Question

{Use of Tech} Fuel economy Sup​​pose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.a. Graph and interpret the mileage function.

Accepted Answer

To graph the mileage function m(g) = 50g - $$25.8g^2$$ + $$12.5g^3$$ - $$1.6g^4$$, first identify the domain of the function, which is 0 ≤ g ≤ 4. This means we will only consider values of g within this interval. Next, calculate key points of the function within the domain. This includes finding the value of m(g) at the endpoints g = 0 and g = 4, as well as any critical points where the derivative m'(g) = 0, which will help identify local maxima or minima. To find the critical points, take the derivative of m(g) with respect to g: m'(g) = 50 - 51.6g + $$37.5g^2$$ - $$6.4g^3. $$Set m'(g) = 0 and solve for g to find the critical points within the interval [0, 4]. Evaluate the function m(g) at the critical points and endpoints to determine the behavior of the function. This will help in understanding how the mileage changes as the amount of gas changes. Plot the calculated points and sketch the graph of m(g) over the interval [0, 4]. Interpret the graph by analyzing how the mileage varies with the amount of gas, noting any points where the mileage is maximized or minimized.

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Key Concepts

Function Analysis

Graphing Techniques

Critical Points and Interpretation