2. The silver blue-green coloring of the Colorado blue spruce makes it a perfect Christmas tree. For a tree farmer in Upstate New York, when a seedling is planted at time t 0,the height of the
tree is 1foot. If h(t) is the height of the tree, in feet, at time in feet, at time t years after it is
planted, then the rate of growth is proportional to the difference between its maximum height,
10ft, and its current height and is given by
dh
dt
3
(10 - h)
20
a. Is the tree growing faster when it is 2ft or when it is 4ft? explain your reasoning.
HINT: The steepness of the curve will determine the rate of increase (2marks)
dez
b. Find
dh
in terms of h. Use this expression to explain why the graph of the solution curve
is concave down.
(2marks)
HINT: What is the interpretation of the second derivative of a function?
c. Find the equation of the tangent line to the graph of h at the point where t = 0. Use the
tangent line to approximate the height of the tree at t = 1year. Is the estimate an
overestimate or an underestimate? Explain your reasoning.
(3marks)