Question

Sylvia has an apple orchard. One season, her 100 trees yielded 120 apples per tree. She wants to increase her production by adding more trees to the orchard. However, she knows that for every 10 additional trees she plants, she will lose 4 apples per tree (i.e., the yield per tree will decrease by 4 apples). How many trees should she have in the orchard to maximize her production of apples?

Answer 1

Answer:

She should have 200 trees

Step-by-step explanation:

Given,

Original number of trees = 100,

Also, original number of apples per tree = 120,

∵ for every 10 additional trees there is a lose 4 apples per tree,

i.e. if number of trees = 100 + 10x, apples per tree = 120 - 4x

Thus, the total apples = number of trees × apples per tree

Y(x) = (100 + 10x) × (120 - 4x)

Differentiating w. r. t. x,

Y'(x) = (100 + 10x)(-4) + (120 - 4x) (10) = -400 - 40x + 1200 - 40x = 800 - 80x

Again differentiating w. r. t. x,

Y''(x) = -80

For maxima or minima,

Y'(x) = 0

$\implies 800 - 80x =0$

$800 = 80x$

$\implies x =\frac{800}{80}=10$

For x = 10,

Y''(x) = negative,

Hence, Y(x) is maximum at x = 10,

i.e. the number of trees for maximize the production = 100 + 10(10) = 100 + 100 = 200

Answer 2

She should plant 200 trees to maximize apple production.

Since Sylvia has an apple orchard, and one season, her 100 trees yielded 120 apples per tree, and she wants to increase her production by adding more trees to the orchard but she knows that for every 10 additional trees she plants, she will lose 4 apples per tree, to determine how many trees should she have in the orchard to maximize her production of apples, the following calculation must be performed:

• 100 x 120 = 12000
• 110 x 116 = 12760
• 120 x 112 = 13440
• 130 x 108 = 14040
• 170 x 92 = 15640
• 180 x 88 = 15840
• 200 x 80 = 16000
• 210 x 76 = 15960

Therefore, she should plant 200 trees to maximize apple production.

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