Answer :
To solve this problem, we need to calculate the crop yield at two specific years (year 5 and year 16), then find the average rate of change between these years.
1. Finding the Crop Yield at Year 5:
We are given the function [tex]\( f(x) = -x^2 + 20x + 75 \)[/tex].
To find the yield at year 5:
[tex]\[ f(5) = -(5)^2 + 20(5) + 75 = -25 + 100 + 75 = 150 \][/tex]
Therefore, the crop yield at year 5 is 150 pounds per acre.
2. Finding the Crop Yield at Year 16:
Using the same function [tex]\( f(x) \)[/tex]:
[tex]\[ f(16) = -(16)^2 + 20(16) + 75 = -256 + 320 + 75 = 139 \][/tex]
Therefore, the crop yield at year 16 is 139 pounds per acre.
3. Calculating the Average Rate of Change:
The average rate of change of a function between two points [tex]\((x_1, f(x_1))\)[/tex] and [tex]\((x_2, f(x_2))\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Here, [tex]\(x_1 = 5\)[/tex] and [tex]\(x_2 = 16\)[/tex]:
[tex]\[ \text{Average Rate of Change} = \frac{139 - 150}{16 - 5} = \frac{-11}{11} = -1.0 \][/tex]
Therefore, the average rate of change is [tex]\(-1.0\)[/tex] pound per acre per year.
### Interpretation:
- The crop yield in year 5 is 150 pounds per acre.
- The crop yield in year 16 is 139 pounds per acre.
- The average rate of change over this period is [tex]\(-1.0\)[/tex] pound per acre per year, which means that the crop yield is decreasing by 1 pound per acre each year on average between year 5 and year 16.
Given the multiple-choice options:
- The crop yield increased by 150 pounds per acre from year 5 to year 15.
- The crop yield decreased by 15 pounds per acre from year 5 to year 15.
- The crop yield decreased by 5 pounds per acre from year 5 to year 15.
- The crop yield did not change from year 5 to year 15.
None of these options directly match the analysis, but focusing on the fact that the yield is decreasing, the closest option with minimal interpretation could be a minor misstatement about the exact pounds:
- The crop yield decreased by 11 pounds from year 5 to year 16, not exactly any given years.
- To maintain selection integrity, correct statement attributes change till 5 to 16.
The correct interpretation based on the calculation should be: "The crop yield decreased by 11 pounds per acre from year 5 to year 16."
1. Finding the Crop Yield at Year 5:
We are given the function [tex]\( f(x) = -x^2 + 20x + 75 \)[/tex].
To find the yield at year 5:
[tex]\[ f(5) = -(5)^2 + 20(5) + 75 = -25 + 100 + 75 = 150 \][/tex]
Therefore, the crop yield at year 5 is 150 pounds per acre.
2. Finding the Crop Yield at Year 16:
Using the same function [tex]\( f(x) \)[/tex]:
[tex]\[ f(16) = -(16)^2 + 20(16) + 75 = -256 + 320 + 75 = 139 \][/tex]
Therefore, the crop yield at year 16 is 139 pounds per acre.
3. Calculating the Average Rate of Change:
The average rate of change of a function between two points [tex]\((x_1, f(x_1))\)[/tex] and [tex]\((x_2, f(x_2))\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Here, [tex]\(x_1 = 5\)[/tex] and [tex]\(x_2 = 16\)[/tex]:
[tex]\[ \text{Average Rate of Change} = \frac{139 - 150}{16 - 5} = \frac{-11}{11} = -1.0 \][/tex]
Therefore, the average rate of change is [tex]\(-1.0\)[/tex] pound per acre per year.
### Interpretation:
- The crop yield in year 5 is 150 pounds per acre.
- The crop yield in year 16 is 139 pounds per acre.
- The average rate of change over this period is [tex]\(-1.0\)[/tex] pound per acre per year, which means that the crop yield is decreasing by 1 pound per acre each year on average between year 5 and year 16.
Given the multiple-choice options:
- The crop yield increased by 150 pounds per acre from year 5 to year 15.
- The crop yield decreased by 15 pounds per acre from year 5 to year 15.
- The crop yield decreased by 5 pounds per acre from year 5 to year 15.
- The crop yield did not change from year 5 to year 15.
None of these options directly match the analysis, but focusing on the fact that the yield is decreasing, the closest option with minimal interpretation could be a minor misstatement about the exact pounds:
- The crop yield decreased by 11 pounds from year 5 to year 16, not exactly any given years.
- To maintain selection integrity, correct statement attributes change till 5 to 16.
The correct interpretation based on the calculation should be: "The crop yield decreased by 11 pounds per acre from year 5 to year 16."
