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bija089 [108]
3 years ago
14

The owner of an orange grove estimates that if 24 trees are planted per acre, then each mature tree will yield 600 oranges per y

ear. For each additional tree planted per acre, the number of oranges produced by each tree decreases by 12 per year. How many trees should be planted per acre to obtain the most oranges per year
Mathematics
1 answer:
tia_tia [17]3 years ago
8 0

Answer:

16428 oranges

Explanation:

Total yield = number of trees × number of oranges in each tree

Initial yield = 600×24= 14400 oranges

To find the equation needed, let x = additional trees and y= total yield

Number of trees = 24 +x

Number of oranges in each tree = 600-12x

Equation of total yield y= (24+x)(600-12x)

y= 14400-288x+600x-12x²

y= -12x²+312x+14400

Using a graphing calculator, from the graph drawn for this quadratic equation, we notice that it is a parabola. Therefore to find the maximum value, we should find the maximum point which is at the vertex of the parabola, we use the formula x= -b/2a

A quadratic equation is such: ax²+bx+c

Therefore x =-312/2×-12

x= -312/-24

x= 13

So we can conclude that in order to maximise oranges from the trees, the person needs to plant an additional 13 trees. Substituting from the above:

24+x=24+13= 37 trees in total

y= -12x²+312x+14400= -12×13²+312×13+14400= -2028+4056+14400

=16428 oranges in total yield

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Answer:

Statements                                                      Reasons

AC+CD=AD and AB+BD=AD         Segment Addition Postulate

AC+CD=AB+BD                            Transitive/Substitution Property

AC=BD                                         Given

BD+CD=AB+BD                            Substitution Property

CD=AB                                         Subtraction Property

AB=CD                                         Symmetric Property

Step-by-step explanation:

By segment addition postulate, we can say the following two equations:

AC+CD=AD and AB+BD=AD.

By either substitution/transitive property, you can say AC+CD=AB+BD.

You are given AC=BD, so we use substitution and write AC+CD=AB+AC.

After using subtraction property (subtracting both sides by AC), you obtain CD=AB.

By symmetric property, you may say AB=CD.

So let's write it into the 2 column-proof you have there:

Statements                                                      Reasons

AC+CD=AD and AB+BD=AD        Segment Addition Postulate

AC+CD=AB+BD                            Transitive/Substitution Property

AC=BD                                          Given

BD+CD=AB+BD                             Substitution Property

CD=AB                                          Subtraction Property

AB=CD                                          Symmetric Property

Properties/Postulates used:

Transitive property which says:

If a=b and b=c, then a=c.

Substitution property which says:

If a=b, then b can be substituted(replaced with) for a.

Subtraction property which says:

a=b implies a-c=b-c.

Segment Addition Postulate says:

If you break a segment into two smaller pieces then the measurement of that segment is equal to the sum of the smaller two segments' measurements.

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