Question

The fuel consumption, in gallons per minute, for an aircraft during 60-minute flight is f(x)=0.1x²-6x+180, where x is the number of minutes into the flight. What is the minimum fuel consumption and at what point in the flight does it occur?

Answer 1

The minimum fuel consumption for the aircraft, given by the quadratic function f(x) = 0.1x² - 6x + 180, is 90 gallons per minute and occurs 30 minutes into the flight.

Step-by-step explanation:

The question asks to determine the minimum fuel consumption for an aircraft during a 60-minute flight, represented by a quadratic function f(x) = 0.1x² - 6x + 180, and to find out at what point into the flight this minimum occurs.

First, we recognize that this is a quadratic function, which graphically represents a parabola. Since the coefficient of x² is positive (0.1), the parabola opens upwards and the vertex represents the minimum point. To find the vertex, we can use the vertex formula x = -b/(2a), where a is the coefficient of x² and b is the coefficient of x. Plugging the coefficients from the function into the formula, we get x = -(-6)/(2*0.1) = 30 minutes.

Now, substituting x back into the function, f(30) = 0.1(30) ² - 6(30) + 180 = 0.1(900) - 180 + 180 = 90 - 180 + 180 = 90 gallons per minute. So, the minimum fuel consumption is 90 gallons per minute, which occurs 30 minutes into the flight.