Question

The dimensions of aluminum foil in a box for sale in supermarkets are \(66^{2} / 3\) yards by 12 inches. The mass of the foil is \(0.83 \mathrm{~kg} .\) If its density is \(2.70 \mathrm{~g} / \mathrm{cm}^{3},\) then what is the thickness of the foil in inches?

Short answer

Answer: To find the thickness of the aluminum foil, follow these steps: 1. Convert the given dimensions to meters: length = \((66^{2} /3)*0.9144\) meters, and width = \(12*0.0254\) meters. 2. Calculate the volume of the foil using the mass and density: \(V = \frac{0.83 * 10^3}{2.70 * 10^3}\) m³. 3. Determine the thickness using the calculated volume: \(t = \frac{(0.83 * 10^3)/(2.70 * 10^3)}{((66^{2}/3)*0.9144)*(12*0.0254)}\) meters. 4. Convert the thickness to inches: Thickness (in inches) = \(\frac{t}{0.0254}\) inches.

Step-by-step solution

Convert dimensions to a consistent unit system (metric)

First, we need to convert the given dimensions of the aluminum foil to meters, since the density is given in g/cm³. We have the dimensions as \(66^{2} / 3\) yards by 12 inches. 1 yard = 0.9144 meters, 1 inch = 2.54 cm = 0.0254 meters. So, the length of foil is \((66^2/3)\) yards, which is \((66^2/3)*0.9144\) meters. The width of foil is 12 inches, which is \(12*0.0254\) meters.

Calculate the volume of the aluminum foil

The volume of the foil (V) can be calculated using the given mass (m) and density (ρ). The formula for volume is: \(V = \frac{m}{ρ}\) Given that the mass (m) is \(0.83\) kg and the density (ρ) is \(2.70\) g/cm³, first convert the mass to grams and density to kg/m³. Mass (m) = \(0.83 * 10^3\) g Density (ρ) = \(2.70 * 10^3\) kg/m³ Now, calculate the volume: \(V = \frac{m}{ρ} = \frac{0.83 * 10^3}{2.70 * 10^3}\) m³

Determine the thickness of the foil using the calculated volume

The volume of the foil can also be expressed as the product of its length, width, and thickness (t). \(V = length * width * thickness\) Using the volume calculated in Step 2, and the converted dimensions from Step 1, we can solve for the thickness (t). \(t = \frac{V}{(length * width)}\) Then, plug in the values: \(t = \frac{(0.83 * 10^3)/(2.70 * 10^3)}{((66^2/3)*0.9144)*(12*0.0254)}\) meters

Convert the thickness to inches

Finally, to convert the thickness back to inches, divide the calculated thickness (in meters) by 0.0254. Thickness (in inches) = \(\frac{t}{0.0254}\) inches Thus, we find the thickness of the aluminum foil in inches.