The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively `1.5%` and `1%`, the maximum error in determining the density is:

Options

A. `6%`
B. `2.5%`
C. `3.5%`
D. `4.5%`

Answer

To find the maximum error in determining the density of a cube given the relative errors in measuring its mass and length, we can follow these steps:

### Step 1: Understand the formula for density
The density \( D \) of a material is defined as:
\[ D = \frac{m}{V} \]
where \( m \) is the mass and \( V \) is the volume of the cube.

### Step 2: Calculate the volume of the cube
For a cube with side length \( s \), the volume \( V \) is given by:
\[ V = s^3 \]

### Step 3: Substitute the volume into the density formula
Thus, the density can be expressed as:
\[ D = \frac{m}{s^3} \]

### Step 4: Determine the relative errors
Given:
- Relative error in mass \( \left( \frac{dm}{m} \right) = 1.5\% = 0.015 \)
- Relative error in length \( \left( \frac{ds}{s} \right) = 1\% = 0.01 \)

### Step 5: Use the formula for maximum error in density
To find the maximum error in density, we differentiate the density formula:
\[ \frac{dD}{D} = \frac{dm}{m} + 3 \frac{ds}{s} \]
This equation states that the relative error in density is the sum of the relative error in mass and three times the relative error in length (since volume depends on the cube of the side length).

### Step 6: Substitute the values into the error formula
Substituting the known values:
\[ \frac{dD}{D} = 0.015 + 3 \times 0.01 \]
Calculating this gives:
\[ \frac{dD}{D} = 0.015 + 0.03 = 0.045 \]

### Step 7: Convert to percentage
To express this as a percentage, multiply by 100:
\[ \text{Maximum error in density} = 0.045 \times 100 = 4.5\% \]

### Final Answer
The maximum error in determining the density is \( 4.5\% \).

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