The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is ______.

 To determine the maximum percentage error in the volume of the cone, we need to analyze how errors in the measurements of the diameter and height propagate to the volume. The volume 

$V$ of a cone is given by:

$$V=\frac{1}{3}\pi r^{2}h$$

where:

• $r$ is the radius of the base,

• $h$ is the height of the cone.

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Step 1: Express Volume in Terms of Diameter

The diameter $d$ of the base is related to the radius by $r=\frac{d}{2}$. Substituting this into the volume formula:

$$V=\frac{1}{3}\pi {\left(\frac{d}{2}\right)}^{2}h=\frac{1}{12}\pi d^{2}h$$

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Step 2: Relative Errors in Measurements

The least count of the scale is $2\text{mm}=0.2\text{cm}$. The diameter $d$ and height $h$ are both measured as $20.0\text{cm}$. The absolute error in each measurement is $\mathrm{\Delta }d=\mathrm{\Delta }h=0.2\text{cm}$.

The relative errors in $d$ and $h$ are:

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Step 3: Error Propagation in Volume

The volume $V$ depends on $d^2$ and $h$. Using the formula for error propagation, the relative error in $V$ is:

$$\frac{\mathrm{\Delta }V}{V}=2\cdot \frac{\mathrm{\Delta }d}{d}+\frac{\mathrm{\Delta }h}{h}$$

Substitute the relative errors:

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Step 4: Maximum Percentage Error

The maximum percentage error in the volume is:

$$\frac{\mathrm{\Delta }V}{V}\times 100=3\mathrm{\%}$$

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Final Answer:

The maximum percentage error in the determination of the volume is:

$$\boxed{{\displaystyle 3\mathrm{\%}}}$$