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).
To find the formula for w(t), we can use the given information that the antler weight starts at 0 at 10 months and reaches 1.05 pounds at 70 months.
First, we can find the slope of the linear relationship by using the formula for the slope of a line:
slope (m) = (y2 - y1) / (x2 - x1)
In this case, (x1, y1) = (10, 0) and (x2, y2) = (70, 1.05). Plugging these values into the formula, we get:
m = (1.05 - 0) / (70 - 10) = 0.015
Next, we can find the value of the constant (C) by using the point-slope formula:
y - y1 = m(x - x1)
Using the point (10, 0) and the slope we found earlier, we get:
y - 0 = 0.015(x - 10)
Simplifying, we have:
y = 0.015x - 0.15
Therefore, the formula for w(t) is:
w(t) = 0.015t - 0.15
Now let's answer the remaining questions:
b. To find the antler weight after 40 months, we can simply plug in t = 40 into the formula for w(t):
w(40) = 0.015(40) - 0.15
w(40) = 0.6 - 0.15
w(40) = 0.45 pounds
So, the antler will weigh 0.45 pounds after 40 months.
c. To find the age at which the antler will weigh 0.7 pounds, we need to solve the equation w(t) = 0.7 for t. Plugging in the formula for w(t):
0.015t - 0.15 = 0.7
Add 0.15 to both sides:
0.015t = 0.85
Divide both sides by 0.015:
t = 56.67
So, the antler will weigh 0.7 pounds at approximately 56.67 months, or around 4 years and 8 months.
d. The weight increase per month can be found by taking the derivative of the w(t) formula with respect to t. The derivative of 0.015t is 0.015, so the weight is increasing by 0.015 pounds per month.
Therefore, the weight is increasing by 0.015 pounds each month.