A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 104 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.
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To find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements, we need to determine the dimensions (length, width, and height) of the box.
Let's assume that the length of the box is L, and the width and height are both W. Since the box has square ends, the width and height would be the same.
According to the given condition, the length plus the girth should not exceed 104 inches. The girth of a box with square ends is the perimeter of the square base, which can be calculated as 4 times the width (W). Therefore, the equation for the given condition is:
L + 4W ≤ 104
Now, let's express the volume of the box in terms of L and W. The volume of a rectangular box is given by:
Volume = Length × Width × Height
In our case, the height is also W since the box has square ends. Therefore, the equation for the volume is:
Volume = L × W × W
Volume = LW²
To maximize the volume, we can express the length (L) in terms of the width (W) using the earlier condition:
L ≤ 104 - 4W
Substituting this expression for L in the volume equation, we get:
Volume = (104 - 4W) × W²
To find the maximum volume, we can take the derivative of the volume equation with respect to W, set it equal to zero, and solve for W. However, this involves calculus and is not suitable for an explanation here.
Instead, we can use numerical methods or trial and error to find the width (W) that maximizes the volume. We can start with different values of W (e.g., 1, 2, 3, etc.) and calculate the corresponding volume using the volume equation.
By trying different values of W, we can find the width that maximizes the volume. Once we have the optimized width, we can substitute it back into the condition (L + 4W ≤ 104) to determine the corresponding length (L). Finally, we can calculate the maximum volume using the width (W), length (L), and height (W).