Question

A rock is dropped from the top of a 500-foot cliff. After 1 second, the rock is traveling 46 feet per second. After 4 seconds, the rock is traveling 142 feet per second. Let y be the rate of descent and x be the number of seconds since the rock was dropped.

a. Write a linear equation that relates time x to rate y. Hint: Use the ordered pairs (1,46) and (4,142).
b. Use this equation to determine the rate of travel of the rock 5 seconds after it was dropped.

Answer 1

To relate time to rate, we can use the slope-intercept form of a linear equation. The equation is y = 32x + 14. After 5 seconds, the rate of travel of the rock is 174 feet per second.

Step-by-step explanation:

To write a linear equation that relates time x to rate y, we can use the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept. We are given two points: (1, 46) and (4, 142). Using these points, we can calculate the slope: m = (y2 - y1) / (x2 - x1) = (142 - 46) / (4 - 1) = 32.

Using the slope-intercept form, we can substitute the slope and one of the points to find the y-intercept: 46 = 32(1) + b. Solving for b, we get b = 14. Therefore, we can write the linear equation as y = 32x + 14.

To determine the rate of travel of the rock 5 seconds after it was dropped, we can substitute x = 5 into the linear equation: y = 32(5) + 14

= 174 feet per second.