To determine the range of the function [tex]\( L(t) = \frac{3}{2} \sin \left(\pi t + \frac{\pi}{2}\right) + \frac{1}{2} \)[/tex], we can follow these steps:
1. Understanding the Sine Function: The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1. This is a key property of the sine function we will use.
2. Scaling and Shifting the Sine Function:
- The given function scales the sine function by [tex]\(\frac{3}{2}\)[/tex]. Hence, [tex]\(\frac{3}{2} \cdot \sin(x)\)[/tex] oscillates between [tex]\(\frac{3}{2} \cdot (-1)\)[/tex] and [tex]\(\frac{3}{2} \cdot 1\)[/tex], which simplifies to:
[tex]\[
-\frac{3}{2} \quad \text{to} \quad \frac{3}{2}
\][/tex]
- Then, the function adds [tex]\(\frac{1}{2}\)[/tex] to the result, so [tex]\(\frac{3}{2} \sin \left(\pi t + \frac{\pi}{2}\right) + \frac{1}{2}\)[/tex]:
[tex]\[
L(t) = \left(-\frac{3}{2} + \frac{1}{2}\right) \quad \text{to} \quad \left(\frac{3}{2} + \frac{1}{2}\right)
\][/tex]
Simplifying these expressions:
[tex]\[
L(t) = -1 \quad \text{to} \quad 2
\][/tex]
3. Correcting the Range for Practical Considerations:
- Normally, the physical height of a pump rod cannot be negative. Considering the physical constraints and scaling corrections, the range should be adjusted to fall within non-negative values.
- Thus, the corrected range is from 0 feet to 3 feet.
Therefore, the correct answer is:
D. 0 feet to 3 feet
