Question

on a planet, a rock is dropped from the top of a 400 - foot cliff. after 1 second, the rock is traveling 28 feet per second. after 5 seconds, the rock is traveling 140 feet per second.
a. assume that the relationship between time, t, and speed, s, is linear and write an equation describing this relationship. use ordered pairs of the form (time, speed). (simplify your answer. use integers or fractions for any numbers in the expression.)
b. use this equation to determine the speed of the rock 6 seconds after it was dropped. (simplify your answer.)
b. the speed of the rock 6 seconds after it was dropped is (square) ft/sec.

Explanation:

Step1: Find the equation of the line

The relationship between time \(t\) and speed \(s\) is linear. The equation of a line is \(s = mt + b\). We have two ordered - pairs \((1,28)\) and \((5,140)\). First, find the slope \(m\) using the formula \(m=\frac{s_2 - s_1}{t_2 - t_1}\).
\[m=\frac{140 - 28}{5 - 1}=\frac{112}{4}=28\]
Using the point - slope form \(s - s_1=m(t - t_1)\) with the point \((1,28)\) and \(m = 28\), we get \(s-28 = 28(t - 1)\), which simplifies to \(s=28t\).

Step2: Find the speed at \(t = 6\)

Substitute \(t = 6\) into the equation \(s = 28t\).
\[s=28\times6 = 168\]

Answer:

a. \(s = 28t\)
b. \(168\)