Question 1191404: Profit per tree grown and sold by a tree grower depends upon the height of a tree at the
time of sale. Taking ha as a tree height in metres, the profit per tree in thousands shillings
is approximated by : P(h) = (10+2h) - 0.1h
(a) What tree height provides maximum profit per tree?
(b) What is the maximum profit per tree?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the tree height that maximizes profit and the maximum profit:
**(a) Finding the Tree Height for Maximum Profit:**
1. **Recognize the Function:** The profit function P(h) = (10 + 2h) - 0.1h² is a quadratic function in the form of P(h) = a + bh - ch², where a=10, b=2 and c=0.1. Since the coefficient of the h² term (-0.1) is negative, the parabola opens downward, meaning there's a maximum point.
2. **Find the Vertex:** The vertex of a parabola gives the maximum (or minimum) value of the function. The h-coordinate of the vertex is given by:
h = -b / (2a)
In our case, a= -0.1 and b=2
h = -2 / (2 * -0.1)
h = -2 / -0.2
h = 10
Therefore, a tree height of 10 meters provides the maximum profit.
**(b) Calculating the Maximum Profit:**
Substitute the value of h (10 meters) back into the profit function P(h):
P(10) = (10 + 2 * 10) - 0.1 * 10²
P(10) = (10 + 20) - 0.1 * 100
P(10) = 30 - 10
P(10) = 20
Since the profit is in thousands of shillings, the maximum profit per tree is 20,000 shillings.
**Answer:**
(a) A tree height of 10 meters provides the maximum profit per tree.
(b) The maximum profit per tree is 20,000 shillings.
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