Problem 1 - College Algebra with Corequisite Support

Question

Terry is skiing down a steep hill. Terry's elevation, \(E(t),\) in feet after \(t\) seconds is given by \(E(t)=3000-70 t .\) Write \(a\) complete sentence describing Terry's starting elevation and how it is changing over time.

Step-by-Step Solution

Answer

Terry starts at 3000 feet and loses 70 feet each second while skiing.

Step 1: Understand the Equation

We are given the equation for Terry's elevation over time as \(E(t) = 3000 - 70t\). This is a linear equation where \(E\) represents elevation, \(t\) represents time in seconds, and the coefficients and constants have specific meanings.

Step 2: Analyze the Equation Structure

The equation \(E(t) = 3000 - 70t\) is in the standard form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(b = 3000\) and represents Terry's initial elevation (elevation at \(t = 0\)). The slope \(m = -70\) indicates the rate of change of Terry's elevation over time.

Step 3: Determine Starting Elevation

The starting elevation is given by the y-intercept of the equation, which is \(b = 3000\). This means Terry starts at an elevation of 3000 feet at \(t = 0\) seconds.

Step 4: Determine Rate of Change

The slope \(-70\) in the equation represents the rate of change of elevation with respect to time. This means Terry's elevation decreases by 70 feet for every second that passes.

Step 5: Compose the Sentence

With the determined information, we can describe the situation as follows: "Terry starts skiing at an elevation of 3000 feet, and his elevation decreases by 70 feet every second."

Key Concepts

Slope

In the context of linear equations, the slope is a critical concept that defines how steep a line is, and in practical scenarios, describes how one quantity changes in relation to another. Think of it as the tilt or incline of a line. When looking at the equation of a line like \(y = mx + b\), the slope is the "\(m\)."

• The equation \(E(t) = 3000 - 70t\) has a slope of \(-70\). This tells us that for every additional second \(t\) that Terry continues skiing, his elevation decreases.
• A negative slope indicates a decrease or reduction—they mean we're going downhill, just like Terry is skiing downhill. Conversely, a positive slope would suggest an incline.
• The magnitude of the slope (ignoring the negative sign) tells us the rate at which the change occurs. Here, the change is 70 feet per second.

The direction and magnitude of a slope provide both qualitative and quantitative insights, which are crucial in interpreting real-world phenomena.

Y-intercept

The y-intercept is the point where a line intersects the y-axis on a graph, and it provides the initial value of the dependent variable (in this case, the elevation) when the independent variable (time \(t\)) is zero. It's essentially where things start, often at the beginning of an observation or measurement.

• In the equation \(E(t) = 3000 - 70t\), the y-intercept is 3000.
• This means that when Terry starts skiing, at \(t = 0\), his elevation is 3000 feet.
• The y-intercept gives the starting condition before any change happens due to the slope.

Understanding the y-intercept is crucial because it sets the stage for interpreting what happens next—in Terry's case, how his skiing journey begins.

Rate of Change

Rate of change is a concept that explains how a quantity changes over time, or in another respect to a variable. It is fundamentally linked to the slope in linear equations, depicting how a variable responds as another variable changes.

• The rate of change here is represented by the slope of \(-70\) in the elevation equation \(E(t) = 3000 - 70t\).
• This tells us that Terry's elevation decreases by 70 feet every second as he skis downhill.
• Knowing the rate of change helps predict future states—how long until Terry reaches the bottom, or how high he will be after a specific time has passed.

Thus, the rate of change provides a dynamic view of how systems evolve over time, offering important predictions and insights into future behaviors.

Elevation

Elevation, in simple terms, refers to how high a point is relative to a base level, usually sea level. In the context of Terry's skiing adventure, it offers insights into his position on the mountain as time progresses.

• Initially, Terry starts at an elevation of 3000 feet, as given by the y-intercept in the linear equation.
• With each passing second, this elevation changes with the rate of \(-70\) feet per second due to the slope.
• Tracking elevation over time can help determine Terry's altitude at any given second, enabling understanding of his descent pattern.

Thus, elevation is a vital metric both for assessing current states and predicting further developments in various applications like geography, skiing, or even flight.