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Tessa is comparing the cost of a fresh lobster dinner at two different restaurants. The first restaurant charges $40 for the meal, plus $2 per kilogram for the lobster she picks. At the second restaurant, Tessa would pay $3 per kilogram for the lobster, in addition to $35 for the meal. Tessa realizes that, in theory, dinner at both restaurants could cost the same amount if the lobster had a certain weight. What is the weight? Write a system of equations, graph them, and type the solution. graph

1 answer

To solve the problem, we need to set up a system of equations based on Tessa's dinner costs at the two restaurants.

Let \( x \) be the weight of the lobster in kilograms.

Cost equations:

  1. First Restaurant:

    • Total cost = Fixed cost + Cost per kg of lobster
    • \( C_1 = 40 + 2x \)

  2. Second Restaurant:

    • Total cost = Fixed cost + Cost per kg of lobster
    • \( C_2 = 35 + 3x \)

Setting up the equations:

To find out when the costs are equal, we can set \( C_1 \) equal to \( C_2 \): \[ 40 + 2x = 35 + 3x \]

Solving the equation:

Now we can solve for \( x \):

  1. Subtract \( 2x \) from both sides: \[ 40 = 35 + x \]
  2. Subtract 35 from both sides: \[ 40 - 35 = x \] \[ x = 5 \]

Solution:

The weight of the lobster for which both restaurant dinners would cost the same is \( \textbf{5 kilograms} \).

System of equations:

To summarize, the system of equations is: \[ \begin{align*} C_1 & = 40 + 2x \ C_2 & = 35 + 3x \end{align*} \]

When graphed, the point where both lines intersect represents the cost being equal, which occurs at \( x = 5 \) kg.

Graphing:

You can graph the two equations, but since I can't create graphical plots directly here, I'll describe how it would look:

  1. On a graph, the x-axis would represent the weight of the lobster \( x \), and the y-axis would represent the cost \( C \).
  2. The line for the first restaurant \( C_1 = 40 + 2x \) would start at (0, 40) and have a slope of 2.
  3. The line for the second restaurant \( C_2 = 35 + 3x \) would start at (0, 35) and have a steeper slope of 3.
  4. The point of intersection of these two lines occurs at \( (5, C) \), where \( C \) can be calculated by substituting \( x = 5 \) into either equation to find that both costs equal \( 50 \).

If you would like help with the actual graphing process or software suggestions to visualize this, please let me know!