Question

A rectangular bar of platinum has a width of 5 centimeters, a height of 8 centimeters, and an unknown length. You measure the platinum bar and it is exactly 1 kilogram. Find the length of the bar if you know its density is 21.4g/cm^3. Round your answer to the nearest hundredth.

Answer 1

To find the length of the platinum bar, we can use the relationship between mass, volume, and density. The formula is:

\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]

From this, we can rearrange the formula to find the volume:

\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]

First, we need to convert the mass of the platinum bar from kilograms to grams since the density is given in grams per cubic centimeter.

\[
1 \text{ kg} = 1000 \text{ g}
\]

Now we can calculate the volume using the given density of platinum, which is 21.4 g/cm³:

\[
\text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.73 \text{ cm}^3
\]

Now that we have the volume, we can determine the length of the bar. The volume \( V \) of a rectangular bar is calculated using the formula:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

Here, the width is 5 cm and the height is 8 cm. Let \( L \) be the unknown length. Therefore, we can set up the equation:

\[
46.73 = L \times 5 \times 8
\]

Calculating \( 5 \times 8 \):

\[
5 \times 8 = 40
\]

Now, substitute this back into the equation:

\[
46.73 = L \times 40
\]

To solve for \( L \), divide both sides by 40:

\[
L = \frac{46.73}{40} \approx 1.16825
\]

Rounding this to the nearest hundredth gives us:

\[
L \approx 1.17 \text{ cm}
\]

Thus, the length of the platinum bar is approximately \( \boxed{1.17} \) cm.