Question

A laboratory experiment requires \(12.0 \mathrm{~g}\) of aluminum wire \(\left(d=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right)\). The diameter of the wire is 0.200 in. Determine the length of the wire, in centimeters, to be used for this experiment. The volume of a cylinder is \(\pi r^{2} \ell,\) where \(r=\) radius and \(\ell=\) length.

Short answer

Calculate the volume: $$ V \approx 4.44\,\text{cm}^3 $$ #tag_title#Step 2: Calculate the length of the aluminum wire#tag_content# We can treat the wire as a cylinder with a given diameter and calculate its length using the formula for the volume of a cylinder: $$ V = \pi r^2h $$ where, - V = volume - r = radius of the cylinder (half of the diameter) - h = height, or length, of the cylinder We already determined the volume of the wire in Step 1 to be approximately 4.44 cm³. We are also given the diameter of the wire as 0.10 cm, so its radius is: $$ r = \frac{0.10}{2} = 0.05\,\text{cm} $$ Now, we can rearrange the formula for the volume of a cylinder to solve for the length of the wire: $$ h = \frac{V}{\pi r^2} $$ Insert the values for V, r, and $\pi$: $$ h = \frac{4.44}{\pi (0.05)^2} $$ Calculate the length: $$ h \approx 178.01\,\text{cm} $$ The length of the aluminum wire is approximately 178.01 cm.

Step-by-step solution

Calculate the volume of the aluminum wire

Since we have the mass and density of the aluminum wire, we can find the wire's volume using the formula: $$ V = \frac{m}{d} $$ where, - V = volume - m = mass of aluminum wire - d = density of aluminum Given, - m = 12.0 g - d = 2.70 g/cm³ Insert the given values into the formula: $$ V = \frac{12.0}{2.70} $$