A fish swimming in a lake is modelled by the equation h(t) = 6cos (pi/7 t) where h(t) is the height of the fish above the lake's surface (in meters) at a given time t, in minutes. On the interval 1<=t<=11, when is the fish at the surface of the lake?

when is cosθ = 0?

To find the time when the fish is at the surface of the lake, we need to find the values of t where the height of the fish, h(t), is equal to zero.

Given the equation: h(t) = 6cos(pi/7t)

Setting h(t) equal to zero:
0 = 6cos(pi/7t)

To solve for t, we need to isolate the variable t. Divide both sides of the equation by 6:
0/6 = cos(pi/7t)

Since the cosine function of an angle is zero when the angle is equal to (2n + 1)π/2, where n is an integer, we can solve for t using the equation:
pi/7t = (2n + 1)π/2

To find the values of t, we can solve for n using the given interval 1 ≤ t ≤ 11.

When n = 0:
pi/7t = π/2
t = 7/2

When n = 1:
pi/7t = 3π/2
t = 21/2

Both of these values fall within the given interval 1 ≤ t ≤ 11.

Therefore, the fish is at the surface of the lake at t = 7/2 minutes and t = 21/2 minutes.

To find when the fish is at the surface of the lake, we need to determine the values of t that make h(t) equal to the surface level of the lake.

The surface level of the lake is where h(t) is equal to zero. In other words, we need to solve the equation h(t) = 0.

The given equation is h(t) = 6cos (π/7 t).

Setting h(t) equal to zero, we have:

0 = 6cos (π/7 t)

To solve for t, we need to isolate the variable t. Divide both sides of the equation by 6:

0/6 = cos (π/7 t)

0 = cos (π/7 t)

Now, let's find the values of t that satisfy the equation cos (π/7 t) = 0.

The cosine function equals zero at certain angles. In this case, we need to find the angles whose cosines are zero within the interval 1<=t<=11.

The cosine is equal to zero at the following angles (in radians): π/2, 3π/2, 5π/2, etc.

To find the corresponding values of t, we need to solve the equation:

π/7 t = π/2, 3π/2, 5π/2, etc.

Solving for t, we have:

t = (2π/7)(1/2), (2π/7)(3/2), (2π/7)(5/2), etc.

Simplifying further, we get:

t = π/7, 3π/7, 5π/7, etc.

However, we need to consider the interval 1<=t<=11. Thus, we only need to consider the values of t within this interval.

In our case, the only value of t that falls within the given interval is:

t = π/7

Therefore, the fish is at the surface of the lake at t = π/7 minutes.