Question

suppose a particle moves along a straight line. the position at time ( t ) is given by ( s(t)=7t - 2t^{2},tgeq0 ), where ( t ) is measured in seconds and ( s(t) ) is measured in meters. complete parts (a) - (d) below. (iv) how far to the left on the straight line does the particle travel? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the particle goes at most meters to the left. b. the graph of ( s(t) ) has no minimum. the particle will continue indefinitely to the left.

Explanation:

Step1: Analyze the function \(s(t)=7t - 2t^{2}\)

This is a quadratic function of the form \(y = ax^{2}+bx + c\) (here \(x=t\), \(a=-2\), \(b = 7\), \(c = 0\)). The vertex of a quadratic function \(y=ax^{2}+bx + c\) is at \(t=-\frac{b}{2a}\). But to find the minimum (since \(a=-2<0\), the parabola opens down - we actually need to consider the behavior as \(t\to+\infty\)).
As \(t\to+\infty\), \(s(t)=7t-2t^{2}=t(7 - 2t)\to-\infty\)

Answer:

B. The graph of \(s(t)\) has no minimum. The particle will continue indefinitely to the left.