question 5, 4.2.14 part 1 of 2 hw score: 80%, 4 of 5 points points: 0 of 1 save an orange grove produces a…

question 5, 4.2.14 part 1 of 2 hw score: 80%, 4 of 5 points points: 0 of 1 save an orange grove produces a profit of $96 per tree when there are 1100 trees planted. because of overcrowding, the profit per tree (for every tree in the grove) is reduced by 4 cents per tree for each one additional tree planted. how many trees should be planted in order to maximize the total profit of the orange grove? what is the maximum profit? how many trees should be planted in order to maximize the total profit of the orange grove?

question 5, 4.2.14 part 1 of 2 hw score: 80%, 4 of 5 points points: 0 of 1 save an orange grove produces a profit of $96 per tree when there are 1100 trees planted. because of overcrowding, the profit per tree (for every tree in the grove) is reduced by 4 cents per tree for each one additional tree planted. how many trees should be planted in order to maximize the total profit of the orange grove? what is the maximum profit? how many trees should be planted in order to maximize the total profit of the orange grove?

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of additional trees planted. The number of trees is $n = 1100 + x$. The profit per - tree is $p=96 - 0.04x$.

Step2: Form the profit function

The total profit $P(x)$ is the product of the number of trees and the profit per - tree. So $P(x)=(1100 + x)(96-0.04x)$. Expand the function: [ \begin{align*} P(x)&=1100\times96-1100\times0.04x+96x - 0.04x^{2}\ &=105600-44x + 96x-0.04x^{2}\ &=- 0.04x^{2}+52x + 105600 \end{align*} ]

Step3: Find the vertex of the quadratic function

For a quadratic function $y = ax^{2}+bx + c$ ($a=-0.04$, $b = 52$, $c = 105600$), the $x$ - coordinate of the vertex is given by $x=-\frac{b}{2a}$. [ x=-\frac{52}{2\times(-0.04)}=\frac{52}{0.08}=650 ]

Step4: Calculate the number of trees

The number of trees to be planted to maximize the profit is $n=1100 + x$. Substitute $x = 650$ into the equation, we get $n=1100 + 650=1750$.

Answer:

1750