Question

A cylindrical rod formed from silicon is \(16.8 \mathrm{~cm}\) long and has a mass of \(2.17 \mathrm{~kg}\). The density of silicon is \(2.33 \mathrm{~g} / \mathrm{cm}^{3}\). What is the diameter of the cylinder? (The volume of a cylinder is given by \(\pi r^{2} h\), where \(r\) is the radius, and \(h\) is its length.)

Short answer

The diameter of the cylindrical silicon rod is \(4.74 \mathrm{~cm}\).

Step-by-step solution

Write down the knowns and unknowns

We are given: - Mass of the rod, m = \(2.17 \mathrm{~kg}\) - Length of the rod, h = \(16.8 \mathrm{~cm}\) - Density of silicon, ρ = \(2.33 \mathrm{~g/cm^3}\) - Volume formula for a cylinder, V(volume) = \(\pi r^{2} h\) - We need to find the diameter (or the radius) of the cylindrical rod.

Convert mass from kg to g

Since the density unit is g/cm³ but the mass is given in kg, we will need to convert the mass to g in order to use it with density. 1 kg = 1000 g So, mass of the rod in g, m = \(2.17 \mathrm{~kg}\) × 1000 = \(2170 \mathrm{~g}\)

Write down the density formula and calculate the volume

The density formula is: ρ = m / V We can solve for V(volume) as, V = m / ρ Now, substitute the known values: V = \(2170 \mathrm{~g}\) / \(2.33 \mathrm{~g/cm^3}\) = \(930.9 \mathrm{~cm}^3\)

Substitute the volume formula into the cylinder formula

We know the volume formula of the cylinder is V = \(\pi r^{2} h\). Let's substitute the volume, V = \(930.9 \mathrm{~cm}^3\), and the length of rod, h = \(16.8 \mathrm{~cm}\), into the formula: \(930.9 \mathrm{~cm}^3 = \pi r^{2} (16.8 \mathrm{~cm})\)

Solve for r

Now, to solve for radius (r), we need to divide both sides of the equation by \(16.8 \mathrm{~cm}\) and \(\pi\): \(r^{2} = \frac{930.9 \mathrm{~cm}^3}{16.8 \mathrm{~cm} \times \pi}\) \(r^{2} = \frac{930.9 \mathrm{~cm}^3}{52.9 \pi \mathrm{~cm}}\) \(r^{2} = 5.63 \mathrm{~cm}^2\) Then, take the square root to find r: \(r = \sqrt{5.63 \mathrm{~cm}^2}\) = \(2.37 \mathrm{~cm}\)

Calculate the diameter

Finally, we have the radius, r = \(2.37 \mathrm{~cm}\). The diameter (D) of the cylindrical rod is simply double the radius: D = 2r D = 2 × \(2.37 \mathrm{~cm}\) = \(4.74 \mathrm{~cm}\) So, the diameter of the cylindrical silicon rod is \(4.74 \mathrm{~cm}\).