Problem 10 - Algebra 2

Question

The weight \(P\) in pounds that a beam can safely carry is inversely proportional to the distance \(D\) in feet between the supports of the beam. For a certain type of wooden beam, \(P=\frac{9200}{D} .\) Use a graphing calculator and the Intersect feature to find the distance between supports that is needed to carry each given weight. 500 \(\mathrm{lb}\)

Step-by-Step Solution

Answer

The x-coordinate at the intersection point of \( P = \frac{9200}{D} \) and \( P = 500 \) gives us the required distance between the supports to carry a weight of 500 lbs. Use a graphing calculator to identify this intersection point.

Step 1: Formulate the equation

First, set up the equation to represent the problem. In this case, we have \( P = \frac{9200}{D} \). Now let's set \( P = 500 \) since the beam needs to carry 500 lbs.

Step 2: Graph the equations

Next, graph both the equations \( P = \frac{9200}{D} \) and \( P = 500 \) on a graphing calculator. \( P = 500 \) will be a horizontal line at \( P = 500 \).

Step 3: Identify the point of intersection

Use the intersect feature on the graphing calculator to find the point where the lines \( P = \frac{9200}{D} \) and \( P = 500 \) intersect. The x-coordinate at this intersection point represents the distance \( D \), and the y-coordinate is the weight \( P \).

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool that helps visualize mathematical functions and their relationships. By plotting equations, it allows you to see patterns and find solutions that might be difficult to calculate manually. To tackle the given problem, you can use a graphing calculator to represent the equation \( P = \frac{9200}{D} \) alongside a constant function, \( P = 500 \), which is the specific weight the beam needs to carry.
This function uses the 'Y=' function on your graphing calculator to input the equations.
Additionally, the graph for \( P = 500 \) is a straight horizontal line, because it depicts a constant weight that does not vary with the distance between supports.

Intersect Feature

The intersect feature on a graphing calculator is an efficient way to find where two or more graphs meet. This feature can determine the precise points of intersection, crucial for solving systems of equations and understanding relationships between variables.
In the context of our exercise, once the equations \( P = \frac{9200}{D} \) and \( P = 500 \) are graphed, the intersect feature assists in pinpointing the exact distance \( D \) where the beam can carry exactly 500 lb.

• Select the intersect feature on your calculator.
• Use the arrow keys to move closer to the intersection and press 'Enter.'
• The calculator will display the intersecting point where the x-coordinate corresponds to the needed distance \( D \).

Distance Between Supports

The distance between the supports \( D \) is critical to the beam's ability to carry weight safely. The problem illustrates an inverse relationship, meaning as \( D \) increases, the weight \( P \) the beam can carry decreases. This is depicted by the equation \( P = \frac{9200}{D} \).
In simpler terms, the closer the supports, the more weight the beam holds. To solve the exercise, you look for the point on the graph where these supports can be set at a specific distance while holding a 500 lb load. This is the intersect point identified on the graph.

Weight Carrying Capacity

The weight carrying capacity of a beam, signified here as \( P \), is inversely proportional to the distance \( D \) between the supports. This means if the distance grows longer, the capacity decreases unless other factors come into play.
The equation \( P = \frac{9200}{D} \) describes this relationship, highlighting the fixed product of the carrying capacity and the distance. For the given beam type, 9200 in this equation shows the maximum potential product of weight and distance.
This constant dictates how the beam can safely hold lighter loads at wider distances, but reduces as loads increase proportionally.