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^3. Round your answer to the nearest hundredth
Answers
GPT-4o mini
To find the length of the platinum bar, we can use the relationship between mass, volume, and density. The formula is given by:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this formula, we can rearrange it to find the volume:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
### Step 1: Calculate the Volume of the platinum bar
We know that the mass \(M\) of the bar is 1 kilogram, which can be converted to grams:
\[
M = 1 \, \text{kg} = 1000 \, \text{g}
\]
The density \(D\) of the platinum bar is given as \(21.4 \, \text{g/cm}^3\). Now we can substitute these values into the volume formula:
\[
\text{Volume} = \frac{M}{D} = \frac{1000 \, \text{g}}{21.4 \, \text{g/cm}^3}
\]
Calculating the volume:
\[
\text{Volume} \approx \frac{1000}{21.4} \approx 46.73 \, \text{cm}^3
\]
### Step 2: Find the length of the bar
The volume of a rectangular prism (the shape of the bar) is given by:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
We know the width and the height of the bar:
- Width \(W = 5 \, \text{cm}\)
- Height \(H = 8 \, \text{cm}\)
Let the unknown length be \(L\). Then we can express the volume as:
\[
\text{Volume} = L \times 5 \, \text{cm} \times 8 \, \text{cm}
\]
Setting the two volume expressions equal to each other:
\[
L \times 5 \times 8 = 46.73
\]
Calculating \(5 \times 8\):
\[
40L = 46.73
\]
Now solve for \(L\):
\[
L = \frac{46.73}{40} \approx 1.16825 \, \text{cm}
\]
### Step 3: Round the answer
Finally, rounding \(L\) to the nearest hundredth:
\[
L \approx 1.17 \, \text{cm}
\]
Thus, the length of the platinum bar is approximately **1.17 centimeters**.
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this formula, we can rearrange it to find the volume:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
### Step 1: Calculate the Volume of the platinum bar
We know that the mass \(M\) of the bar is 1 kilogram, which can be converted to grams:
\[
M = 1 \, \text{kg} = 1000 \, \text{g}
\]
The density \(D\) of the platinum bar is given as \(21.4 \, \text{g/cm}^3\). Now we can substitute these values into the volume formula:
\[
\text{Volume} = \frac{M}{D} = \frac{1000 \, \text{g}}{21.4 \, \text{g/cm}^3}
\]
Calculating the volume:
\[
\text{Volume} \approx \frac{1000}{21.4} \approx 46.73 \, \text{cm}^3
\]
### Step 2: Find the length of the bar
The volume of a rectangular prism (the shape of the bar) is given by:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
We know the width and the height of the bar:
- Width \(W = 5 \, \text{cm}\)
- Height \(H = 8 \, \text{cm}\)
Let the unknown length be \(L\). Then we can express the volume as:
\[
\text{Volume} = L \times 5 \, \text{cm} \times 8 \, \text{cm}
\]
Setting the two volume expressions equal to each other:
\[
L \times 5 \times 8 = 46.73
\]
Calculating \(5 \times 8\):
\[
40L = 46.73
\]
Now solve for \(L\):
\[
L = \frac{46.73}{40} \approx 1.16825 \, \text{cm}
\]
### Step 3: Round the answer
Finally, rounding \(L\) to the nearest hundredth:
\[
L \approx 1.17 \, \text{cm}
\]
Thus, the length of the platinum bar is approximately **1.17 centimeters**.