = 5.5.
The vertex of the parabola is located at (5.5, V(5.5)).
Let's calculate V(5.5):
V(5.5) = (5.5^2)(5.5 - 5)(5.5 - 6) = (30.25)(0.5)(-0.5) = -3.78125
Rounding to the nearest cubic foot, the maximum volume for the domain 0 < h < 6 is approximately -4 cubic feet.