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) = h(h – 5)(h – 6). Graph the function. What is the maximum volume for the domain 0 < h < 6? Round to the nearest cubic foot.
(1 point)
Responses
29 ft3
29 ft 3
27 ft3
27 ft 3
24 ft3
24 ft 3
6 ft3
To find the maximum volume for the function V(h) = h(h – 5)(h – 6) in the given domain 0 < h < 6, we can take the derivative of the function and find the critical points.
First, let's find the derivative of V(h):
V'(h) = 3h^2 - 22h + 30
To find the critical points, we set V'(h) equal to zero and solve for h:
3h^2 - 22h + 30 = 0
Using the quadratic formula, we can solve for h:
h = (22 ± sqrt((-22)^2 - 4(3)(30)))/(2(3))
h = (22 ± sqrt(484 - 360))/6
h = (22 ± sqrt(124))/6
h ≈ (22 ± 11.14)/6
The two possible solutions for h are h ≈ (22 + 11.14)/6 ≈ 5.86 and h ≈ (22 - 11.14)/6 ≈ 1.19.
Since the domain is 0 < h < 6, the only critical point within this domain is h ≈ 1.19.
Now, we need to evaluate V(h) at the critical point and at the endpoints of the domain to find the maximum volume.
V(1.19) ≈ 1.19(1.19 – 5)(1.19 – 6) ≈ 24.6 ft^3
V(0) ≈ 0(0 – 5)(0 – 6) ≈ 0 ft^3
V(6) ≈ 6(6 – 5)(6 – 6) ≈ 0 ft^3
Therefore, the maximum volume in the given domain is approximately 24 ft^3. So the correct answer is:
24 ft^3