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$f\left(t\right)=11.65(1-{e}^{-\frac{t}{1.27}})$ where f(t) is the speed of the sprinter per second after t seconds.

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

mhalmantus

Skilled2021-08-19Added 105 answers

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$f\left(t\right)=11.65(1-{e}^{-\frac{t}{1.27}})$

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$x=9.86,f\left(9.86\right)=11.65(1-{e}^{-\frac{9.86}{1.27}})$

Using the calculator,$f\left(9.86\right)=11.65\times (1-0.00042484781)$

$f\left(9.86\right)=11.65$ 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,$f\left(t\right)=11.65(1-{e}^{-\frac{t}{1.27}})=10$

Divide each side by 11.65,$(1-{e}^{-\frac{t}{1.27}})=\frac{10}{11.65}$

Using calculator,$(1-{e}^{-\frac{t}{1.27}})=0.858369$

$1-0.858369={e}^{-\frac{t}{1.27}}$

${e}^{-\frac{t}{1.27}}=0.1416$

Taking natural logarithm on each side, In$\left({e}^{-\frac{t}{1.27}}\right)=In0.1416$

Using calculator,$(-\frac{t}{1.27}){\mathrm{log}}_{e}e={\mathrm{log}}_{e}0.1416$

$-\frac{t}{1.27}=-1.9547$

Divide by$-1,\frac{t}{1.27}=1.9547$

Using calculator,$t=1.9547\times 1.27$

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.

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

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

Let

Using the calculator,

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,

So,

Divide each side by 11.65,

Using calculator,

Taking natural logarithm on each side, In

Using calculator,

Divide by

Using calculator,

Rounded to nearest hundredth,

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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