To find the length of the platinum bar, we can use the relationship between mass, density, and volume.
The formula for density is:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Rearranging this formula gives:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
Given:
- Mass = 1 kg = 1000 g (since 1 kg = 1000 g)
- Density = 21.4 g/cm³
Now, we can calculate the volume of the platinum bar:
\[ \text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.73 \text{ cm}^3 \]
Next, the volume of a rectangular prism (like our rectangular bar) is given by the formula:
\[ \text{Volume} = \text{Width} \times \text{Height} \times \text{Length} \]
Plugging in the known values:
\[ 46.73 \text{ cm}^3 = 5 \text{ cm} \times 8 \text{ cm} \times \text{Length} \]
Calculating the area of the base:
\[ 5 \text{ cm} \times 8 \text{ cm} = 40 \text{ cm}^2 \]
Now we have:
\[ 46.73 \text{ cm}^3 = 40 \text{ cm}^2 \times \text{Length} \]
To find the length, divide both sides by 40 cm²:
\[ \text{Length} = \frac{46.73 \text{ cm}^3}{40 \text{ cm}^2} \approx 1.16825 \text{ cm} \]
Rounding to the nearest hundredth:
\[ \text{Length} \approx 1.17 \text{ cm} \]
Thus, the length of the platinum bar is approximately 1.17 cm.