Solve. A rock is dropped from the top of a 400 -foot...

Solve. A rock is dropped from the top of a 400 -foot building. After 1 second, the rock is traveling 32 feet per second. After 3 seconds, the rock is traveling 96 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 $(1,32)$ and $(3,96)$.] b. Use this equation to determine the rate of travel of the rock 4 seconds after it was dropped.

Solve.
A rock is dropped from the top of a 400 -foot building. After 1 second, the rock is traveling 32 feet per second. After 3 seconds, the rock is traveling 96 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 $(1,32)$ and $(3,96)$.]
b. Use this equation to determine the rate of travel of the rock 4 seconds after it was dropped.

Key Concepts

Linear Equation

A linear equation represents a constant relationship between two variables, where the change in one variable is proportional to the change in the other. This is typically expressed in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. In many applications, this form is used to model phenomena that change at a constant rate.

Ordered Pairs

Ordered pairs are used to convey the values of two related quantities in a coordinate system. They consist of two numbers written in the form (x, y), where x is the independent variable and y is the dependent variable. Ordered pairs serve as the foundational data points for plotting graphs and for deriving equations that describe relationships.

Slope

The slope of a line quantifies the rate of change between two variables and is calculated as the ratio of the change in the dependent variable to the change in the independent variable between any two points. It indicates the steepness and the direction of the line, with positive slopes indicating an increasing relationship and negative slopes indicating a decreasing relationship.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis, representing the value of the dependent variable when the independent variable is zero. It provides an essential initial condition in the linear equation, forming the base value from which changes in the dependent variable are measured.

Predicting Values Using a Linear Model

Once a linear relationship has been established, the model can be used to predict unknown values. By substituting a given value of the independent variable into the equation, the corresponding value of the dependent variable can be calculated. This predictive capability is especially useful in applications involving consistent rates of change, such as uniformly accelerated motion.

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