To find the length of the platinum bar, we can use the formula for density, which is given by:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Rearranging this formula gives us:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
-
Convert the mass from kilograms to grams: The mass of the platinum bar is 1 kilogram, which is equivalent to \(1000\) grams.
-
Use the density to find the volume: The density of platinum is \(21.4 , \text{g/cm}^3\). Plugging in the values, we get:
\[ \text{Volume} = \frac{1000 , \text{g}}{21.4 , \text{g/cm}^3} \approx 46.731 , \text{cm}^3 \]
-
Now, we can find the volume of a rectangular bar with given dimensions: The volume \(V\) of a rectangular bar is given by:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given the width \(w = 5 , \text{cm}\) and the height \(h = 8 , \text{cm}\), we can write:
\[ V = L \times 5 , \text{cm} \times 8 , \text{cm} \]
where \(L\) is the length of the bar we need to find. From the volume we calculated earlier, we have:
\[ 46.731 , \text{cm}^3 = L \times 5 , \text{cm} \times 8 , \text{cm} \]
-
Solving for length \(L\):
\[ 46.731 = L \times 40 \]
\[ L = \frac{46.731}{40} \approx 1.168275 \]
-
Rounding to the nearest hundredth:
\[ L \approx 1.17 , \text{cm} \]
Thus, the length of the platinum bar is approximately 1.17 cm.