Question

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).