An orchard needs to produce 5,000 oranges. The following
function represents the average number of oranges that
number of trees must produce to meet the goal
How do the mathematical range and reasonable range
compare?
mathematical: [YER[y=0} reasonable: Ve Rly> 0)
F(x) = 5,000
mathematical: {VER|y20} reasonable: {y ER|Y20)
mathematical: {YER=0} reasonable: tye wy>0}
mathematical: {VERY=0) reasonable:ye Wy20)

answer:

x=# of new trees planted

(100+x)(300-2x)=y

30,000+300x-200x-2x2=y

-2x2+100x+30,000=0

divide all terms by 2

-x2+50x+15,00=0

this is a parabola that opens downward so it has a maximum point, its vertex given by (h,k)

h=-b/2a and k=f(h)

where does -b/2a come from ?

ax2+bx+c=y

complete the square

ax2+bx   =y-c

a(x2+(b/a)x)  =y-c

a(x2+(b/a)x)  =y-c+a(   )

(b/2a)2=b2/4a2

add this to both sides

a(x2+(b/a)x)+(b2/4a2))=y-c+a(b2/4a2)

a(x+[b/2a])2=y-c+a(b2/4a2)

a(x+[b/2a])2-(b2/4a)+c=y

a(x-[-b/2a])2+([4ac-b2]/4a)=y

compare this to y=a(x-h)2+k and see that h=-b/2a and k=(4ac-b2)/4a

using -b/2a

-b/2a=-50/-2=25

the orchard should contain 25 trees to maximize revenue

Step-by-step explanation:

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step-by-step explanation:

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