Question

The speed of each of the runners by modeling the

Modeling
ANSWERED
asked 2021-08-18
The speed of each of the runners by modeling the system of linear equations.

Expert Answers (1)

2021-08-19

Procedure used:
"odel and solve system of linear equations
1) Identify the category of given problem.
2) Recognize and algebraically name the unknowns.
3) Translate the problem statement as a system of linear equations using the variables from Step 2.
S 4) Solve the system as obtained in Step 3.
5) State the answer statemen."
Calculation:
Step 1:
The given problem is a uniform motion problem where speed of the runner is to be calculated using the following formula:
time \(\displaystyle={\frac{{{d}{i}{s}{\tan{{c}}}{e}}}{{{s}{p}{e}{e}{d}}}}\)
Step 2:
There are two unknowns in the problem. Consider x be the speed of first runner and hence, from the given information, \(\displaystyle{x}+{2}\) be the speed of second runner.
Step 3:
The faster runner runs with the speed of \(\displaystyle{x}+{2}\) with the distance of 12 miles and second runner runs with x speed and the distance covered is 9 miles. It is summarized in Table.
\(\begin{array}{|c|c|} \hline & Distance & Speed & Time \\ \hline First\ runner & 12 & x+2 & t \\Second\ runner & 9 & x & t\\ \hline \end{array}\)
For first runner, use distance \(\displaystyle={s}{p}{e}{e}{d}\times{t}{i}{m}{e}\) and obtain the equation:
\(\displaystyle{12}={\left({x}+{2}\right)}\times{t}\)
\(\displaystyle{t}={\frac{{{12}}}{{{x}+{2}}}}\)
For second runner, use distance \(\displaystyle={s}{p}{e}{e}{d}\times{t}{i}{m}{e}\) and obtain the equation:
\(\displaystyle{S}{9}={\left({x}\right)}\times{t}\)
\(\displaystyle{t}={\frac{{{9}}}{{{x}}}}\)
Thus, the system of linear equations is:
\(\displaystyle{t}={\frac{{{12}}}{{{x}+{2}}}}{\left({1}\right)}\)
\(\displaystyle{t}={\frac{{{9}}}{{{x}}}}{\left({2}\right)}\)
Step 4:
Substitute equation (1) in equation (2).
\(\displaystyle{\frac{{{12}}}{{{x}+{2}}}}={\frac{{{9}}}{{{x}}}}\)
Thus, solve the equations to find the variable x:
\(\displaystyle{\frac{{{12}}}{{{x}+{2}}}}={\frac{{{9}}}{{{x}}}}\)
\(\displaystyle{12}{x}={9}{\left({x}+{2}\right)}\)
\(\displaystyle{12}{x}={9}{x}+{18}\)
Solve the above equation to obtain the value of variable x as:
\(\displaystyle{12}{x}-{9}{x}={18}\)
\(\displaystyle{3}{x}={18}\)
\(\displaystyle{x}={\frac{{{18}}}{{{3}}}}\)
S \(\displaystyle{x}={6}\)
Thus, the value of x is 6miles/hour.
Therefore, the speed of second runner is \(\displaystyle{x}={6}\) miles/hour.
S Therefore, the speed of first runner is,
\(\displaystyle{x}+{2}={6}+{2}={8}\) miles/hour
Step 5:
Thus, the speed of first runner is 8 miles/hour and the speed of second runner is 6 miles/hour.

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