we'll let x = the number of weeks.
when x = 0, the farmer has 600 bushels to sell.
when x = 1, the farmer has 700 bushels to sell.
each week, the farmer has 100 more bushels to sell.
the formula for that would be b = 100 * x + 600
b is the number of bushels.
x is the number of weeks.
if you apply the formula, you can see that:
when x = 0, b = 600
when x = 1, b = 700
when x = 2, b = 800
etc.
now the price that he can get for each bushels is 1.00 today and .90 in a week and .80 in another week.
the formula for that would be p = (1 - .10 * x) * b
p is the price that he gets for the potatoes in week x.
when x = 0, p = 1 * b
when x = 1, p = (1-.10*1) * b = .90 * b
when x = 2, p = (1 - .10*2) * b = .80 * b
etc.
so you have two formulas:
the first formuls is b = 100 * x + 600
the second formula is p = (1 - .10 * x) * b
i will combine these formula later, but for now, i wish to show you how you would analyze this with just the information you already have.
we'll do an analysis for x = 0, 1, 2, 3, and 4.
since p depends on b, we'll evaluate for b first, and then for p.
you'll get a table that looks like this:
x b p
0 600 600
1 700 630
2 800 640
3 900 630
4 1000 600
from the table, you can see that the price the farmer gets for the potatoes will peak at the end of 2 weeks.
an example of how we got the number for b and p is shown below:
when x = 2, b = 100 * x + 600 which becomes:
b = 100 * 2 + 600 which becomes:
b = 200 + 600 which becomes:
b = 800.
when x = 2:
p = (1-.10*x) * b
since b = 800 when x = 2, the formula becomes:
p = (1 - .10*2)*800 which becomes:
p = .8 * 800 which becomes:
p = 640.
in fact, what you see is the values given by a quadratic equation.
how did we get that quadratic equation?
look below for the derivation of the quadratic equation and the graph of that equation.
go back to where we derived both formulas:
the first formuls is b = 100 * x + 600
the second formula is p = (1 - .10 * x) * b
the second formula depends on the value of b from the first formula.
we can replace b in the second formula with the value of b from the first formula.
we will get:
p = (1 - .10 * x) * b becomes:
p = (1 - .10 * x) * (100 * x + 600)
b was replaced with its equivalent value of (100 * x + 600)
we can graph this formula as is by replacing p with y to get:
y = (1 - .10 * x) * (100 * x + 600)
we can also convert it to ax^2 + bx + c form by performing the multiplication indicated.
when we perform the multiplication indicated, we get:
y = (1 - .10 * x) * (100 * x + 600) becomes:
y = (1 * 100 * x) + (1 * 600) - (.10 * x * 100 * x) - (.10 * 600 * x)
we simplify this to get:
y = 100x + 600 - 10x^2 - 60x
we combine like terms to get:
y = 40x + 600 - 10x^2
we rearrange the expression on the right in descending order of the power of each term to get:
y = -10x^2 + 40x + 600
the two equations of:
[y = (1 - .10 * x) * (100 * x + 600)] and [y = -10x^2 + 40x + 600] are equivalent.
the graph of both equations will be identical.
this means if you show both equations on the same graph, their graphs will superimpose on each other and look like one graph rather than 2.
the graph of both of these equations is shown below:
the black line is the graph of both of these equations.
the orange line shows the value of x = 2.
the intersection point of the black line and the orange line shows the value of y when the value of x is equal to 2.
this intersection point agrees with the table.
the value of y is 640 when the value of x = 2.
this means the price the farmer gets for the potatoes is max at the end of 2 weeks and is equal to 640 dollars.
you can also find the max point by using the formula that tells you:
x value of max/min point of a quadratic equation is:
x = -b/2a
in the equation of y = -10x^2 + 40x + 600:
a = -10
b = 40
c = 600
x value of max/min point is -40 / -20 = 2.
y value when x value is 2 is equal to -10*2^2 + 40*2 + 600.
that becomes -40 + 80 + 600 which becomes:
40 + 600 which becomes:
y = 640 when x = 2.