Question 1078355
.
<pre>
For the number of oranges per one tree, "n", as the function of the number of trees per acre, "t", the condition gives this formula:

n = 630 - 15*(t-20).

Then the number of oranges per acre is 

O(t) = n*t = t*(630 - 15*(t-20)) = 630t - 15t^2 + 300t = -15t^2 + 930t.


They ask to find the maximum of this quadratic function of t.


From the general theory, the maximum is achieved at t = {{{-b/(2a)}}} = {{{-930/(2*(-15))}}} = {{{930/30}}} = 31.


So, 31 tree per acre provide the maximal total number of oranges per acre.


This maximum is equal to O(31) = 31*(630-15*(31-20)) = 31*(630-15*(-11)) = 21*(630+15*11) = 16695.


<U>Answer</U>.  31 trees per acre provide the maximal number of oranges of 16695.
</pre>

 *** Solved ***



{{{graph( 330, 330, -10.5, 40.5, -1000.5, 20000.5,
          x*(630 - 15*(x-20))
)}}}


Plot y = t*(630 - 15*(t-20))



On finding the maximum of a quadratic function and associated solved problems see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/Using-quadratic-functions-to-solve-problems-on-maximizing-profit.lesson>Using quadratic functions to solve problems on maximizing revenue/profit</A>



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic "<U>Finding minimum/maximum of quadratic functions</U>".