Problem 11 - 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. 1200 \(\mathrm{lb}\)

Step-by-Step Solution

Answer

The distance \(D\) between the supports necessary to carry a weight of 1200 pounds is approximately 7.67 feet.

Step 1: Understand the problem

We have an equation which gives us the weight \(P\) a beam can support in an inverse relation to the distance \(D\) between the supports. This relationship can be expressed as \(P = \frac{9200}{D}\). So we are supposed to use this equation to find the value of \(D\) (distance between supports), when \(P\) (weight that the beam can safely carry) is 1200 pounds.

Step 2: Set up the equation

The equation becomes \(1200 = \frac{9200}{D}\), because we are given that \(P\) is 1200 pounds.

Step 3: Solve for \(D\)

Rearrange the equation to find the value of \(D\). Multiply both sides of the equation by \(D\) to get \(1200D = 9200\), and then divide by 1200 to solve for \(D\).

Step 4: Calculate \(D\)

By performing the division \(D = \frac{9200}{1200}\), we should get the distance between supports needed for the beam to safely carry 1200 pounds.

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools that aid in visualizing mathematical equations. They allow students to see the relationship between variables, like in our beam example with weight and distance. By plotting the equation \( P = \frac{9200}{D} \) on a graphing calculator, you can see how as the distance (\( D \)) increases, the weight (\( P \)) decreases. This graphical representation helps you better understand inverse relationships.
To find the distance (\( D \)) needed to carry a particular weight (\( P = 1200 \) pounds), the Intersect feature can be used. Here's how:

• First, load the equation \( P = \frac{9200}{D} \) to your graphing calculator.
• Then, plot the horizontal line \( P = 1200 \).
• Use the Intersect feature to find where these two graphs meet, which provides the value for \( D \).

Understanding how to utilize these features on a graphing calculator enables you to easily solve complex equations.

Equation Solving

Solving equations involves manipulating them to find unknown variables, like \( D \) in our exercise. Given the formula \( P = \frac{9200}{D} \), and knowing that \( P = 1200 \) pounds, you'd set up the equation as follows: \( 1200 = \frac{9200}{D} \). Our goal is to isolate \( D \).
To do this, you must rearrange the equation:

• Multiply both sides by \( D \) to eliminate the denominator: \( 1200D = 9200 \).
• Then, divide both sides by 1200 to solve for \( D \): \( D = \frac{9200}{1200} \).

By performing the division, you'll discover the needed distance between supports. This method of rearranging and performing algebraic operations is key to solving many types of math problems.

Distance Calculation

Distance calculation in our context means determining how far the supports of a beam should be for it to hold a specific weight. In problems with inverse variation, like \( P = \frac{9200}{D} \), distance impacts weight-carrying capacity.
The calculation involves solving for \( D \),specifically, understanding that as the distance increases, the weight capacity decreases, and vice versa.
From the equation \( 1200 = \frac{9200}{D} \),you find \( D = \frac{9200}{1200} \), simplifying this to calculate the actual distance.
This computation is vital in engineering, ensuring structures are safe and functional. Calculating accurately ensures that beams will carry the necessary weight without risking structural failure.

Weight Capacity

Weight capacity refers to how much weight a structure or object can bear. For our beam example, weight capacity dictates the permissible load the beam can hold given a specific distance between its supports. This relationship is described by inverse variation: as distance \( D \) increases, weight capacity \( P \) decreases, represented by \( P = \frac{9200}{D} \).
Understanding weight capacity is crucial in design and construction, aiding decisions regarding material choice and spacing of supports to ensure safe weight distribution.
By calculating weight capacity, engineers can predict how structures will behave under specific loads, adjusting parameters to maximize safety. This knowledge ensures structures are built to withstand anticipated forces without failure, keeping users secure and confident in the construct's reliability.