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 turned out to be 5 and it's right. I found it by using the formula 5=a+bsin(c0) and since anytime sin is times by 0, you get 0, then a=5.

This is a 9th grade math problem by the way.

So, now you know that

d(t) = 5+b sin(ct)
Now plug in the values you know:

b sin(12c) = 0
b sin(1300c) = 2
b sin(1400c) = 3.46

we can see that 12c is a multiple of π.
So,
sin(1300c) = sin(1296c+4c) = sin(4c)
sin(1400c) = sin(1392c+8c) = sin(8c)

Hmmm. I'm stuck. 4c and 8c are both multiples of π/3, which have the same value for the sine. So, how can be be different?

Am I off base here?

To find the values of b and c, we can use the given information and solve for these constants step-by-step.

Step 1: Use the known values to form a system of equations:

From the given information:
When t = midnight, d = 5: 5 = a + bsin(c0) --> equation (1)
When t = noon, d = 5: 5 = a + bsin(c12) --> equation (2)
When t = 1300, d = 7: 7 = a + bsin(c13) --> equation (3)
When t = 1400, d = 8.46: 8.46 = a + bsin(c14) --> equation (4)

Step 2: Substitute the value of a (which is 5) into equations (1), (2), (3), and (4):

Equation (1): 5 = 5 + bsin(c0) --> equation (5)
Equation (2): 5 = 5 + bsin(c12) --> equation (6)
Equation (3): 7 = 5 + bsin(c13) --> equation (7)
Equation (4): 8.46 = 5 + bsin(c14) --> equation (8)

Step 3: Simplify equations (5), (6), (7), and (8):

Equation (5): bsin(c0) = 0 --> equation (9)
Equation (6): bsin(c12) = 0 --> equation (10)
Equation (7): bsin(c13) = 2 --> equation (11)
Equation (8): bsin(c14) = 3.46 --> equation (12)

Step 4: Solve equations (9) and (10) to find the possible values of b:

From equation (9): bsin(c0) = 0
Since b is non-zero, sin(c0) must be equal to 0. Therefore, c0 = 0 degrees or any multiple of 180 degrees.

From equation (10): bsin(c12) = 0
Similarly, sin(c12) = 0 as b is non-zero. Hence, c12 = 180 degrees or any multiple of 180 degrees.

Step 5: Use the value of c12 to solve equations (11) and (12):

From equation (11): bsin(c13) = 2
Since sin(c13) is non-zero, b = 2 / sin(c13)

From equation (12): bsin(c14) = 3.46
Substitute the value of b from equation (11): (2 / sin(c13)) * sin(c14) = 3.46
Simplify to find sin(c13) * sin(c14) = 3.46 / 2
Find the possible values of c13 * c14 by taking inverse sine(sine^-1) of (3.46 / 2).

Step 6: Substitute the values of b and c into equation (1) to determine the remaining constant:

From equation (1): 5 = a + 2 / sin(c13) * sin(c0)
Substitute the known values of a = 5 and sin(c0) = 0.

Simplify the equation to find a possible value of b.

With these steps, you should be able to find the values of b and c by using the given information and solving the resulting equations.

To find the values of b and c, we can use the given information and the formula provided.

We know that at midnight (t = 0), the depth of water is d = 5. Substituting these values into the formula, we get:

5 = a + bsin(c * 0)
5 = a + bsin(0)
5 = a + 0 (sin(0) is 0)
a = 5

Therefore, as you correctly found, a = 5.

Now, let's use the other given information to find the values of b and c.

At noon (t = 12), the depth of water is also d = 5:

5 = a + bsin(c * 12)
5 = 5 + bsin(12c)

Since sin(12c) must be 0 (to give a sum of 5), we can determine that 12c is a multiple of π (pi) or π/2 radians. Note that sin(x) is 0 at multiples of π or π/2.

Therefore, we have two possibilities:
1. 12c = π * n, where n is an integer (e.g., 1, 2, 3, ...)
2. 12c = (π/2) * n, where n is an odd integer (e.g., 1, 3, 5, ...)

Next, let's use the information when t = 13:00 (1 PM), and the depth of the water is d = 7:

7 = a + bsin(c * 13)
7 = 5 + bsin(13c)

Similarly, we can conclude that 13c is also a multiple of π or π/2. This gives us two possibilities again:
1. 13c = π * m, where m is an integer
2. 13c = (π/2) * m, where m is an odd integer

Lastly, when t = 14:00 (2 PM) and d = 8.46:

8.46 = a + bsin(c * 14)
8.46 = 5 + bsin(14c)

Using the same logic, we have two possibilities for 14c:
1. 14c = π * k, where k is an integer
2. 14c = (π/2) * k, where k is an odd integer

To find the values of b and c, we need to find which possibilities are consistent with all the given information. We can use some trial and error or apply a logical approach to narrow down the options and find the correct values.

Note: It would be helpful to have additional information or constraints to determine the precise values of b and c in this scenario.