The size of an antler on a deer depends linearly on the age of the animal. For a mule deer in the Cache la poudre deer herd in Colorado, the antler begins growing at age 10 months and reaches a weight of 1.05 pounds after 70 months. Let w(t) be the weight of the antler for a deer of t months.
a. Find the formula for w(t).
b. What will the antler weight after 40 months ?
c. At what age will the antler weigh 0.7 pound ?
d. How much is the weight increasing each month ?
Being linear means...
weight= m*time + constant
You are given that weight is 0 at t=10, and at t=70, weight is 1.05
0=m*10 + C
1.05=m70 + C
solve those for m,and C.
There exists a shortcut for such problems.
If you know that W(t) = 0 at t = 10, then you know that W(t) must contain a factor (t-10). You can fix the prefactor of (t-10) by using the value at t = 70.
This generalizes to more complicated problems where you are given 3 values and have to find a quandratic equation, or more points and have to find a higher order polynomial.
Suppose you know that W(t) is an n-th degree polynomial and:
W(t1) = W1
W(t2) = W2
W(t3) = W3
etc...
W(tn) = Wn
Then from the first data point W(t1)= W1, it follows that
W(t) - W1
equals zero at t = t1. This means that
W(t) - W1 contans a factor (t - t1).
You then define the polynomial
P2(t) = (W(t) - W1)/(t-t1)
and evaluate that for all the remaining data points. The value at the first is undetermined 0/0. Now P2(t) is one degree less than W(t), so you have enough equations to fix this polynomial.
You then use the same trick, i.e. you define:
P3(t) = (P2(t) - W2)/(t - t2)
etc. etc. until you get to the trivial case. Then you work your way back to find W(t).
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