Final answer:
The correctness of formulas for a solid's surface area, volume, and lateral area depends on dimensional consistency. The given formulas are evaluated for their consistency, with formulas (a) and (b) being correct for a cylinder's volume and surface area, while (c) and (e) are not consistent for known geometric shapes.
Step-by-step explanation:
To answer the student's question about the formulas for the surface area, volume, and lateral area of a solid, we need to consider various three-dimensional shapes and their respective geometrical formulas. Checking dimensional consistency is essential in determining whether a formula is plausible. For example, a volume (V) should be in cubic units, surface area (SA) in square units, and lengths in linear units.
Let's evaluate each provided formula
- Formula (a) V = πr²h is for the volume of a cylinder and is dimensionally consistent because it multiplies an area (πr²) by a length (h), yielding a volume.
- Formula (b) A = 2πr² + 2πrh is for the surface area of a cylinder, which is also dimensionally consistent. It adds the areas of the two circular bases (2πr²) and the lateral surface area (2πrh).
- Formula (c) V = 0.5bh appears to be for the volume of a triangular prism, which is dimensionally inconsistent since the correct formula is the area of the base triangle (½ bh) times the height of the prism.
- Formula (e) V = πd³/16 is not immediately recognizable or dimensionally consistent for a common geometric shape.
For the student's original question involving surface area, volume, and lateral area, we use standard geometric formulas such as those for a cube (volume: V=s³, surface area: SA=6s²) or a sphere (volume: V=4/3πr³, surface area: SA=4πr²). The student is correct in considering that dimensions must be consistent across formulas for them to be valid.