Answer :
To solve this problem, we need to use the formula for the volume of a cylinder and the definition of density. We also have two rates to deal with, the rate of growth in height and the rate of growth in diameter, and we need to take both into account.
First, we recall the formula for the volume of a cylinder, V = π * (d/2)^2 * h. We notice that the tree trunk resembles a cylinder, so this formula is relevant.
Given the variables:
- the height (h) of the tree is 150m,
- the diameter (d) of the tree is 4m,
- the change in height annually (dh/dt) is 1m per year,
- the change in diameter annually (dd/dt) is 0.04m per year (4cm converted to meters), and
- the density (ρ) of the tree is 4000kg/m^3.
Next, we’ll calculate the rate of change of the mass (dm/dt). Our change in mass over time (dm/dt) is equal to density times the rate of change in volume over time (dV/dt), since mass = density * volume.
The volume of the tree is changing due to both the change in its height and diameter, so we’ll have to calculate each of these changes separately, using the formula for the volume of a cylinder.
First, we calculate the contribution to volume change from the height increase:
dV/dt(h) = ρ * BV/dh = ρ * π * (d/2)^2 * dh/dt.
After plugging in the given values, we get:
dV/dt(h) = 4000 * π * (4/2)^2 * 1 = 4000 * π * 4 * 1.
Second, we calculate the contribution to the volume from the diameter increase:
dV/dt(d) = ρ * BV/dd = ρ * π * h * d * dd/dt.
Plugging in the given values, we get:
dV/dt(d) = 4000 * π * 150 * 4 * 0.04 = 4000 * π * 600 * 0.04.
Now, to find the total rate of change of the mass of the tree, we add these two contributions together:
dm/dt = dV/dt(h) + dV/dt(d).
After calculating, we find that the rate of mass of the tree increasing is approximately 351858.377 Kg per year.
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