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 21.4 g/cm3 . Round your answer to the nearest hundredth.(1 point)
cm

Answer 1

To find the length of the platinum bar, we can use the relationship between mass, volume, and density. The formula for density (\( \rho \)) is given by:

\[
\rho = \frac{\text{mass}}{\text{volume}}
\]

We can rearrange this formula to find the volume:

\[
\text{volume} = \frac{\text{mass}}{\rho}
\]

### Step 1: Convert mass to grams
Since the density is given in grams per cubic centimeter, we need to convert the mass from kilograms to grams.

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

### Step 2: Calculate the volume of the platinum bar
Using the density of platinum (21.4 g/cm\(^3\)), we can calculate the volume of the bar:

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

### Step 3: Calculate the length of the bar
The volume \( V \) of a rectangular bar is given by the formula:

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

Given that the width is 5 cm and the height is 8 cm, we can substitute these values into the equation:

\[
46.73 \text{ cm}^3 = \text{length} \times 5 \text{ cm} \times 8 \text{ cm}
\]

This simplifies to:

\[
46.73 = \text{length} \times 40
\]

Now, solving for length:

\[
\text{length} = \frac{46.73}{40} \approx 1.16825 \text{ cm}
\]

### Step 4: Round the length to the nearest hundredth
Rounding 1.16825 to the nearest hundredth gives:

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

Thus, the length of the bar is approximately **1.17 cm**.