Question 278152
Let x = number of new trees to be planted.


So adding 'x' trees to the already established 30 trees gives us {{{30+x}}} trees total. Now because "each new tree will reduce the average yield per tree by about 10 apples over the growing season", this means that each tree will produce {{{400-10x}}} apples. To see why, let x=1 which means that 400-10(1)=400-10=390. So planting one new tree means that each tree only produces 390 apples, which is what we expect. Try other values of 'x' to confirm this expression.



Since we have {{{30+x}}} trees which each produce {{{400-10x}}} apples, this means that there will be a total of {{{(30+x)(400-10x)}}} apples total after 'x' trees are planted.



Note: if there is a part that doesn't make sense explained above, it is wise to go over it again or ask me about it. Translation is a critical element to solving word problems.



So let {{{y=(30+x)(400-10x)}}}



{{{y=(30+x)(400-10x)}}} Start with the given equation.



{{{y=12000-300x+400x-10x^2}}} FOIL



{{{y=-10x^2+100x+12000}}} Rearrange and combine like terms.



Now the goal is to maximize 'y' which is the total number of apples. To do this, we need to find the y coordinate of the vertex. Remember that the vertex is the highest/lowest point on a graph. In this case, it is the highest point on the parabola.




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=-10x^2+100x+12000}}}, we can see that {{{a=-10}}}, {{{b=100}}}, and {{{c=12000}}}.



{{{x=(-(100))/(2(-10))}}} Plug in {{{a=-10}}} and {{{b=100}}}.



{{{x=(-100)/(-20)}}} Multiply 2 and {{{-10}}} to get {{{-20}}}.



{{{x=5}}} Divide.



So the x-coordinate of the vertex is {{{x=5}}}. Note: this means that the axis of symmetry is also {{{x=5}}}.



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



{{{y=-10x^2+100x+12000}}} Start with the given equation.



{{{y=-10(5)^2+100(5)+12000}}} Plug in {{{x=5}}}.



{{{y=-10(25)+100(5)+12000}}} Square {{{5}}} to get {{{25}}}.



{{{y=-250+100(5)+12000}}} Multiply {{{-10}}} and {{{25}}} to get {{{-250}}}.



{{{y=-250+500+12000}}} Multiply {{{100}}} and {{{5}}} to get {{{500}}}.



{{{y=12250}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=12250}}}.



So the vertex is *[Tex \LARGE \left(5,12250\right)].



Since the y-coordinate of the vertex is {{{y=12250}}}, this means that the maximum production of apples is 12,250 which occurs when 5 new trees are planted (since x=5 at the vertex).