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Most cars get their best gas mileage when traveling at relatively modest speed. The gas mileage M for a certain new car is modeled by the function. ๐(๐ ) = โ 1/ 28*(๐ ^2) + (3 + ๐)๐  + (31 + ๐)  ,  note:  (m=8 )       , where ๐  is the speed in mi/h and M is measured in mi/gal   .

a. What is the carโs best gas mileage, and at what speed is it attained? Justify your answer?

b. Sketch the quadratic function manually showing all steps.

Jun 25, 2021

#1
+163
+2

i would recommend doing b, then going on with a;

though, before we continue any further, lets just work the function a bit:

$M(s)=-\frac{1}{28}(s^2)+(3+8)s+(31+8)$

$M(s)=-\frac{1}{28}s^2+\left(3+8\right)s+31+8$

$M(s)=-\frac{1}{28}s^2+11s+39$

used desmos and got this image:

now its asking us for the cars best gas mileage, and the speed its in

in our graph, the x axis is going to be speed, and the y axis is going to be the gas mileage

thus, by looking at the coordinate of the vertex, or the max, we can see that the best mileage is at $\boxed{886 \ \text{miles per gallon}}$, whilst the speed this is attained is at $\boxed{154 \ \text{miles per hour}}$.

:D

Jun 25, 2021
#2
+36430
+2

As I posted yesterday (when the value of 'm' was not given in the original posting) , the best mileage will occur at speed , s = - b/2a    where b = 3+m = 11    and a = - 1/28

- 11 / ( 2* (- 1/28)) = 154    mi/hr      ( not very realistic....but that is the answer given the equation in the question)

substituting this value into the equation to find the mileage results in:   (-1/28) ( 154^2) + 11(154) + 39 = 886 m/gal    ( again, not very realistic)

Jun 25, 2021