Question 215089
To find the fuel efficiency with the least carbon dioxide pollution, simply find the vertex of {{{y=32x^2-1952x+42000}}}




In order to find the vertex, we first need to find the x-coordinate of the vertex.



To find the x-coordinate of the vertex, use this formula: {{{x=(-b)/(2a)}}}.



{{{x=(-b)/(2a)}}} Start with the given formula.



From {{{y=32x^2-1952x+42000}}}, we can see that {{{a=32}}}, {{{b=-1952}}}, and {{{c=42000}}}.



{{{x=(-(-1952))/(2(32))}}} Plug in {{{a=32}}} and {{{b=-1952}}}.



{{{x=(1952)/(2(32))}}} Negate {{{-1952}}} to get {{{1952}}}.



{{{x=(1952)/(64)}}} Multiply 2 and {{{32}}} to get {{{64}}}.



{{{x=61/2}}} Reduce.



So the x-coordinate of the vertex is {{{x=61/2}}}. Note: this means that the axis of symmetry is also {{{x=61/2}}}.



Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.



{{{y=32x^2-1952x+42000}}} Start with the given equation.



{{{y=32(61/2)^2-1952(61/2)+42000}}} Plug in {{{x=61/2}}}.



{{{y=32(3721/4)-1952(61/2)+42000}}} Square {{{61/2}}} to get {{{3721/4}}}.



{{{y=29768-1952(61/2)+42000}}} Multiply {{{32}}} and {{{3721/4}}} to get {{{29768}}}.



{{{y=29768-59536+42000}}} Multiply {{{-1952}}} and {{{61/2}}} to get {{{-59536}}}.



{{{y=12232}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=12232}}}.



So the vertex is *[Tex \LARGE \left(\frac{61}{2},12232\right)] which in decimal form is (30.5, 12232)



Since the y-coordinate is the max/min value, this means that the lowest carbon emission is 12,332 pounds per 15000 miles.