Question 570984: 1.) A citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Find the total number of trees the grower should plant to maximize yield.
2.) A person is standing 3 ft away from a street light that is 15.6 ft tall. How long is his shadow if he is 5.2 ft tall?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! CITRUS GROWER PROBLEM
The yield per tree,  can be expressed as  with  being the number of trees.
The total yield,  will be  .
We need to express as a function of .
-->  --> 
So 
The total yield function,
 , is a quadratic function, which would graph as a parabola.
Because the leading coefficient is negative, the function has a maximum, corresponding to the vertex of the parabola, where the parabola crosses its axis of symmetry.
We need to find the axis of symmetry, which will give us the x-coordinate for the maximum.
The equation for axis of symmetry/x-coordinate of the maximum is
 --> 
The number of trees for maximum total yield in the same acreage is  .
SHADOW PROBLEM
 <--- This sketch illustrates the situation.
Consider two similar right triangles. The legs of the smaller right triangle are the height of the person (p) and the length of the person’s shadow. The legs of the larger right triangle are the height of the street light (sl) , and the distance from its base to the end of the person’s shadow. The hypotenuse of the larger triangle is the light beam coming from the light bulb, grazing the person's head, and touching the ground just at the end of the person's shadow.
Let x be the length of the person’s shadow. Because those right triangles are similar, there is a proportion for the ratio of horizontal leg to vertical leg:
 -->  -->  -->  -->  -->  --> 
The length of the shadow is  feet.
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