Linear expansion shows how much a length changes in proportion to the original length, the temperature change, and the material’s coefficient of linear expansion. The actual change in length is the original length times the fractional change, which is alpha times deltaT. So deltaL = L0 * (alpha * deltaT) = alpha * L0 * deltaT. This reflects why longer pieces change more and why bigger temperature increases or larger alpha produce a bigger deltaL. Units line up as well: alpha is 1 per degree, deltaT is degrees, and L0 is a length, giving deltaL a length unit. If deltaT is negative, the material shrinks. Expressions that omit L0 or alpha lose either the length scale or the material property, leading to an incorrect result. For example, with L0 = 2 m, alpha = 1e-5 /°C, deltaT = 50°C, deltaL = 1e-5 × 2 × 50 = 0.001 m (1 mm).

Heat conduction is the transfer of thermal energy within a material or between materials that are in direct contact, driven by a temperature difference. The speed of that transfer is governed by a property called thermal conductivity. A material with high thermal conductivity allows energy to move quickly from the hotter part to the cooler part, so it conducts heat readily. In metals, free electrons act as fast energy carriers, making conduction especially efficient; in nonmetals, energy moves mainly through lattice vibrations (phonons) and is less efficient. Vacuum cannot conduct heat because there are no particles to transfer energy by collisions, though heat can still be transferred by radiation. Insulators have low thermal conductivity and resist heat flow, while not all liquids are perfect conductors—many conduct heat poorly, though some vary in conductivity. So the statement that describes heat conduction best is that materials with high thermal conductivity conduct heat readily.

A basic dimension defines an exact value for location or size without providing a tolerance itself. Its purpose is to establish the theoretically exact geometry, which is then controlled by a GD&T tolerance such as true position. In other words, basic dimensions specify where features should be or how large they should be, but the allowable variation comes from the tolerance applied to the feature’s location or form. A practical use is locating a group of holes on a flange. You would mark the hole centers and the bolt circle diameter with basic dimensions to indicate their exact intended positions. Then you apply a true position tolerance to each hole relative to datum features, which tells you how far the actual hole centers may deviate from those exact locations while still mating correctly with the other parts. This distinction is why basic dimensions are not toleranced themselves. They set the precise target, and the true position tolerance defines the permissible deviation from that target.

Understanding casting cost and quality trade-offs helps explain why sand casting fits this description. Sand casting uses simple, inexpensive sand molds and patterns, so setup is quick and tooling costs are low, making it versatile for a wide range of shapes and sizes and suitable for low to medium production runs. The trade-off is in the finish and accuracy: the mold surface formed by loose sand is rough on a microscopic level, and small shifts in the mold or solidification can lead to looser dimensional tolerances. Finishing steps like machining or grinding are often needed to achieve precise dimensions. Other methods use more expensive tooling and processes that yield smoother surfaces and tighter tolerances, but at a higher cost and with less flexibility for quick changes. Therefore, sand casting is the best match for being inexpensive with a versatile setup but rougher finishes and looser tolerances.