Question

Write an equation of the form y = k+a sin mx for a curve that has a maximum point at
(30, 6) and that has 2 as its y-intercept.

Answer 1

To find an equation in the form y = k + a sin mx, we need to determine the values of k, a, and m.

We know that the curve has a maximum point at (30, 6). This means that when x = 30, y = 6. Plugging these values into the equation, we get:

6 = k + a sin (30m)

Since sin (30m) can take on any value between -1 and 1, we can see that k + a must be greater than or equal to 5.

Additionally, the curve has a y-intercept of 2. This means that when x = 0, y = 2. Plugging these values into the equation, we get:

2 = k + a sin (0)

Since sin (0) = 0, we can see that k + a must equal 2.

Combining these conditions, we have the following system of equations:

k + a sin (30m) ≥ 5
k + a = 2

To find the values of k and a, we can subtract the second equation from the first:

(k + a sin (30m)) - (k + a) ≥ 5 - 2

a sin (30m) - a ≥ 3

Since a cannot be zero, we can divide by a:

sin (30m) - 1 ≥ 3/a

sin (30m) ≥ 1 + 3/a

To minimize the value of sin (30m), we want 30m to be equal to 90 degrees or π/2 radians. Therefore, we have:

30m = π/2

m = π/60

Now that we have the value of m, we can determine the value of k + a using the second equation:

k + a = 2

Since a cannot be zero, k must be 2 - a.

Substituting this into the equation from the first condition, we get:

(2 - a) + a sin (30m) ≥ 5

2 - a + a sin (π/2) ≥ 5

2 - a + a ≥ 5

2 ≥ 5

This is not possible, so the system of equations does not have a solution. Therefore, there is no curve of the form y = k + a sin mx that satisfies the given conditions.