the gas mileage m(x) (in mpg) for a certain vehicle can be approximated by m(x)=-0.028x² + 2.597x - 35.02…

the gas mileage m(x) (in mpg) for a certain vehicle can be approximated by m(x)=-0.028x² + 2.597x - 35.02, where x is the speed of the vehicle in mph. part 1 of 2 (a) determine the speed at which the car gets its maximum mileage. round your answer to the nearest mph. the gas mileage is at a maximum when the car travels at 46 mph. part: 1 / 2 part 2 of 2 (b) determine the maximum gas mileage. round your answer to one decimal place. the maximum gas mileage is mpg.

the gas mileage m(x) (in mpg) for a certain vehicle can be approximated by m(x)=-0.028x² + 2.597x - 35.02, where x is the speed of the vehicle in mph. part 1 of 2 (a) determine the speed at which the car gets its maximum mileage. round your answer to the nearest mph. the gas mileage is at a maximum when the car travels at 46 mph. part: 1 / 2 part 2 of 2 (b) determine the maximum gas mileage. round your answer to one decimal place. the maximum gas mileage is mpg.

Answer

Explanation:

Step1: Identify the function

The gas - mileage function is $m(x)=-0.028x^{2}+2.597x - 35.02$, which is a quadratic function in the form $y = ax^{2}+bx + c$ with $a=-0.028$, $b = 2.597$, and $c=-35.02$.

Step2: Recall the formula for the x - coordinate of the vertex of a quadratic function

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. We already know from part (a) that the speed $x = 46$ mph (since $x=-\frac{2.597}{2\times(-0.028)}\approx46$).

Step3: Substitute the value of x into the gas - mileage function

Substitute $x = 46$ into $m(x)=-0.028x^{2}+2.597x - 35.02$. $m(46)=-0.028\times46^{2}+2.597\times46-35.02$ First, calculate $-0.028\times46^{2}=-0.028\times2116=-59.248$. Then, calculate $2.597\times46 = 119.462$. Now, $m(46)=-59.248 + 119.462-35.02$. $m(46)=(-59.248+119.462)-35.02=60.214 - 35.02=25.2$.

Answer:

$25.2$