An ordinary gasoline can measuring \(30.0 \mathrm{~cm}\) by \(20.0 \mathrm{~cm}\) by \(15.0 \mathrm{~cm}\) is evacuated with a vacuum pump. Assuming that virtually all of the air can be removed from inside the can and that atmospheric pressure is \(14.7 \mathrm{psi},\) what is the total force (in pounds) on the surface of the can? Do you think that the can will withstand the force?

Understanding surface area is critical in diverse scientific calculations, not least when dealing with forces exerted by atmospheric pressure on objects. The surface area of an object like a gasoline can is simply the sum of all the areas of the sides that enclose it. For a rectangular object, we consider six sides - pairs of identical rectangles. The surface area is calculated by multiplying the length and width for the top and bottom, the length and height for the front and back, and finally, the width and height for the sides, summing these products together to get the total area.

For a gasoline can that measures 30.0 cm by 20.0 cm by 15.0 cm, the calculation would appear like this: \[ A = 2(30.0 \times 20.0) + 2(30.0 \times 15.0) + 2(20.0 \times 15.0) \] Simplifying the expression gives us the total surface area in square centimeters. However, since atmospheric pressure is often measured in psi (pounds per square inch), it's then necessary to convert this area to square inches—a process we'll look at shortly.

The relationship between pressure and force is a fundamental principle of physics that tells us how an external pressure impacts an object. Pressure is defined as the amount of force exerted per unit area, and it's a measure of how concentrated that force is. To determine the force applied on the surface of an object, we use the simple equation \( F = PA \), where \( F \) represents the force, \( P \) signifies the pressure, and \( A \) expresses the surface area over which the pressure is applied.

For the evacuated gasoline can scenario, once the surface area is known in the correct units, calculating the total force coming from atmospheric pressure is straightforward. With the given pressure of 14.7 psi, this force is the result of atmospheric pressure working against each square inch of the can's external surface. Indeed, the larger the surface area, the greater the total force will be.

In sciences and engineering tasks, a clear understanding of unit conversion is vital because consistency in units is key to accurate calculations. The conversion factor between inches and centimeters, for example, is crucial when dealing with surface areas and pressures in differing units. There are 2.54 centimeters in an inch, so to convert square centimeters to square inches, we square this factor, resulting in (2.54)^2 = 6.4516 square centimeters per square inch.

Care must be taken when converting to ensure accuracy. To convert the surface area of our gasoline can from square centimeters to square inches, we divide the area calculated in centimeters squared by 6.4516. This adjusted figure then integrates seamlessly into the pressure equation, providing the final force in pounds (lbs), which is a coherent unit for comparing against the strength and resilience of the structure in question.