preprekomW

2021-08-18

To find:
a) The speed of the sprinter while he running as crossed the finish line
b) After how many seconds was he running at the rate of 10 m per sec
Using the model function $f\left(t\right)=11.65\left(1-{e}^{-\frac{t}{1.27}}\right)$ where f(t) is the speed of the sprinter per second after t seconds.

mhalmantus

Concept:
Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables.
Calculation:
The given model function is $f\left(t\right)=11.65\left(1-{e}^{-\frac{t}{1.27}}\right)$
where f(t) is the speed of the sprinter per second after t seconds.
a) The sprinter finished the race in 9.86 seconds.
So, substituting $x=9.86$, we het the speed of the sprinter while he crossing the finish line.
Let $x=9.86,f\left(9.86\right)=11.65\left(1-{e}^{-\frac{9.86}{1.27}}\right)$
Using the calculator, $f\left(9.86\right)=11.65×\left(1-0.00042484781\right)$
$f\left(9.86\right)=11.65$ meters per second if rounded to nearest hundredth.
The speed of the sprinter while he running as crossed the finish line is 11.645 m/sec.
b) The speed of the sprinter is given by 10 m/sec.
So, $f\left(t\right)=10$
So, $f\left(t\right)=11.65\left(1-{e}^{-\frac{t}{1.27}}\right)=10$
Divide each side by 11.65, $\left(1-{e}^{-\frac{t}{1.27}}\right)=\frac{10}{11.65}$
Using calculator, $\left(1-{e}^{-\frac{t}{1.27}}\right)=0.858369$
$1-0.858369={e}^{-\frac{t}{1.27}}$
${e}^{-\frac{t}{1.27}}=0.1416$
Taking natural logarithm on each side, In $\left({e}^{-\frac{t}{1.27}}\right)=In0.1416$
Using calculator, $\left(-\frac{t}{1.27}\right){\mathrm{log}}_{e}e={\mathrm{log}}_{e}0.1416$
$-\frac{t}{1.27}=-1.9547$
Divide by $-1,\frac{t}{1.27}=1.9547$
Using calculator, $t=1.9547×1.27$
Rounded to nearest hundredth, $t\approx 2.48$ seconds.
So, after 2.48 seconds his speed was 10 m/sec.
Final statement:
a) The speed of the sprinter while he running as crossed the finish line is 11.65 m/sec.
b) After 2.48 seconds was he running at the rate of 10 m per sec.

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