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On the west coast of Canada, crows eat whelks (a shellfish). To open the whelks, the crows drop them from the air onto a rock. If the shell does not smash the first time, the whelk is dropped again.1 The average number of drops, n, needed when the whelk is dropped from a height of x meters is approximated by n(x)=1+27x2. (a) Give the total vertical distance the crow travels upward to open a whelk as a function of drop height, x. The total vertical distance the crow travels is meters. (b) Crows are observed to drop whelks from the height that minimizes the total vertical upward distance traveled per whelk. What is this height? Round your answer to one decimal place. The dropping height is Number meters.

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jolis1796

Answer:

a) d= 2x + (54/x)

b) x = √27 m ≈ 5.2 m

Step-by-step explanation:

a) Given

The average number of drops = n  

Height in meters from the point where the whelk is dropped = x

If   n(x) = 1 + (27/x²)

Then, the total vertical distance the crow travels upward to open a whelk is the height (x) times the average number of times the crow has to fly upwards to try again

d = 2xn = 2x*(1 + (27/x²)) = 2x + (54/x)

b) To minimize the distance consider the first derivative

d' = (2x + (54/x))' = 2 - (54/x²)

and d' = 0  give us (x must be positive)

2 - (54/x²) = 0  ⇒  x = √27 m ≈ 5.2 m

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