The silvery 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 1 foot. If h(t) is the height of the tree, in feet, at time t years after it is planted, then the rate of growth is proportional to the difference between its maximum height, 10 feet, and its current height is given by
dh/dt = 3/20(10-h)
(a) Is the tree growing faster when it is 2 feet or when it is 4 feet? Explain your reasoning.
(b) find d^2h/dt^2 in terms of h. Use this expression to explain why the graph of the solution curve is concave down.
(c) Find an 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 = 1. Is the estimate an overestimate or an underestimate? Explain your reasoning.
(d) Find the particular solution to this differential equation with initial value h(0) = 1.
(e) If a tree is harvested when its height is 8 feet, at what time t will a blue spruce be ready to sell?

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