Illustrate a traditional wooden water wheel rotating in a gentle stream. The stream should cover around half the height of the wheel, with bulrushes growing around the edges of the water. Symbolize the pathway of a single nail on the rim of the water wheel over a 24 second period as a multi-colored dotted line. The colors should change according to time, starting from a cool blue at the beginning and transitioning to a warm red at the end of the 24 second cycle. Show parts of the path going below the water surface.

the height, in metres, of a nail in a water wheel above the surface of the water , as a function of time, can be modelled by the function h(t) = -4sin pie/4 (t-1)+2.5, where t is the time in seconds. During what periods of time is the nail below the water in the first 24s that the wheel is rotating?

I have the answer i would just like to know the steps of how to solve for this sort of question.

h(t) = -4sin (π/4)(t-1)+2.5

The period of this function is 2π/(π/4) = 8
So we want to see where the sine curve falls below the x-axis for 3 periods of the curve, starting at t=0

0 = -4sin (π/4)(t-1)+2.5
4sin (π/4)(t-1) = 2.5
sin (π/4)(t-1) = .625

make sure your calculator is set to radians
(π/4)(t-1) = .67513 or (π/4)(t-1) = π - .67513

case1: (π/4)(t-1) = .67513
t-1 = .8596
t = 1.8596

case2: (π/4)(t-1) = π - .67513
(π/4)(t-1) = 2.46646..
t-1 = 3.1404
t = 4.1404

my Wolfram graph confirms this:
http://www.wolframalpha.com/input/?i=plot+h(t)+%3D+-4sin+((%CF%80%2F4)(t-1))+%2B2.5

so the nail is below the x-axis or 4.1404-1.8596 or
2.28 seconds for each cycle

my edited graph shows that this happens 3 times in your domain,
http://www.wolframalpha.com/input/?i=plot+h(t)+%3D+-4sin+((%CF%80%2F4)(t-1))+%2B2.5+for+t%3D0+to+24

So for your domain, the nail is underwater for a total of 3(2.28..) seconds or 6.842 seconds
or
the nail is below:
between 1.860 and 4.140 seconds
between 9.860 and 12.140 seconds and
between 17.860 and 20.140 seconds

depending on how you interpret your question.
(note that each interval for the last interpretation is 2.28 seconds for a total of 6.842 seconds)

again, my second Wolfram graph confirms this

Well, solving this kind of question involves finding the time intervals where the height of the nail is less than or equal to zero (indicating that it is below the surface of the water).

To determine these time intervals, you need to set the equation h(t) = 0 and solve for t.

So, starting with the equation:
-4sin(pi/4(t-1)) + 2.5 = 0

First, subtract 2.5 from both sides:
-4sin(pi/4(t-1)) = -2.5

Next, divide both sides by -4:
sin(pi/4(t-1)) = 0.625

To solve for t, you need to find the inverse trigonometric function of 0.625. In this case, the inverse sine function (sin^(-1)) would be appropriate.

Using the inverse sine function, you can write:
pi/4(t-1)=sin^(-1)(0.625)

Now, solve for t by isolating it:
t-1 = (4/pi) * sin^(-1)(0.625)

Finally, add 1 to both sides to obtain the value of t:
t = 1 + (4/pi) * sin^(-1)(0.625)

Now, you can use a calculator to find the approximate value of t.

Once you have the value of t, you can determine the periods of time when the nail is below the water by looking at the values within the first 24 seconds.

To determine the periods of time during which the nail is below the water, we need to find when the height function h(t) is less than or equal to zero.

The given function for the height of the nail above the water is:

h(t) = -4sin((π/4)(t-1)) + 2.5

Step 1: Set up the inequality:
-4sin((π/4)(t-1)) + 2.5 ≤ 0

Step 2: Solve for the sin term:
sin((π/4)(t-1)) ≥ 2.5/4

Step 3: Use the inverse sine function (sin^(-1)) to find the possible values for (π/4)(t-1):

(π/4)(t-1) ≥ sin^(-1)(2.5/4)

Step 4: Simplify the inequality:
(t-1) ≥ (4/π)sin^(-1)(2.5/4)

Step 5: Solve for t:
t ≥ (4/π)sin^(-1)(2.5/4) + 1

Step 6: Convert to degrees (optional):
t ≥ (180/π)sin^(-1)(2.5/4) + 1

Step 7: Substitute the value of sin^(-1)(2.5/4) into the equation and calculate the value of t.

You can now substitute different values of t from 0 to 24 seconds into the equation and check whether the nail is below the water during those time intervals by comparing the calculated height h(t) with zero.

Please note that the given function h(t) is periodic, so there may be multiple time intervals when the nail is below the water during the first 24 seconds of rotation.

To find the periods of time during which the nail is below the water, you need to determine the values of t that make the height, h(t), less than or equal to zero.

Here's a step-by-step approach:

1. Start with the given function: h(t) = -4sin(pi/4)(t - 1) + 2.5.

2. Set h(t) less than or equal to zero, since we're looking for when the nail is below the water surface:

-4sin(pi/4)(t - 1) + 2.5 ≤ 0.

3. Solve the inequality:

First, subtract 2.5 from both sides: -4sin(pi/4)(t - 1) ≤ -2.5.

Then, divide by -4sin(pi/4): t - 1 ≥ -2.5 / (-4sin(pi/4)).

Note: Since sin(pi/4) is positive, we do not need to flip the inequality sign.

4. Simplify the right side of the inequality:

t - 1 ≥ -2.5 / (-4sin(pi/4)).

Since sin(pi/4) = 1/sqrt(2), simplify the right side further: -2.5 / (-4sin(pi/4)) = -2.5 / (-4/sqrt(2)).

To simplify the expression, rationalize the denominator by multiplying the numerator and denominator by sqrt(2):

-2.5 / (-4/sqrt(2)) = -2.5 * (sqrt(2)/(-4)) = 0.625 * sqrt(2).

Therefore, t - 1 ≥ 0.625 * sqrt(2).

5. Solve for t:

Add 1 to both sides: t ≥ 0.625 * sqrt(2) + 1.

6. To determine the periods within the first 24 seconds when the nail is below the water, set up an inequality with the interval:

0 ≤ t ≤ 24.

7. Find the intersection of the inequality t ≥ 0.625 * sqrt(2) + 1 and the interval 0 ≤ t ≤ 24.

Since the inequality t ≥ 0.625 * sqrt(2) + 1 represents when the nail is below the water, the solution is the intersection of this inequality and the given interval:

0 ≤ t ≤ 0.625 * sqrt(2) + 1, within the interval 0 ≤ t ≤ 24.

This represents the periods of time during the first 24 seconds when the nail is below the water's surface.