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 + 10)(–h + 8). What is the maximum volume for the domain
0 < h < 10? Round to the nearest cubic foot.
(1 point)
Responses
10 ft3
10 ft 3
107 ft3
107 ft 3
105 ft3
105 ft 3
110 ft3
To find the maximum volume of the shipping boxes, we need to find the maximum value of the function V(h) within the given domain 0 < h < 10.
Step 1: Find the derivative of V(h)
V'(h) = 3h^2 - 36h + 80
Step 2: Set the derivative equal to zero and solve for h
3h^2 - 36h + 80 = 0
Step 3: Use the quadratic formula to solve for h
h = (-(-36) ± √((-36)^2 - 4*3*80)) / (2*3)
h = (36 ± √(1296 - 960)) / 6
h = (36 ± √336) / 6
Step 4: Simplify the expression
h = (36 ± √(16*21)) / 6
h = (36 ± 4√21) / 6
h = 6 ± 2√21
Since the domain is 0 < h < 10, we can ignore the value h = 6 + 2√21 because it is greater than 10.
Step 5: Evaluate V(h) for the remaining value of h
V(6 - 2√21) = (6 - 2√21)(-6 + 2√21)(-6 + 4√21)
V(6 - 2√21) ≈ 107.26 ft³
Therefore, the maximum volume for the domain 0 < h < 10 is approximately 107 ft³.
To find the maximum volume, we need to find the maximum value of the function V(h) within the given domain.
First, let's simplify the function V(h) = h(-h + 10)(-h + 8) by expanding it:
V(h) = h(-h^2 + 18h - 80)
V(h) = -h^3 + 18h^2 - 80h
To find the maximum value, we can take the derivative of V(h) and set it equal to zero:
V'(h) = -3h^2 + 36h - 80
0 = -3h^2 + 36h - 80
Next, we can solve the quadratic equation for h:
0 = -3(h^2 - 12h + 26.67)
Using the quadratic formula, we get:
h = (12 ± √(12^2 - 4(-3)(26.67))) / (2(-3))
h = (12 ± √(144 + 320.04)) / -6
h = (12 ± √464.04) / -6
h = (12 ± 21.55) / -6
Since the domain is 0 < h < 10, we can discard the negative root and get:
h = (12 + 21.55) / -6
h ≈ -5.59 (discard)
Therefore, the maximum volume occurs at h ≈ 5.59.
Now we can substitute this value of h into the function to find the maximum volume:
V(h) = h(-h + 10)(-h + 8)
V(5.59) = 5.59(-5.59 + 10)(-5.59 + 8)
V(5.59) ≈ 105.06 cubic feet
Rounded to the nearest cubic foot, the maximum volume for the domain 0 < h < 10 is 105 ft³.
Therefore, the correct answer is "105 ft3".
To find the maximum volume for the given function V(h) = h(–h + 10)(–h + 8), we can use calculus. The process involves finding the critical points and determining whether they correspond to a maximum or minimum value.
First, we need to find the critical points by taking the derivative of the function.
V'(h) = (–h + 10)(–h + 8) + h(–1)(–h + 8) + h(–h + 10)(–1)
= (–h + 10)(–h + 8) + h(–h + 8) + (–h + 10)(–1)
= (–h + 10)(–h + 8 + h) + (–h + 10)(–1)
= (–h + 10)(8) + (–h + 10)(–1)
= (–h + 10)(8 - 1)
= (–h + 10)(7)
= –7h + 70
Next, we set the derivative equal to zero to find the critical points:
–7h + 70 = 0
–7h = –70
h = –70/–7
h = 10
Now we can check if this critical point corresponds to a maximum or minimum. We can do this by taking the second derivative:
V''(h) = –7
Since the second derivative is negative, the critical point represents a maximum value.
Therefore, the maximum volume occurs when h = 10 feet.
To confirm this maximum volume, we can substitute h = 10 into the original function:
V(10) = 10(–10 + 10)(–10 + 8)
= 10(0)(–2)
= 0
Hence, the maximum volume for the given domain 0 < h < 10 is 0 cubic feet.
None of the given answer options (10 ft3, 107 ft3, 105 ft3, and 110 ft3) match the correct answer.