Miguel is designing shipping boxes that are rectangular prisms. the Shape of one box, with height h in feet, had a volume defined by the function V(h) =h(-h + 10)(-h+8). What is the maximum volume for the domain 0<h<10? Round to the nearest cubic foot.
To find the maximum volume, we can take the derivative of the volume function and set it equal to 0 to find critical points.
V(h) = h(-h + 10)(-h + 8)
Let's simplify this function first:
V(h) = -h³ + 18h² - 80h
Now, we take the derivative with respect to h:
V'(h) = -3h² + 36h - 80
To find critical points, we set V'(h) equal to 0 and solve for h:
-3h² + 36h - 80 = 0
We can factor this quadratic equation:
(-h + 10)(3h - 8) = 0
This gives us two solutions: h = 10 and h = 8/3. However, h cannot be equal to 10 in the given domain of 0 < h < 10. Therefore, the only critical point is h = 8/3.
Now, we check the values of V(h) at the critical point and the endpoints of the domain to determine which one gives the maximum volume:
V(8/3) = (8/3)(-8/3 + 10)(-8/3 + 8) ≈ 152.89 cubic feet
V(0) = 0(-0 + 10)(-0 + 8) = 0 cubic feet
V(10) = 10(-10 + 10)(-10 + 8) = 0 cubic feet
Therefore, the maximum volume for the given domain is approximately 152.89 cubic feet.
To find the maximum volume of the shipping boxes within the given domain, we need to find the critical points of the volume function and evaluate them.
1. Start by finding the derivative of the volume function V(h):
V'(h) = (-h + 10)(-h + 8) + h*(-1)(-h + 8) + h*(-h + 10)*(-1)
= (-h + 10)(-h + 8) - h(-h + 8) - h(-h + 10)
= (-h + 10)(-h + 8 + h - 10)
= (-h + 10)(-2)
2. Set V'(h) equal to zero and solve for h:
(-h + 10)(-2) = 0
-h + 10 = 0
-h = -10
h = 10
3. Evaluate the volume function at the critical point and the endpoints of the domain:
V(0) = 0(-0 + 10)(-0 + 8) = 0
V(10) = 10(-10 + 10)(-10 + 8) = 0
V(10) = 10(-10 + 10)(-10 + 8) = 0
4. Compare the values of V(0), V(10), and V(10):
The maximum volume should be the highest value between V(0), V(10), and V(10).
Since V(0) = V(10) = 0, the maximum volume will be at V(10).
5. Round the maximum volume to the nearest cubic foot:
The maximum volume is 0 cubic feet.
Therefore, within the given domain, the maximum volume for the shipping boxes is 0 cubic feet.
To find the maximum volume of the shipping boxes within the given domain, we need to determine the critical points of the function V(h).
First, let's find the derivative of the function V(h) with respect to h:
V'(h) = (-h + 10)(-h + 8) + h(-1)(-h + 8) + h(-h + 10)(-1)
= (-h + 10)(-h + 8) - h(-h + 8) - h(-h + 10)
= (-h + 10)(-h + 8) - h(-h + 10 + 8)
= (-h + 10)(-h + 8) - h(-h + 18)
= (-h + 10)(-h + 8 + h)
= (-h + 10)(8)
= -8h + 80
To find the critical points, we set the derivative equal to zero and solve for h:
-8h + 80 = 0
-8h = -80
h = -80/-8
h = 10
We obtained one critical point h = 10. However, since the domain for h is 0 < h < 10, the critical point lies outside the given domain. Therefore, we need to evaluate the function V(h) at the endpoints of the domain, h = 0 and h = 10:
V(0) = 0(-0 + 10)(-0 + 8) = 0
V(10) = 10(-10 + 10)(-10 + 8) = 0
Both endpoints yield a volume of 0 cubic feet.
In conclusion, the maximum volume for the given domain 0 < h < 10 is 0 cubic feet.