Question

when a stone is dropped in a pond, ripples are formed and travel in concentric circles away from where the stone was dropped. the equation of the least - squares regression line is \\( \widehat { \text { area } } = 0.010 + 3.141 ( \text { time } ^ { 2 } ) \\). what is the predicted area, in \\( mathrm { cm } ^ { 2 } \\), of the circle 8 seconds after the stone was dropped?
\\( 25.14 \mathrm { cm } ^ { 2 } \\)
\\( 50.27 \mathrm { cm } ^ { 2 } \\)
\\( 64.01 \mathrm { cm } ^ { 2 } \\)
\\( 201.03 \mathrm { cm } ^ { 2 } \\)

Explanation:

Step1: Substitute Time value into the equation

Given the regression line equation \(\widehat{\text{Area}}=0.010 + 3.141(\text{Time}^{2})\), and Time \(t = 8\) seconds.
Substitute \(t = 8\) into the equation: \(\widehat{\text{Area}}=0.010+3.141\times(8)^{2}\)

Step2: Calculate \(8^{2}\)

\(8^{2}=64\)
So the equation becomes \(\widehat{\text{Area}}=0.010 + 3.141\times64\)

Step3: Calculate \(3.141\times64\)

\(3.141\times64=(3 + 0.141)\times64=3\times64+0.141\times64=192+9.024 = 201.024\)

Step4: Calculate the final value of \(\widehat{\text{Area}}\)

\(\widehat{\text{Area}}=0.010+201.024=201.034\approx201.03\)

Answer:

\(201.03\mathrm{cm}^{2}\) (the fourth option)