Use the table to answer the question.

x 320
0
1
2
3
4
5


Cason is scheduled to get his wisdom teeth removed. The oral surgeon gives him a local anesthetic to numb his mouth before surgery. Cason is given 320 milligrams of the anesthetic, which metabolizes at a rate of 25% per hour. The situation can be modeled using the exponential equation f(x)=320(0.75)x
. Finish the chart to determine which graph displays the correct plotted points for the situation.

Option #1:

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.



Option #2:

A curve with an arrow on the top is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 350 in increments of 50. The y-axis for milligrams ranges from 0 to 7 in increments of 1.



Option #3:

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.

(1 point)
The graph with the correct plotted points is Option #
.

1 answer

To find which graph displays the correct plotted points for the situation where Cason is given 320 milligrams of the anesthetic that metabolizes at a rate of 25% per hour, we can calculate the values of the function \( f(x) = 320(0.75)^x \) for \( x = 0, 1, 2, 3, 4, 5 \).

Here are the calculations:

  • For \( x = 0 \): \( f(0) = 320(0.75)^0 = 320 \)

  • For \( x = 1 \): \( f(1) = 320(0.75)^1 = 320 \times 0.75 = 240 \)

  • For \( x = 2 \): \( f(2) = 320(0.75)^2 = 320 \times 0.5625 = 180 \)

  • For \( x = 3 \): \( f(3) = 320(0.75)^3 = 320 \times 0.421875 = 135 \)

  • For \( x = 4 \): \( f(4) = 320(0.75)^4 = 320 \times 0.31640625 \approx 101.25 \)

  • For \( x = 5 \): \( f(5) = 320(0.75)^5 = 320 \times 0.2373046875 \approx 75.0 \)

This results in the following points plotted against hours (x-values):

  • \( (0, 320) \)
  • \( (1, 240) \)
  • \( (2, 180) \)
  • \( (3, 135) \)
  • \( (4, 101) \)
  • \( (5, 75) \)

The values show that the function decreases over time, starting at 320 milligrams and going down with each passing hour.

Now looking at the options given:

  • Option #1 has a decreasing curve with correct axes for hours and milligrams.
  • Option #2 seems incorrectly oriented, as it has hours on the y-axis and milligrams on the x-axis.
  • Option #3 is the same as Option #1.

Thus, the correct graph with the plotted points is Option #1 (or #3, as they are effectively the same).

So the final answer is: The graph with the correct plotted points is Option #1.