Problem 34 - Calculus Single Variable

Question

A door designed to be \(885 \mathrm{~mm}\) wide and \(1475 \mathrm{~mm}\) high is to fit in a \(895 \mathrm{~mm} \times 1485 \mathrm{~mm}\) rectangular frame. The clearance between the door and the frame can be no greater than \(7 \mathrm{~mm}\) on any side for an acceptable seal to result. Under anticipated temperature ranges, the door will expand by at most \(0.2 \%\) in height and width. The frame does not expand significantly under these temperature ranges. At all times, even when the door has expanded, there must be a \(1 \mathrm{~mm}\) clearance between the door and the frame on each of the four sides. The door does not have to be manufactured precisely to specifications, but there are limitations. In what interval must the door width lie? The height?

Step-by-Step Solution

Answer

Door width: 881 to 891.23 mm, Door height: 1471 to 1480.05 mm.

Step 1: Calculate Maximum Door Expansion

First, determine how much the door can expand by remembering that it can expand by up to 0.2% in width and height. For a door that is initially 885 mm wide, the expansion is \(0.2\% \times 885 = 1.77 \text{ mm}\). Similarly, for a door that is 1475 mm high, the expansion is \(0.2\% \times 1475 = 2.95 \text{ mm}\). Therefore, the maximum expanded dimensions of the door are 886.77 mm in width and 1477.95 mm in height.

Step 2: Define Frame and Clearance Requirements

The frame is 895 mm wide and 1485 mm high. To ensure there remains at least a 1 mm clearance on each side, we subtract 2 mm (1 mm for each side) from each frame dimension. This calculation gives us the maximum allowable width and height for the door when expanded: \(895 - 2 = 893 \text{ mm width} \) and \(1485 - 2 = 1483 \text{ mm height} \).

Step 3: Determine Acceptable Door Dimensions

Using the calculations from Steps 1 and 2, the door's width when expanded must be less than or equal to 893 mm and the height must be less than or equal to 1483 mm. Therefore, before expansion, the door can be sized such that its expanded dimensions fit within these maximum values: allowing for up to 1.77 mm width expansion and up to 2.95 mm height expansion.

Step 4: Solve for the Interval of Door Dimensions

For the width, the pre-expansion width must not surpass \(893 - 1.77 = 891.23 \text{ mm} \). The minimum width can be deduced from the maximum allowed clearance (7 mm), so the smallest acceptable width is \(895 - 2 \times 7 = 881 \text{ mm}\). Therefore, the acceptable interval for the width is \(881 \leq \text{Width} \leq 891.23\).For the height, the pre-expansion height must not surpass \(1483 - 2.95 = 1480.05 \text{ mm}\). Similarly, computing with the maximum allowed clearance leads to a minimum height of \(1485 - 2 \times 7 = 1471 \text{ mm}\). Thus, the acceptable interval for the height is \(1471 \leq \text{Height} \leq 1480.05\).

Key Concepts

Thermal Expansion

When designing structures like doors that are subject to temperature changes, understanding thermal expansion is crucial. Thermal expansion refers to the increase in the dimensions of an object as its temperature rises. This happens because the heat causes the particles within the material to move more vigorously and occupy more space.
Understanding thermal expansion is essential for door design, especially if you want it to fit perfectly within a frame, without expanding excessively when temperatures vary.

• Materials: Different materials expand at different rates. Metals, for instance, tend to expand more than wood.
• Measurement: The percentage of expansion can be applied using the formula: Change in length = Original length \(\times\) Expansion percentage.

In our exercise, the door expands by no more than 0.2% in both height and width. This means calculating a small change but one significant enough to impact the fit of the door in its frame.

Clearance Requirements

Clearance is the intentional space between two objects, like a door and its frame. This space ensures that even as the door expands or if there are slight manufacturing differences, it will still fit well.
For the door in question, the clearance requirements dictate that when fully expanded:

• There must be a minimum of 1 mm space on all four sides.
• The total clearance should not exceed 7 mm on any edge, ensuring a snug but functional fit.

Clearance requirements are crucial for allowing the door to operate smoothly without being too loose or tight. In any design involving moving parts, such as doors, clearances provide operational tolerances.
Understanding these requirements ensures a balance between functionality and thermal expansion, guiding the manufacturing specifications.

Rectangular Frames

Frames provide structural support and define the shape and fit for doors and other objects. A door frame essentially sets the maximum boundary within which the door must fit. Ensuring the door fits well involves considering both the frame's dimensions and how the door is centered within it.
In this scenario, the rectangular frame is 895 mm wide and 1485 mm high. The door must accommodate to fit when expanded and still maintain the required clearance. This requires precise calculations to ensure that both the width and height lie within specific ranges.
To ensure that the door fits the frame perfectly, you consider:

• Calculating Maximum Fit: Determine the maximum expanded size that still maintains clearance.
• Accounting for Expansion: Adjust dimensions for potential thermal expansion, ensuring that when it occurs, the door remains operational within the frame.

The rectangular frame sets fixed boundaries, guiding the overall design and placement of the door, impacting material choice and construction specifics to ensure these parameters are met.