Question

Consider a \(70-\mathrm{kg}\) woman who has a total foot imprint area of \(400 \mathrm{cm}^{2}\). She wishes to walk on the snow, but the snow cannot withstand pressures greater than 0.5 kPa. Determine the minimum size of the snowshoes needed (imprint area per shoe) to enable her to walk on the snow without sinking.

Short answer

Answer: The minimum size of each snowshoe needed is 6867 cm^2.

Step-by-step solution

Calculate the woman's weight

First, we need to find the woman's weight by multiplying her mass by the acceleration due to gravity (9.81 m/s^2). Let W represent her weight. W = mass x gravity W = 70 kg * 9.81 \frac{m}{s^2} W = 686.7 N

Convert the given pressure to Pa

We are given the maximum pressure the snow can withstand in kPa. We need to convert it to Pa. 0.5 kPa = 0.5 * 1000 Pa = 500 Pa

Use the pressure formula to find the necessary area

We'll use the pressure formula, \(P = \frac{F}{A}\), to find the necessary area of the snowshoes. Rearranging it to solve for A, we get \(A = \frac{F}{P}\). A = \frac{686.7 N}{500 Pa} A = 1.3734 m^2

Divide the total area by 2 to find the area per shoe

Since the woman will be wearing two snowshoes, we must find the minimum effective area for each snowshoe to distribute her weight. Divide the total area by 2. Area per shoe = \frac{1.3734 m^2}{2} Area per shoe = 0.6867 m^2

Convert the area to cm^2

To present the result in cm^2, we need to convert the minimum area per shoe from m^2 to cm^2. 0.6867 m^2 * 10000 \frac{cm^2}{m^2} = 6867 cm^2 The minimum size of the snowshoes needed (imprint area per shoe) to enable her to walk on the snow without sinking is 6867 cm^2 per shoe.