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.
. Round your answer to the nearest hundredth.
Answers
GPT-4o mini
To find the length of the rectangular bar of platinum, we can use the relationship between mass, volume, and density.
First, we can convert the mass of the platinum bar from kilograms to grams:
\[
\text{Mass} = 1 \text{ kg} = 1000 \text{ g}
\]
Next, we use the formula for density:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
Rearranging this formula to solve for volume, we have:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Substituting in the values we know:
\[
\text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3}
\]
Calculating the volume:
\[
\text{Volume} \approx 46.73913 \text{ cm}^3
\]
Next, we know the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the values for width and height (5 cm and 8 cm respectively), we have:
\[
46.73913 \text{ cm}^3 = \text{Length} \times 5 \text{ cm} \times 8 \text{ cm}
\]
Calculating the area of the base:
\[
5 \text{ cm} \times 8 \text{ cm} = 40 \text{ cm}^2
\]
Now we can rewrite the equation for volume as:
\[
46.73913 \text{ cm}^3 = \text{Length} \times 40 \text{ cm}^2
\]
Solving for Length:
\[
\text{Length} = \frac{46.73913 \text{ cm}^3}{40 \text{ cm}^2}
\]
Calculating the length:
\[
\text{Length} \approx 1.1685 \text{ cm}
\]
Rounding to the nearest hundredth:
\[
\text{Length} \approx 1.17 \text{ cm}
\]
Thus, the length of the bar is approximately **1.17 cm**.
First, we can convert the mass of the platinum bar from kilograms to grams:
\[
\text{Mass} = 1 \text{ kg} = 1000 \text{ g}
\]
Next, we use the formula for density:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
Rearranging this formula to solve for volume, we have:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Substituting in the values we know:
\[
\text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3}
\]
Calculating the volume:
\[
\text{Volume} \approx 46.73913 \text{ cm}^3
\]
Next, we know the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the values for width and height (5 cm and 8 cm respectively), we have:
\[
46.73913 \text{ cm}^3 = \text{Length} \times 5 \text{ cm} \times 8 \text{ cm}
\]
Calculating the area of the base:
\[
5 \text{ cm} \times 8 \text{ cm} = 40 \text{ cm}^2
\]
Now we can rewrite the equation for volume as:
\[
46.73913 \text{ cm}^3 = \text{Length} \times 40 \text{ cm}^2
\]
Solving for Length:
\[
\text{Length} = \frac{46.73913 \text{ cm}^3}{40 \text{ cm}^2}
\]
Calculating the length:
\[
\text{Length} \approx 1.1685 \text{ cm}
\]
Rounding to the nearest hundredth:
\[
\text{Length} \approx 1.17 \text{ cm}
\]
Thus, the length of the bar is approximately **1.17 cm**.