According to an article by Thomas H. McMahon in the July 1975 issue of Scientific American, a tree’s height varies directly with the radius of the base of its trunk. He expressed this relation using the formula h = kr^2/3 where k is a constant, h is the tree’s height, and r is the tree’s radius.
Now suppose you own a stand of trees whose pulp can be used for making paper. The amount of wood pulp you can produce from a tree increases as the tree’s volume increases. The model approximates a tree without its branches as a right circular cone. The formula for the volume of the tree then becomes V = (1/3)πr^2h.
Substituting the formula for height of a tree in the formula for volume of a tree, the new formula for volume becomes ________________.
1. V=(1/3)πr(kr^2/3)
2. V=(1/3)πkr^3
3. V=(1/3)πkr^8/3
4. V=(1/3)πr^2
well, shucks -- just plug it in
(1/3)πr^2h = (1/3)πr^2*(kr^2/3) = (1/3)πk r^8/3
To find the new formula for volume by substituting the formula for the height of a tree in the formula for volume, we can replace the 'h' variable with the expression 'kr^2/3' from the height formula.
The formula for volume of a tree is V = (1/3)πr^2h. Substituting 'kr^2/3' for 'h', we get:
V = (1/3)πr^2 * (kr^2/3)
Now, we can simplify the equation further by multiplying the terms inside the parentheses:
V = (1/3)πr^2 * (k * r^2/3)
Next, we can rewrite the equation by combining the exponents:
V = (1/3)π * k * (r^(2 + 2/3))
Simplifying the exponent by adding 2 and 2/3:
V = (1/3)π * k * (r^(8/3))
Therefore, the answer is:
3. V = (1/3)πkr^(8/3)