An object is attached by a string to the end of a spring. When the weight is released it starts oscillating vertically in a periodic way that can be modeled by a trigonometric function.

The object's average height is −20 cm (measured from the top of the spring). It achieves that average height every 2 seconds, and its maximum and minimum heights are each 5 cm from its average height.
Find the formula of the trigonometric function that models the height H of the weight t seconds after it reached its maximum height. Define the function using radians.

The answer says, "We know the object reaches its highest point at t=0, so let's use a cosine function."

How do I determine the maximum value is at t=0? This problem doesn't mention at what time the maximum value of the function occurs, only that it is 5 cm from the average height. Since the problem doesn't state where the max occurs, periodic behavior could be modeled by either a sine or cosine function.

The problem states:

Find the formula of the trigonometric function that models the height H of the weight t seconds after it reached its maximum height.

Your function is supposed to model the height starting when it is at its maximum. So, let that be t=0.

You are correct that the problem statement does not explicitly state when the maximum value occurs. However, by convention, the maximum value of a cosine function is achieved at t=0, while the maximum value of a sine function is achieved at t=π/2 or at an equivalent time in radians.

In this problem, since the object starts its motion from its highest point (maximum height), it is reasonable to assume that the maximum value of the function occurs at t=0. This is because when the weight is released from its highest point, it starts to oscillate downwards.

If the maximum value occurred at t=π/2, for example, it would mean that the object starts from its lowest point and oscillates upwards, which is not the case according to the problem statement.

Therefore, based on the given information, it is appropriate to use a cosine function to model the height of the weight as a function of time.

To determine whether the maximum value occurs at t=0 or not, we can consider the given information about the object's behavior.

In the problem statement, it is mentioned that the object achieves its average height every 2 seconds. Since the average height is -20 cm and the maximum and minimum heights are each 5 cm from the average height, we can deduce the following:

- The maximum height is -20 cm + 5 cm = -15 cm.
- The minimum height is -20 cm - 5 cm = -25 cm.

Now, let's consider the behavior of a cosine function. In a cosine function, the maximum value occurs when the argument of cosine is 0 (cos(0) = 1). This means that if we use a cosine function to model the height of the weight, the maximum height should occur at t=0.

If the maximum value does not occur at t=0, it would imply that the weight reaches its average height (or maximum height) at a different time than every 2 seconds, contradicting the given information.

Hence, based on the information provided in the problem, we can conclude that the maximum value of the trigonometric function occurs at t=0, and using a cosine function for modeling the height of the weight is appropriate.