What would the values of A, B, and C be in this problem?
The tides in a particular bay can be modeled with an equation of the form d= A cos (Bt) + C, where t represents the number of hours since high-tide and d represents the depth of water in the bay. The maximum depth of water is 36 feet, the minimum depth is 22 feet and high-tide is hitting every 12 hours.
max - min = 36-22 = 14
so the amplitude is 7
what do we have to add to 7 to get 36 ? ----- 29
period of cosine curve = 2π/k
12 = 2π/k
k = π/6
so d = 7 cos (πt/6) + 29
check:
at t=0 , d = 7cos 0 + 29 = 36 ---> high tide
at t=3 , d = 7cos π/2 + 29 = 29 ---> makes sense
at t=6 , d = 7cos π + 29 = -7+29 = 22 ---> low tide
at t=9 , d = 7cos 3π/2 + 29 = 29 --->makes sense
at t=12, d = 7cos 2π + 29 = 7(1)+29 = 36 ---> back to high tide
What will the graph look like for two full periods?
first period:
mark off from 0 to 12, with marks at 0, 3, 6, 9, and 12
at 0 , d=36
at 3, d= 29
at 6, d = 22
at 9, d = 29
at 12, d = 36 --- end of first period, now repeat that
draw a smoth cosine curve .
(I am somewhat surprised you even asked that question)
To find the values of A, B, and C in this problem, we need to examine the given information.
1. The maximum depth of water is 36 feet: This represents the amplitude of the cosine function. In the equation, A represents the amplitude, so A = 36.
2. The minimum depth is 22 feet: This represents the vertical shift (or the C value) of the cosine function. In the equation, C represents the vertical shift, so C = 22.
3. High-tide is hitting every 12 hours: This represents the period of the cosine function. The period of a cosine function is given by 2π/B, where B represents the coefficient of t. In this case, the period is 12 hours, so 2π/B = 12. Solving for B, we can rewrite the equation as B = 2π/12 = π/6.
Therefore, the values of A, B, and C in the given problem are A = 36, B = π/6, and C = 22.