Problem 52 - Algebra 2

Question

The time \(t\) in seconds for a trapeze to complete one full cycle is given by the function \(t=1.11 \sqrt{\ell}\) , where \(\ell\) is the length of the trapeze in feet. a. Graph the equation on your calculator. Make a sketch of the graph. b. How long is a full cycle if the trapeze is 15 ft. long? 30 ft. long?

Step-by-Step Solution

Answer

The graph is a square root function in the first quadrant. For a 15ft trapeze, one full cycle takes approximately 4.30 seconds. For a 30ft trapeze, one full cycle takes approximately 6.08 seconds.

Step 1: Sketching the graph

Begin by inputting the function \(t=1.11 \sqrt{\ell}\) into the graphing calculator. The graph will be a curve starting at the origin, shaped like half of a parabola on its side. Because \(\ell\) cannot be negative (it represents a length), the graph will only exist in the first quadrant.

Step 2: Calculating for a 15ft trapeze

Substitute \(\ell = 15\) into the function: \(t = 1.11 \sqrt{15}\). Solve for \(t\).

Step 3: Calculating for a 30ft trapeze

Now, substitute \(\ell = 30\) into the function: \(t = 1.11 \sqrt{30}\). Again, solve for \(t\).

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools that allow us to visualize mathematical functions efficiently. To graph a function like the one given in the exercise, begin by inputting it directly into the calculator. The function here, \( t=1.11 \sqrt{\ell} \), represents the time it takes for a trapeze to complete a cycle based on its length. This equation is perfect for plotting on a graphing calculator because it shows how changes in \( \ell \) (the length of the trapeze) affect \( t \) (the time). Once the function is entered, you will see a graphical representation on the screen. This graph typically appears as a curve since the equation involves a square root, resembling the right side of a parabola extending from the origin. Graphing calculators are particularly helpful since they allow you to observe the function across different domains and ranges by adjusting the view settings, giving a clear visual interpretation that aids understanding of complex algebraic expressions.

Function Evaluation

Function evaluation is a process used to determine the output of a function for a given input. In this context, the function \( t=1.11 \sqrt{\ell} \) can be evaluated by plugging in different values for \( \ell \). This process will help find the time \( t \) needed for various trapeze lengths. Evaluating a function involves simply substituting the chosen value of the independent variable \( \ell \) into the function and solving for the dependent variable \( t \). For example, if \( \ell = 15 \), the calculation becomes \( t = 1.11 \sqrt{15} \). A calculator can be used to find the square root and multiply by 1.11, giving an interpreted value for \( t \). Repeating this process with different lengths, such as \( \ell = 30 \), will allow for the exploration of how time changes with length. This step-by-step evaluation helps solidify the understanding of the functional relationship between trapeze length and cycle time.

Parabola

A parabola is a specific type of graph known for its distinctive curve shape, resembling an 'U' or an upside-down 'U'. In this exercise, the function \( t=1.11 \sqrt{\ell} \) when graphed, appears similar to a parabola, though it's actually a part of a side-facing parabola that only exists in the first quadrant.

• This means the curve extends from the origin to the right, consistently rising.
• Only the positive values of \( \ell \) are considered because \( \ell \) represents a length and cannot be negative.

This is why the graph is described as half of a parabola.
The usage of square roots in the equation, mathematically controls the opening of the curve. Parabolas have many applications, especially in physics and engineering, due to their properties and predictable patterns. Understanding why this graph forms a parabola-like shape helps link algebraic equations to their visual representations, making concepts easier to grasp in fields involving quadratic functions.