A farmer sells almonds according to the chart below. The more pounds of almonds that are sold, the lower the cost per pound is for the entire order of almonds.

Almond Prices
[tex]\[
\begin{tabular}{|c|c|}
\hline Number of Pounds & Cost (per Pound) \\
\hline up to 10 & \$20 \\
\hline more than 10, and up to 20 & \$15 \\
\hline more than 20, and up to 30 & \$10 \\
\hline more than 30 & \$5 \\
\hline
\end{tabular}
\][/tex]

The farmer graphs a step function to model the situation, where [tex]\(y\)[/tex] represents the cost per pound for an order of [tex]\(x\)[/tex] pounds of almonds.

Which of the following describes the farmer's graph for the step that includes 20.75 pounds, and what is the total cost of 20.75 pounds of almonds?

A. A horizontal segment with an open circle at [tex]\((20,10)\)[/tex] and a closed circle at [tex]\((30,10)\)[/tex]; \[tex]$207.50

B. A horizontal segment with a closed circle at \((20,10)\) and an open circle at \((30,10)\); \$[/tex]207.50

C. A horizontal segment with an open circle at [tex]\((21,10)\)[/tex] and a closed circle at [tex]\((30,10)\)[/tex]; \$311.25



Answer :

To solve this problem, let's break it down step-by-step.

First, we need to determine the cost per pound for ordering 20.75 pounds of almonds based on the given pricing structure:

1. Identify the appropriate price bracket:
- From the chart:
- For up to 10 pounds, the cost per pound is [tex]$20. - For more than 10, and up to 20 pounds, the cost per pound is $[/tex]15.
- For more than 20, and up to 30 pounds, the cost per pound is [tex]$10. - For more than 30 pounds, the cost per pound is $[/tex]5.

Since 20.75 pounds falls into the "more than 20, and up to 30" category, the cost per pound will be [tex]$10. 2. Calculate the total cost: - Multiply the number of pounds by the cost per pound to get the total cost: \[ \text{Total cost} = \text{Weight} \times \text{Cost per pound} = 20.75 \, \text{pounds} \times 10 \, \text{dollars per pound} = 207.5 \, \text{dollars} \] 3. Describe the step function for the graph: - The step function represents the cost per pound for varying quantities of almonds. - For the interval "more than 20, and up to 30 pounds": - At \(x = 20\), the price changes from the previous interval, so it will be an open circle. - At \(x = 30\), the interval ends, so it will be a closed circle. - The y-value (cost per pound) for this interval is 10 dollars. So, the graph will have a horizontal segment at y = 10 with an open circle at (20, 10) indicating the transition to this segment, and a closed circle at (30, 10) indicating the continuation of the segment up to 30 pounds. Putting it all together, based on the steps outlined, the correct answer is: ``` a horizontal segment with an open circle at (20,10) and a closed circle at (30,10) ; \$[/tex] 207.50
```