To determine Aruba's production possibilities frontier (PPF) when 100 labor hours are available, we need to calculate how many coolers and radios Aruba can produce depending on how they allocate their labor between the two products.
Given:
- Labor hours needed to produce a cooler in Aruba: 2 hours
- Labor hours needed to produce a radio in Aruba: 5 hours
- Total labor hours available: 100 hours
We can find the maximum number of coolers and radios that can be produced:
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Maximum coolers (when no radios are produced): \[ \text{Maximum coolers} = \frac{\text{Total labor hours}}{\text{Labor hours per cooler}} = \frac{100}{2} = 50 \text{ coolers} \]
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Maximum radios (when no coolers are produced): \[ \text{Maximum radios} = \frac{\text{Total labor hours}}{\text{Labor hours per radio}} = \frac{100}{5} = 20 \text{ radios} \]
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Finding combinations of coolers and radios:
- If \(x_c\) is the number of coolers and \(x_r\) is the number of radios produced, we have the labor constraint: \[ 2x_c + 5x_r \leq 100 \]
Rearranging gives us: \[ x_r = \frac{100 - 2x_c}{5} \]
Now, let's look at a few combinations:
- If \(x_c = 0\) then \(x_r = 20\)
- If \(x_c = 10\) then \(x_r = \frac{100 - 20}{5} = 16\)
- If \(x_c = 20\) then \(x_r = \frac{100 - 40}{5} = 12\)
- If \(x_c = 30\) then \(x_r = \frac{100 - 60}{5} = 8\)
- If \(x_c = 40\) then \(x_r = \frac{100 - 80}{5} = 4\)
- If \(x_c = 50\) then \(x_r = 0\)
From these calculations, we can plot the PPF for Aruba using the points:
- (0, 20)
- (10, 16)
- (20, 12)
- (30, 8)
- (40, 4)
- (50, 0)
The correct representation of this PPF would ideally be a downward-sloping curve between these points, showing the trade-off in production between coolers and radios.
Without the specific options (a, b, c, d), I can't select a definite answer, but based on the points we've calculated, you should look for a curve that connects these coordinates, illustrating the trade-offs as Aruba reallocates labor from coolers to radios and vice versa.
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