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.
Answers
GPT-4o mini
To find the length of the platinum bar, we will use the formula for mass, density, and volume.
The relationship between these quantities is given by:
\[
\text{density} = \frac{\text{mass}}{\text{volume}}
\]
We can rearrange this formula to find the volume:
\[
\text{volume} = \frac{\text{mass}}{\text{density}}
\]
We know the mass of the platinum bar is 1 kilogram (which is 1000 grams) and the density is 21.4 g/cm³.
Now, we can calculate the volume of the platinum bar:
\[
\text{volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.7 \text{ cm}^3
\]
Now, we know that the volume of a rectangular prism is given by the formula:
\[
\text{volume} = \text{length} \times \text{width} \times \text{height}
\]
In this case, the width is 5 cm and the height is 8 cm. Let \( l \) be the length of the bar. We can set up the equation:
\[
l \times 5 \text{ cm} \times 8 \text{ cm} = 46.7 \text{ cm}^3
\]
Simplifying this, we get:
\[
l \times 40 \text{ cm}^2 = 46.7 \text{ cm}^3
\]
To find the length \( l \):
\[
l = \frac{46.7 \text{ cm}^3}{40 \text{ cm}^2} \approx 1.1675 \text{ cm}
\]
Rounding this to the nearest hundredth, we get:
\[
l \approx 1.17 \text{ cm}
\]
Thus, the length of the platinum bar is approximately **1.17 cm**.
The relationship between these quantities is given by:
\[
\text{density} = \frac{\text{mass}}{\text{volume}}
\]
We can rearrange this formula to find the volume:
\[
\text{volume} = \frac{\text{mass}}{\text{density}}
\]
We know the mass of the platinum bar is 1 kilogram (which is 1000 grams) and the density is 21.4 g/cm³.
Now, we can calculate the volume of the platinum bar:
\[
\text{volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.7 \text{ cm}^3
\]
Now, we know that the volume of a rectangular prism is given by the formula:
\[
\text{volume} = \text{length} \times \text{width} \times \text{height}
\]
In this case, the width is 5 cm and the height is 8 cm. Let \( l \) be the length of the bar. We can set up the equation:
\[
l \times 5 \text{ cm} \times 8 \text{ cm} = 46.7 \text{ cm}^3
\]
Simplifying this, we get:
\[
l \times 40 \text{ cm}^2 = 46.7 \text{ cm}^3
\]
To find the length \( l \):
\[
l = \frac{46.7 \text{ cm}^3}{40 \text{ cm}^2} \approx 1.1675 \text{ cm}
\]
Rounding this to the nearest hundredth, we get:
\[
l \approx 1.17 \text{ cm}
\]
Thus, the length of the platinum bar is approximately **1.17 cm**.