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/cm'.
Round your answer to the nearest hundredth.

Answer 1

To find the length of the platinum bar, we can start by using the formula for density:

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

We know the density of platinum is \( 21.4 \, \text{g/cm}^3 \) and the mass of the bar is \( 1 \, \text{kg} \), which is equivalent to \( 1000 \, \text{g} \).

Rearranging the formula to find the volume:

\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{1000 \, \text{g}}{21.4 \, \text{g/cm}^3}
\]

Calculating the volume:

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

Now, the volume of a rectangular bar is given by the formula:

\[
\text{Volume} = \text{Width} \times \text{Height} \times \text{Length}
\]

Substituting the known values for width and height:

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

Calculating the product of the width and height:

\[
5 \times 8 = 40 \, \text{cm}^2
\]

Now we can set up the equation:

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

To find the length, we divide both sides by 40:

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

Rounding to the nearest hundredth:

\[
\text{Length} \approx 1.17 \, \text{cm}
\]

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