Answer:
The hunger function H(t) for Hungry Harry is:
H(t) = 7.5 * cos(π * t) + 22.5
Step-by-step explanation:
To find the hunger function H(t) for Hungry Harry, we can use the information provided and the form of the sinusoidal expression:
H(t) = a * cos(b * t) + d
We are given:
When Harry wakes up at t = 0, his hunger is maximum, and he desires 30 kg of pigs. This gives us the maximum value for the cosine function: 1.
Within 2 hours, his hunger subsides to its minimum, when he only desires 15 kg of pigs. This is the minimum value for the cosine function, which is -1.
So, we have:
H(0) = a * cos(b * 0) + d = 30 kg
H(2) = a * cos(b * 2) + d = 15 kg
Solving for a and d:
a * 1 + d = 30
a * (-1) + d = 15
Now, let's solve this system of equations:
a + d = 30
-a + d = 15
Add the two equations:
(a + d) + (-a + d) = 30 + 15
2d = 45
Now, divide by 2:
d = 45 / 2
d = 22.5 kg
Now that we have d, we can find a using one of the equations:
a + 22.5 = 30
Subtract 22.5 from both sides:
a = 30 - 22.5
a = 7.5 kg
So, we have a = 7.5 and d = 22.5. Now, we need to find b. We know that Harry's hunger subsides to its minimum within 2 hours (t = 2). This corresponds to one complete cycle of the cosine function (from maximum to minimum). In a cosine function, one entire process occurs when the argument of the cosine function (b * t) goes from 0 to 2π radians.
So, we have:
b * 2 = 2π
Now, solve for b:
b = 2π / 2
b = π radians per hour
Therefore, the hunger function H(t) for Hungry Harry is:
H(t) = 7.5 * cos(π * t) + 22.5