Let's go through the step-by-step solution to understand the expression for [tex]\( y \)[/tex] in terms of [tex]\( t \)[/tex].
1. Identify the Variables:
- We have [tex]\( t \)[/tex] as our independent variable.
- [tex]\( y \)[/tex] is our dependent variable expressed as a function of [tex]\( t \)[/tex].
2. Given Expression:
[tex]\[
y = 34.09 t - 4.9 t^2
\][/tex]
This is a quadratic equation where:
- The coefficient of [tex]\( t \)[/tex] (the linear term) is 34.09.
- The coefficient of [tex]\( t^2 \)[/tex] (the quadratic term) is -4.9.
3. Understanding the Equation:
- The expression [tex]\( y = 34.09 t - 4.9 t^2 \)[/tex] represents a parabolic curve.
- The term [tex]\( 34.09 t \)[/tex] indicates that for every unit increase in [tex]\( t \)[/tex], the value of [tex]\( y \)[/tex] increases by 34.09 units initially.
- The term [tex]\( -4.9 t^2 \)[/tex] represents the decelerating effect on the value of [tex]\( y \)[/tex] as [tex]\( t \)[/tex] increases, due to the negative coefficient.
4. Analyzing the Terms:
- When [tex]\( t = 0 \)[/tex], the value of [tex]\( y \)[/tex] is 0.
- For small values of [tex]\( t \)[/tex], [tex]\( 34.09 t \)[/tex] dominates [tex]\( -4.9 t^2 \)[/tex], and [tex]\( y \)[/tex] increases.
- As [tex]\( t \)[/tex] increases further, the [tex]\( -4.9 t^2 \)[/tex] term grows faster than the [tex]\( 34.09 t \)[/tex] term because [tex]\( t^2 \)[/tex] grows more quickly than [tex]\( t \)[/tex].
5. Vertex of the Parabola:
- Since the coefficient of [tex]\( t^2 \)[/tex] is negative, the parabola opens downwards.
- The vertex (maximum point) of this parabola can be found using the vertex formula for a quadratic equation [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[
t_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -4.9 \)[/tex] and [tex]\( b = 34.09 \)[/tex]:
[tex]\[
t_{\text{vertex}} = -\frac{34.09}{2 \cdot (-4.9)} = 3.48 \ (approximately)
\][/tex]
6. Conclusion:
- The function [tex]\( y = 34.09 t - 4.9 t^2 \)[/tex] is a quadratic function in [tex]\( t \)[/tex].
- It increases initially due to the positive linear term but eventually decreases as the quadratic term (which is negative) becomes more significant.
- The maximum value of [tex]\( y \)[/tex] can be found at [tex]\( t_{\text{vertex}} \approx 3.48 \)[/tex].
So, the detailed solution of the given expression confirms that:
[tex]\[
y = -4.9 t^2 + 34.09 t
\][/tex]
where the change in [tex]\( y \)[/tex] with respect to [tex]\( t \)[/tex] first increases and then decreases after a certain point (i.e., the vertex of the parabola).
