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
. Round your answer to the nearest hundredth.(1 point)
cm
Answers
GPT-4o mini
To find the length of the platinum bar, we can use the formula for density, which is defined as:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this, we can rearrange the formula to find the volume:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
1. We know the mass of the platinum bar is 1 kilogram, which is equivalent to 1000 grams.
2. The density of platinum is given as 21.4 g/cm³.
Substituting the values into the volume formula:
\[
\text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.7 \text{ cm}^3
\]
Now we can calculate the volume of the rectangular bar using the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Width} \times \text{Height} \times \text{Length}
\]
The dimensions of the bar are as follows:
- Width = 5 cm
- Height = 8 cm
- Length = \( L \) (unknown)
Setting the two volume expressions equal to each other:
\[
5 \text{ cm} \times 8 \text{ cm} \times L = 46.7 \text{ cm}^3
\]
Calculating the left side:
\[
40 \text{ cm}^2 \times L = 46.7 \text{ cm}^3
\]
Now solve for \( L \):
\[
L = \frac{46.7 \text{ cm}^3}{40 \text{ cm}^2} = 1.1675 \text{ cm}
\]
Rounding to the nearest hundredth:
\[
L \approx 1.17 \text{ cm}
\]
Thus, the length of the bar is approximately \( \boxed{1.17} \) cm.
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this, we can rearrange the formula to find the volume:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
1. We know the mass of the platinum bar is 1 kilogram, which is equivalent to 1000 grams.
2. The density of platinum is given as 21.4 g/cm³.
Substituting the values into the volume formula:
\[
\text{Volume} = \frac{1000 \text{ g}}{21.4 \text{ g/cm}^3} \approx 46.7 \text{ cm}^3
\]
Now we can calculate the volume of the rectangular bar using the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Width} \times \text{Height} \times \text{Length}
\]
The dimensions of the bar are as follows:
- Width = 5 cm
- Height = 8 cm
- Length = \( L \) (unknown)
Setting the two volume expressions equal to each other:
\[
5 \text{ cm} \times 8 \text{ cm} \times L = 46.7 \text{ cm}^3
\]
Calculating the left side:
\[
40 \text{ cm}^2 \times L = 46.7 \text{ cm}^3
\]
Now solve for \( L \):
\[
L = \frac{46.7 \text{ cm}^3}{40 \text{ cm}^2} = 1.1675 \text{ cm}
\]
Rounding to the nearest hundredth:
\[
L \approx 1.17 \text{ cm}
\]
Thus, the length of the bar is approximately \( \boxed{1.17} \) cm.