To determine the minimum height of the flag given the function [tex]\( h(t) = 3 \sin \left(\frac{4 \pi}{5} \left(t - \frac{1}{2}\right) \right) + 12 \)[/tex], let's carefully analyze the function step by step.
1. Identify the Components of the Function:
- The function [tex]\( \sin \left( \cdot \right) \)[/tex] varies between -1 and 1.
- It is given in the form [tex]\( h(t) = 3 \sin(\text{something}) + 12 \)[/tex].
2. Understand the Sine Function Behavior:
- Since [tex]\(\sin x\)[/tex] ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex], multiplying [tex]\(\sin x\)[/tex] by [tex]\(3\)[/tex] will change the range to [tex]\(-3\)[/tex] to [tex]\(3\)[/tex].
3. Determine the Effect of the Sine Function on the Entire Function [tex]\( h(t) \)[/tex]:
- The term [tex]\(3 \sin(\cdot) \)[/tex] will range from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex].
4. Calculate the Range of [tex]\( h(t) \)[/tex]:
- Since [tex]\(h(t) = 3 \sin(\text{something}) + 12\)[/tex], we take the minimum value of [tex]\(3 \sin(\text{something})\)[/tex] which is [tex]\(-3\)[/tex] and add 12 to it.
- Thus, the minimum height is [tex]\(3 \times (-1) + 12 = -3 + 12 = 9\)[/tex].
Therefore, the minimum height of the flag is 9 feet.
