Sure, let's solve the given problem step-by-step. We have the function that models the change in a runner's performance as:
[tex]\[ t = 0.0119s^2 - 0.308s - 0.0003 \][/tex]
We're asked to predict the change in a runner's finishing time with a wind speed ([tex]\( s \)[/tex]) of 3 meters/second.
First, we substitute [tex]\( s = 3 \)[/tex] into the function:
[tex]\[ t = 0.0119(3^2) - 0.308(3) - 0.0003 \][/tex]
Calculating each term inside the expression:
1. Calculate [tex]\( 3^2 \)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
2. Multiply [tex]\( 0.0119 \)[/tex] by [tex]\( 9 \)[/tex]:
[tex]\[ 0.0119 \times 9 = 0.1071 \][/tex]
3. Multiply [tex]\( 0.308 \)[/tex] by [tex]\( 3 \)[/tex]:
[tex]\[ 0.308 \times 3 = 0.924 \][/tex]
Now substitute these results back into the expression:
[tex]\[ t = 0.1071 - 0.924 - 0.0003 \][/tex]
Next, perform the subtraction:
1. Subtract [tex]\( 0.924 \)[/tex] from [tex]\( 0.1071 \)[/tex]:
[tex]\[ 0.1071 - 0.924 = -0.8169 \][/tex]
2. Subtract [tex]\( 0.0003 \)[/tex] from [tex]\( -0.8169 \)[/tex]:
[tex]\[ -0.8169 - 0.0003 = -0.8172 \][/tex]
Therefore, the predicted change in the runner's finishing time for a wind speed of 3 meters/second is approximately [tex]\( -0.8172 \)[/tex].
Finally, we need to round this result to the nearest hundredth:
[tex]\[ -0.8172 \approx -0.82 \][/tex]
So, the predicted change in the runner's finishing time with a wind speed of 3 meters/second, rounded to the nearest hundredth, is [tex]\( -0.82 \)[/tex]. This means the runner is expected to finish [tex]\( 0.82 \)[/tex] seconds faster.
