Question
A person dives into a pool from its edge to swim to the other side. The table shows the depth in feet of the person from the surface of the water after x
seconds. The data can be modeled by a quadratic function.
Time, x (seconds) Depth of person from surface of water d(x) (feet)
1 -1.85
4 -7.28
6 -8.3
8.5 -6.65
10 -4.1
11.5 -0.38
Which function best models the data?
(1 point)
Responses
d(x)=0.05x2+0.74x
d times x is equal to 0 point 0 5 x squared plus 0 point 7 4 x
d(x)=0.05x2+0.74x+9.17
d times x is equal to 0 point 0 5 x squared plus 0 point 7 4 x plus 9 point 1 7
d(x)=0.26x2−3.11x
d times x is equal to 0 point 2 6 x squared minus 3 point 1 1 x
d(x)=0.26x2−3.11x+1
seconds. The data can be modeled by a quadratic function.
Time, x (seconds) Depth of person from surface of water d(x) (feet)
1 -1.85
4 -7.28
6 -8.3
8.5 -6.65
10 -4.1
11.5 -0.38
Which function best models the data?
(1 point)
Responses
d(x)=0.05x2+0.74x
d times x is equal to 0 point 0 5 x squared plus 0 point 7 4 x
d(x)=0.05x2+0.74x+9.17
d times x is equal to 0 point 0 5 x squared plus 0 point 7 4 x plus 9 point 1 7
d(x)=0.26x2−3.11x
d times x is equal to 0 point 2 6 x squared minus 3 point 1 1 x
d(x)=0.26x2−3.11x+1
Answers
GPT-4o mini
To determine which quadratic function best models the given data of depth \(d(x)\) from the surface of the water after \(x\) seconds, we can analyze the provided options and see which one fits best to the data points.
Given the data points:
- \(d(1) = -1.85\)
- \(d(4) = -7.28\)
- \(d(6) = -8.3\)
- \(d(8.5) = -6.65\)
- \(d(10) = -4.1\)
- \(d(11.5) = -0.38\)
We need to see which of the suggested functions:
1. \(d(x) = 0.05x^2 + 0.74x\)
2. \(d(x) = 0.05x^2 + 0.74x + 9.17\)
3. \(d(x) = 0.26x^2 - 3.11x\)
4. \(d(x) = 0.26x^2 - 3.11x + 1\)
can be used to approximate the values of \(d(x)\) for the given \(x\) values.
**Method of Evaluation:**
1. Plug in each \(x\) value into the function.
2. Calculate the depth and compare it with the actual depth from the table.
3. The best model will be the one with the closest values.
Let's briefly evaluate the first function:
For \(d(x) = 0.05x^2 + 0.74x\):
- \(d(1) = 0.05(1^2) + 0.74(1) = 0.05 + 0.74 = 0.79\) (compared to -1.85)
- \(d(4) = 0.05(4^2) + 0.74(4) = 0.8 + 2.96 = 3.76\) (compared to -7.28)
This approach is not matching well.
Next function:
\(d(x) = 0.05x^2 + 0.74x + 9.17\):
- \(d(1) = 0.05(1^2) + 0.74(1) + 9.17 = 0.79 + 9.17 = 9.96\) (not matching)
Next:
\(d(x) = 0.26x^2 - 3.11x\):
- \(d(1) = 0.26(1^2) - 3.11(1) = 0.26 - 3.11 = -2.85\) (compared to -1.85)
- \(d(4) = 0.26(4^2) - 3.11(4) = 4.16 - 12.44 = -8.28\) (compared to -7.28, close)
And finally, the last one:
\(d(x) = 0.26x^2 - 3.11x + 1\):
- \(d(1) = 0.26(1^2) - 3.11(1) + 1 = 0.26 - 3.11 + 1 = -1.85\) (matches exactly)
- \(d(4) = 0.26(4^2) - 3.11(4) + 1 = 4.16 - 12.44 + 1 = -7.28\) (matches)
After testing all the functions, it is clear that the function \(d(x) = 0.26x^2 - 3.11x + 1\) is the one that best models the data given that it matches all known data points for \(x = 1\) and \(x = 4\) perfectly, along with providing closer values for the remaining entries.
**Final Answer:**
\(d(x) = 0.26x^2 - 3.11x + 1\)
Given the data points:
- \(d(1) = -1.85\)
- \(d(4) = -7.28\)
- \(d(6) = -8.3\)
- \(d(8.5) = -6.65\)
- \(d(10) = -4.1\)
- \(d(11.5) = -0.38\)
We need to see which of the suggested functions:
1. \(d(x) = 0.05x^2 + 0.74x\)
2. \(d(x) = 0.05x^2 + 0.74x + 9.17\)
3. \(d(x) = 0.26x^2 - 3.11x\)
4. \(d(x) = 0.26x^2 - 3.11x + 1\)
can be used to approximate the values of \(d(x)\) for the given \(x\) values.
**Method of Evaluation:**
1. Plug in each \(x\) value into the function.
2. Calculate the depth and compare it with the actual depth from the table.
3. The best model will be the one with the closest values.
Let's briefly evaluate the first function:
For \(d(x) = 0.05x^2 + 0.74x\):
- \(d(1) = 0.05(1^2) + 0.74(1) = 0.05 + 0.74 = 0.79\) (compared to -1.85)
- \(d(4) = 0.05(4^2) + 0.74(4) = 0.8 + 2.96 = 3.76\) (compared to -7.28)
This approach is not matching well.
Next function:
\(d(x) = 0.05x^2 + 0.74x + 9.17\):
- \(d(1) = 0.05(1^2) + 0.74(1) + 9.17 = 0.79 + 9.17 = 9.96\) (not matching)
Next:
\(d(x) = 0.26x^2 - 3.11x\):
- \(d(1) = 0.26(1^2) - 3.11(1) = 0.26 - 3.11 = -2.85\) (compared to -1.85)
- \(d(4) = 0.26(4^2) - 3.11(4) = 4.16 - 12.44 = -8.28\) (compared to -7.28, close)
And finally, the last one:
\(d(x) = 0.26x^2 - 3.11x + 1\):
- \(d(1) = 0.26(1^2) - 3.11(1) + 1 = 0.26 - 3.11 + 1 = -1.85\) (matches exactly)
- \(d(4) = 0.26(4^2) - 3.11(4) + 1 = 4.16 - 12.44 + 1 = -7.28\) (matches)
After testing all the functions, it is clear that the function \(d(x) = 0.26x^2 - 3.11x + 1\) is the one that best models the data given that it matches all known data points for \(x = 1\) and \(x = 4\) perfectly, along with providing closer values for the remaining entries.
**Final Answer:**
\(d(x) = 0.26x^2 - 3.11x + 1\)