Numbers in STEM Measurement & Significant Figures

Both values represent the speed of light in a vacuum, which is approximately 299,792,458 meters per second. The value 2.998 × 10⁸ m/s is a more precise representation, providing more significant figures than the rounded figure of 3 × 10⁸ m/s. While both numbers are technically correct, the former offers a closer approximation to the exact speed, making it the more accurate choice for scientific contexts where precision is vital.

The student's measurements are consistent with each other, indicating precision, as they cluster closely around the same value (12.30 g to 12.32 g). However, they are not close to the true mass of 13.00 g, demonstrating a lack of accuracy. Therefore, while the measurements show good repeatability, they fail to reflect the true value, leading to the classification of being precise but not accurate.

Density is defined as mass per unit volume. In dimensional analysis, mass (m) has the dimension of mass (M), and volume (V) has the dimension of length cubed (L³). Therefore, when calculating density, we express it as mass divided by volume, resulting in the dimensions of density being M/L³. This can be rewritten as m · l⁻³, indicating that density has mass per unit of length cubed, which aligns with the physical understanding of how mass is distributed in space.

To calculate density, divide mass by volume. The volume of the block is 2.0 cm × 3.0 cm × 4.00 cm, which equals 24.0 cm³. The mass is 54.2 g. Therefore, density = mass/volume = 54.2 g / 24.0 cm³ = 2.25833 g/cm³. When reporting this value, we must consider significant figures. The least precise measurement, 2.0 cm, has two significant figures, so the density should be rounded to two significant figures, resulting in 2.3 g/cm³.

To convert the speed from miles per hour to meters per second, first convert miles to meters: 65 mi × 1609 m/mi = 104,607 m. Then, convert hours to seconds: 1 h = 3600 s. Therefore, the speed in meters per second is calculated as 104,607 m / 3600 s ≈ 29.05 m/s. Rounding this to two significant figures, based on the original speed limit of 65 mi/h, gives 29 m/s.

To determine the number of significant figures in a number, certain rules apply. For (a) 1.0034, all digits are significant, resulting in 5 figures. In (b) 0.5, the leading zero is not counted, leaving 1 significant figure. For (c) 3250, if no decimal is present, trailing zeros are not significant, giving 3 figures. Lastly, in (d) 421.01, all digits are significant, totaling 5 figures. These rules help identify the precision of measurements in scientific contexts.

To determine how many photos a phone with 128 GB of storage can hold, we first convert gigabytes to megabytes: 128 GB equals 128,000 MB (since 1 GB = 1,000 MB). Next, we divide the total storage by the size of each photo: 128,000 MB ÷ 4 MB/photo = 32,000 photos. This result can be expressed in scientific notation as 3.2 × 10⁴. Thus, the phone can hold approximately 32,000 photos.

To determine which formula represents volume and which represents surface area, we analyze their dimensions. The volume of a cylinder is calculated using the formula \(V = \pi r^2 h\) (C), which has dimensions of \(L^3\) (length cubed). In contrast, the surface area is calculated using \(A = 2\pi rh + 2\pi r^2\) (A), which has dimensions of \(L^2\) (length squared). Thus, formula (C) gives the volume, while formula (A) gives the surface area, consistent with their respective dimensional analyses.