A parcel delivery service will deliver packages only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.
clearly the maximum volume will happen with a value of 112. So,
x^2+y = 112
v = x^2y = x^2(112-x^2)
now just find where dv/dx = 0
oops. I did the girth wrong. It is 4x, not x^2. 4x+y = 112
v = x^2(112-4x)
To find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements, we can use optimization techniques.
Let's start by breaking down the problem into variables:
Let's assume the length of the rectangular box is "l", and each side of the square ends is "x". The girth of the box can be calculated as the sum of the lengths of the sides of the square ends, which is 4x.
Given that the length plus the girth of the box should not exceed 112 inches, we can form the equation:
l + 4x ≤ 112
To find the maximum volume, we need to maximize the volume function, which is given by:
Volume = l * x * x
The constraint equation associates "l" and "x," so we can express the volume in terms of a single variable:
Volume = (112 - 4x) * x * x
To find the maximum volume, we'll differentiate the volume function:
d(Volume)/dx = 112x - 8x^2
Setting this derivative equal to zero and solving, we can find the critical points:
112x - 8x^2 = 0
8x(14 - x) = 0
From solving the equation, we find that there are two critical points: x = 0 and x = 14.
To determine which x value gives the maximum volume, we'll evaluate the second derivative:
d^2(Volume)/dx^2 = 112 - 16x
Evaluating this second derivative at x = 14:
d^2(Volume)/dx^2 = 112 - 16(14) = -32
Since the second derivative is negative at x = 14, we have concave-down behavior, indicating a maximum.
Hence, the maximum volume occurs when x = 14. Substituting this value back into the volume equation:
Volume = (112 - 4(14)) * 14 * 14
Volume = (112 - 56) * 14 * 14
Volume = 56 * 14 * 14
Volume = 10976 cubic inches
Thus, the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements is 10976 cubic inches.