Answer:
a. 1492 kg
b. 20.8 m
Explanation:
Given a tree is initially 20 m high and has a diameter of 0.5 m, you want the mass of the trunk if its density is 380 kg/m³. If it adds a growth ring of 4 mm per year and adds height of 0.2 m in the first year, you want the height of the tree after 5 years, assuming the same amount of wood is added each year.
a. Mass
The volume of the tree trunk is that of a cylinder. The formula is ...
V = (π/4)d²h
V = (π/4)(0.5 m)²(20 m) ≈ 3.9270 m³
The mass is the product of volume and density:
M = Vρ
M = (3.9270 m³)(380 kg/m³) ≈ 1492 kg
The mass of the tree trunk is about 1492 kg.
b. Height
In the first year, the diameter of the tree increases by 8 mm, and the height increases by 0.2 m,. This means the volume of the tree increases to ...
V = (π/4)(0.508 m)²(20.2 m) ≈ 4.0942 m³
The volume increase is the same each year for 5 years, so after 5 years, the volume is ...
3.9270 m³ + 5(4.0942 -3.9270) m³ ≈ 4.7630 m³
At that time, the diameter is about 0.540 m. Solving the volume equation for the height, we find it to be ...
h = 4v/(π·d²)
h = 4(4.7630 m³)/(π·(0.54 m)²) ≈ 20.797 m
The height of the trunk after 5 years is about 20.8 m.
__
Additional comment
We note that the wood added in the first year includes the cylindrical shell represented by the tree ring, and a cylindrical "plug" that is 0.2 m high and equivalent in diameter to the rest of the tree. This seems an odd way for the tree to grow, but may be a reasonable approximation to the actual growth.
The height, diameter, and growth are all given to 1 significant figure. Hence the 4- and 3-significant figures used in the answers may be unsupported by the precision of the given numbers. Keeping 2 significant figures, we might report the initial mass as 1500 kg, and the final height as 21 m.
The graph is drawn by a function formulated to have the correct height at the x-intercept: v(d, h) - (final volume) = 0.
<95141404393>