The depth of water(d metres) in a harbour is given by the formula d = a+bsin(ct) where a, b and c are constants, and t is the time in hours after midnight. It is known that both b and c are non-zero and 20<c<35. If t=midnight, d=5; if t=noon, d=5; if t=1300, d=7; and if t=1400, d=8.46. Using the pieces of information, find the values of a,b and c.
I already found the a, which is 5.
To find the values of b and c in the formula, we can use the given information and solve for the variables.
From the information given:
When t = midnight (0 hours), d = 5.
When t = noon (12 hours), d = 5.
When t = 1300 (13 hours), d = 7.
When t = 1400 (14 hours), d = 8.46.
Using these values, we can form a system of equations to solve for b and c.
1) Equation 1: d = a + b*sin(ct)
When t = 0, d = 5
5 = 5 + b*sin(c*0)
5 = 5 + b*sin(0)
5 = 5 + 0
Therefore, b*sin(0) = 0
2) Equation 2: d = a + b*sin(ct)
When t = 12, d = 5
5 = 5 + b*sin(c*12)
5 = 5 + b*sin(12c)
3) Equation 3: d = a + b*sin(ct)
When t = 13, d = 7
7 = 5 + b*sin(c*13)
7 = 5 + b*sin(13c)
4) Equation 4: d = a + b*sin(ct)
When t = 14, d = 8.46
8.46 = 5 + b*sin(c*14)
8.46 = 5 + b*sin(14c)
Now, let's solve this system of equations to find the values of b and c.
From Equation 1, we have b*sin(0) = 0, which means that either b = 0 or sin(0) = 0. Since the question states that b is non-zero, we conclude that sin(0) must be 0. Therefore, c*0 = 0, and c can take any value since sin(0) remains 0 for any value of c.
From Equations 2, 3, and 4, we have the following equations:
5 = 5 + b*sin(12c) (Equation 2)
7 = 5 + b*sin(13c) (Equation 3)
8.46 = 5 + b*sin(14c) (Equation 4)
Let's subtract 5 from both sides in all equations to simplify:
0 = b*sin(12c) (Equation 2')
2 = b*sin(13c) (Equation 3')
3.46 = b*sin(14c) (Equation 4')
Since sin(0) = 0, it means that for every value of c, sin(12c) = sin(13c) = sin(14c) = 0.
Now, we can conclude that b can take any non-zero value since sin(12c), sin(13c), and sin(14c) are all 0 for any value of c.
Therefore, we can't determine the value of b based on the given information. Only the range that c falls within (20 < c < 35) is provided, so we can't narrow it down further.