A car's gasoline tank has the shape of a right rectangular box with a square base whose sides measure \(62 \mathrm{~cm} .\) Its capacity is \(52 \mathrm{~L}\). If the tank has only 1.5 L remaining, how deep is the gasoline in the tank, assuming the car is parked on level ground?

Since the tank is a right rectangular box with a square base, its total volume can be calculated as the product of the base area and tank height. First, let's convert the capacity from liters to cubic centimeters. Since 1 liter is equal to 1000 cubic centimeters, the tank's capacity is \(52\times1000 = 52,000\mathrm{~cm^3}\). The base area of the tank can be found using the formula for the area of a square: \(A = a^2\), where \(a\) is the length of one side of the square. In this case, the side length is 62 cm, so the base area is \(A = 62^2 = 3844\mathrm{~cm^2}\). Now, we can find the height of the tank (\(h\)) using the formula for the volume of a rectangular box: \(V=A\cdot h\). Since \(V = 52,000 \mathrm{cm^3}\) and \(A = 3844 \mathrm{cm^2}\), we have \(h = \frac{V}{A}\). \(h = \frac{52000}{3844} \approx 13.52 \mathrm{~cm}\) Thus, the height of the tank is approximately 13.52 cm.

We know that there is 1.5 L of gasoline remaining in the tank. To work with the same units, we need to convert this volume to cubic centimeters. Since 1 liter is equal to 1000 cubic centimeters, the remaining gasoline volume is \(1.5 \times 1000 = 1500 \mathrm{~cm^3}\).

With the remaining gasoline volume and the base area, we can calculate the depth of the remaining gasoline (\(d\)) in the tank paralleling the method used to find the height of the tank. We use the formula \(V_d = A \cdot d\), where \(V_d\) is the volume of remaining gasoline, \(A\) is the base area, and \(d\) is the depth of the remaining gasoline. Diving both sides of the equation by \(A\), we get, \(d = \frac{V_d}{A}\). Substituting our known values, we have: \(d = \frac{1500}{3844} \approx 0.39\mathrm{~cm}\) Hence, the depth of gasoline remaining in the tank when parked on level ground is approximately 0.39 cm.