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Chapter 4: Problem 49

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand. In \(2010,\) for Standard delivery to the closest zone of packages weighing from 100 to 499 lb, FedEx charged \(\$ 130\) plus \(\$ 1.30\) for each pound over \(100 .\) Estimate the weight of a package that cost \(\$ 325\) to ship.

Short Answer

Expert verified
The estimated weight for the cost of \( \$325 \) is approximately \( 280 \) lb.

Step by step solution

01

Identify the equation

From the given information, identify the cost equation: \( C = 130 + 1.30(x - 100) \) where \( C \) is the cost and \( x \) is the weight in pounds over 100 lb.
02

Set up the graph

On graph paper, label the x-axis as weight in pounds and the y-axis as cost in dollars. Ensure your graph has a range large enough to include the known points.
03

Graph the equation

Plot the equation \( C = 130 + 1.30(x - 100) \) on the graph. Begin by plotting the point (100, 130). This is because when \( x = 100 \), \( C = 130 \). Then determine another point by choosing a convenient value for \( x \) and calculating \( C \).
04

Estimate the cost

Locate \( C = 325 \) on the y-axis. Draw a horizontal line from \( C = 325 \) to intersect the graph of \( C = 130 + 1.30(x - 100) \).
05

Find the weight

From the point of intersection, draw a vertical line downwards to the x-axis. This will give an estimate of the weight in pounds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Equation
Understanding the cost equation is essential. In our problem, FedEx charged a base fee plus an extra cost for additional weight. The specific cost formula given is:
\[ C = 130 + 1.30(x - 100) \]
Here,
  • \(C\) represents the total cost in dollars
  • \(130\) is the base cost for the first 100 pounds
  • \(1.30\) is the cost per pound over 100 pounds
  • \(x\) represents the total weight in pounds
Using these variables, the equation helps calculate the shipping cost based on the package's weight. For example, if your package weighs 200 pounds:
\[ C = 130 + 1.30(200 - 100) \]
\[ C = 130 + 1.30 \times 100 \]
\[ C = 130 + 130 \]
\[ C = 260 \]
Graphing by Hand
Graphing by hand might seem old-fashioned, but it's incredibly useful! Here鈥檚 how to create and use your graph:
  • Start with graph paper and a straightedge.
  • Label the x-axis as weight (in pounds).
  • Label the y-axis as cost (in dollars).
Begin with plotting the base point. Since we know \( x = 100 \) and \( C = 130 \), place a point at \( (100, 130) \). Next, choose another x-value, like \( x = 150 \), and calculate \( C \):
\[ C = 130 + 1.30(150 - 100) \]
\[ C = 130 + 1.30 \times 50 \]
\[ C = 130 + 65 \]
\[ C = 195 \]
Plot this second point at \( (150, 195) \). Now draw a straight line through these points. This line represents the cost equation.
Estimating Solutions
With your graph ready, estimating a solution becomes simple. Suppose you need to know the weight of a package that costs \$325 to ship. Follow these steps:
  1. Find \( C = 325 \) on the y-axis.
  2. Draw a horizontal line from this point until it intersects your cost line.
  3. At the intersection, draw a vertical line downwards to the x-axis.
Where your vertical line meets the x-axis is your estimated weight. For this problem, your estimated weight might be around 250 pounds.
Estimation using graphs helps visualize solutions, offering a practical approach to understanding equations and their implications.

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