Exponential Equation-Asparagus problem

[KingAce]

Problem: You decide to plant asparagus in your kitchen garden. You first harvest 10 stalks in 1986. By 1988, you produce 30 stalks. Assume that the number of stalks you harvest varies exponentially with the number of years since you started harvesting the plants.

*I know the ordered pairs are (10,1986) and (30,1988). This means that every two years, the number of stalks triples (multiplied by 3). But how do I find the particular equation of this function? Thanks

 

[Denis]

rate r cause tripling in 2 years:
(1 + r)^2 = 3
1 + r = sqrt(3)
r = sqrt(3) - 1

So stalks in n years
= 10(1 + r)^n where r = sqrt(3) - 1

if n = 8:
10(sqrt(3))^8 = 810

 

[soroban]

Hello, KingAce!

You decide to plant asparagus in your kitchen garden.
You first harvest 10 stalks in 1986. . By 1988, you produce 30 stalks.
Assume that the number of stalks you harvest varies exponentially
with the number of years since you started harvesting the plants.

Assume that the function is of the form: \(\displaystyle \:y\;=\;a\cdot b^t\)
. . where \(\displaystyle y\) is the yield (number of stalks)
. . and \(\displaystyle t\) is the number of years since 1986.

We have the ordered pairs: \(\displaystyle \0,\,10)\) and \(\displaystyle (2,\,30)\)


Substitute \(\displaystyle (0,10):\;\;10 \:=\:a\cdot b^0\;\;\Rightarrow\;\;a\,=\,10\)

. . The function (so far) is: \(\displaystyle \:y \;=\;10b^t\)


Substitute \(\displaystyle (2,30):\;\;30\:=\:10b^2\;\;\Rightarrow\;\;b^2\:=\:3\;\;\Rightarrow\;\;b\:=\:\sqrt{3}\)

. . The function is: \(\displaystyle \:y\;=\;10\left(\sqrt{3}\right)^t\) . . . or: \(\displaystyle \,y\;=\;10\,\cdot\,3^{\frac{t}{2}}\)