To determine the year when the men's world record in the 100-meter dash falls below 9.2 seconds, we start with the given linear equation:
[tex]\[ T = -0.01y + 10.6 \][/tex]
Where:
- [tex]\( T \)[/tex] is the time in seconds.
- [tex]\( y \)[/tex] is the number of years after 1900.
We are asked to find the year when [tex]\( T \)[/tex] is less than 9.2 seconds. Thus, we set up the inequality:
[tex]\[ -0.01y + 10.6 < 9.2 \][/tex]
To solve for [tex]\( y \)[/tex]:
1. Subtract 10.6 from both sides of the inequality:
[tex]\[ -0.01y < 9.2 - 10.6 \][/tex]
2. Simplify the right-hand side:
[tex]\[ -0.01y < -1.4 \][/tex]
3. Divide both sides by -0.01. Note that dividing by a negative number will reverse the inequality:
[tex]\[ y > \frac{-1.4}{-0.01} \][/tex]
4. Simplify the division:
[tex]\[ y > 140 \][/tex]
This means that the record time will fall below 9.2 seconds more than 140 years after 1900. Therefore, we add 140 to 1900:
[tex]\[ 1900 + 140 = 2040 \][/tex]
Hence, the world record time for the 100-meter dash will fall below 9.2 seconds in the year 2040.
