Final answer:
To find the required capacity of a gasoline tank based on a given probability density function, integrate the function to get the necessary volume. Use the integral value corresponding to a 90% certainty of the tank not being emptied to determine the tank's capacity. For passenger car tank size reasonability, compare calculated volume with standard tank sizes.
Step-by-step explanation:
The question asks about calculating the capacity of a gasoline tank based on a given probability density function to ensure a specific level of supply reliability.
To calculate the tank capacity required so that the probability of the supply being exhausted is 0.1, one would first have to integrate the probability density function, f(x) = 5(1 - x)⁴, from 0 to a certain value c, such that the area under the curve equals 0.9 (because we want a 10% chance of running out of fuel, meaning there's a 90% chance we won't). Solving the integral of f(x) with respect to x and setting it equal to 0.9 will yield the value c. This value c represents the required capacity in thousands of gallons for the probability criterion to be met.
To answer part (b) of the question that pertains to whether a tank size is reasonable for a passenger car, one would typically compare the tank's volume with the average tank sizes for passenger cars. The given dimensions of the tank can be used to calculate its volume using the formula volume = width × length × depth. With the density of gasoline, the mass (50 kg) can be converted to volume in liters, which then can be compared to the average tank capacity for passenger cars.