A town has noticed that the level of water in the nearby lake is periodic. The fluctuation of the depth of the lake repeats on a yearly basis. The depth of the lake in feet, h, as a function of the day of the year, d, can be modeled by the equation shown, where January 1st is d=0, the start of the cycle. h(d)=2cos(2π365d)+12

Determine whether each statement is a correct interpretation of the equation representing the depth of the lake. Select Yes or No for each statement.
Statement

Yes No
At its deepest the depth of the lake is 12 feet.

At its shallowest the depth of the lake is 10 feet.

On January 1st the lake is at its deepest point.

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step-by-step explanation:

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt