Answer :
To determine which column of distance data shows a quadratic relationship with time, we need to analyze how the distance changes with respect to time squared. A quadratic relationship can be expressed as [tex]\( \text{distance} = k \times (\text{time}^2) \)[/tex], where [tex]\( k \)[/tex] is a constant.
Here, we are given five sets of distance data corresponding to different times. Let's assess each column to see if the distance values correspond to a quadratic relationship.
Column A:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 4.00, 9.00, 16.00, 25.00, 36.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{4.00}{2^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{9.00}{3^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{16.00}{4^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{25.00}{5^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{36.00}{6^2} = 1.0 \)[/tex]
Since [tex]\( k \)[/tex] is always 1.0, Column A follows a quadratic relationship with time.
Column B:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [2.00, 4.00, 6.00, 8.00, 10.00, 12.00, 14.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{4.00}{1^2} = 4.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{6.00}{2^2} = 1.5 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{8.00}{3^2} = 0.89 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{10.00}{4^2} = 0.625 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{12.00}{5^2} = 0.48 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{14.00}{6^2} = 0.39 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column B does not follow a quadratic relationship.
Column C:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [9.00, 18.00, 27.00, 36.00, 45.00, 54.00, 63.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{18.00}{1^2} = 18.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{27.00}{2^2} = 6.75 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{36.00}{3^2} = 4.0 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{45.00}{4^2} = 2.8125 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{54.00}{5^2} = 2.16 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{63.00}{6^2} = 1.75 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column C does not follow a quadratic relationship.
Column D:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 0.50, 0.33, 0.25, 0.20, 0.16]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{0.50}{2^2} = 0.125 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{0.33}{3^2} = 0.0367 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{0.25}{4^2} = 0.0156 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{0.20}{5^2} = 0.008 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{0.16}{6^2} = 0.0044 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column D does not follow a quadratic relationship.
Column E:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 0.25, 0.11, 0.06, 0.04, 0.02]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{0.25}{2^2} = 0.0625 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{0.11}{3^2} = 0.0122 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{0.06}{4^2} = 0.00375 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{0.04}{5^2} = 0.0016 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{0.02}{6^2} = 0.00056 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column E does not follow a quadratic relationship.
Thus, the correct answer is:
A. Column A.
Here, we are given five sets of distance data corresponding to different times. Let's assess each column to see if the distance values correspond to a quadratic relationship.
Column A:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 4.00, 9.00, 16.00, 25.00, 36.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{4.00}{2^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{9.00}{3^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{16.00}{4^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{25.00}{5^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{36.00}{6^2} = 1.0 \)[/tex]
Since [tex]\( k \)[/tex] is always 1.0, Column A follows a quadratic relationship with time.
Column B:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [2.00, 4.00, 6.00, 8.00, 10.00, 12.00, 14.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{4.00}{1^2} = 4.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{6.00}{2^2} = 1.5 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{8.00}{3^2} = 0.89 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{10.00}{4^2} = 0.625 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{12.00}{5^2} = 0.48 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{14.00}{6^2} = 0.39 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column B does not follow a quadratic relationship.
Column C:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [9.00, 18.00, 27.00, 36.00, 45.00, 54.00, 63.00]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{18.00}{1^2} = 18.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{27.00}{2^2} = 6.75 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{36.00}{3^2} = 4.0 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{45.00}{4^2} = 2.8125 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{54.00}{5^2} = 2.16 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{63.00}{6^2} = 1.75 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column C does not follow a quadratic relationship.
Column D:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 0.50, 0.33, 0.25, 0.20, 0.16]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{0.50}{2^2} = 0.125 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{0.33}{3^2} = 0.0367 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{0.25}{4^2} = 0.0156 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{0.20}{5^2} = 0.008 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{0.16}{6^2} = 0.0044 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column D does not follow a quadratic relationship.
Column E:
- Time: [0, 1, 2, 3, 4, 5, 6]
- Distance: [0, 1.00, 0.25, 0.11, 0.06, 0.04, 0.02]
For a quadratic relationship:
- [tex]\( k \)[/tex] at time 1 is [tex]\( \frac{1.00}{1^2} = 1.0 \)[/tex]
- [tex]\( k \)[/tex] at time 2 is [tex]\( \frac{0.25}{2^2} = 0.0625 \)[/tex]
- [tex]\( k \)[/tex] at time 3 is [tex]\( \frac{0.11}{3^2} = 0.0122 \)[/tex]
- [tex]\( k \)[/tex] at time 4 is [tex]\( \frac{0.06}{4^2} = 0.00375 \)[/tex]
- [tex]\( k \)[/tex] at time 5 is [tex]\( \frac{0.04}{5^2} = 0.0016 \)[/tex]
- [tex]\( k \)[/tex] at time 6 is [tex]\( \frac{0.02}{6^2} = 0.00056 \)[/tex]
[tex]\( k \)[/tex] is not consistent, so Column E does not follow a quadratic relationship.
Thus, the correct answer is:
A. Column A.
