Question

On a 7-hour trip home from winter break, Marta was forced to drive on a stretch of highway that had a lot of construction. At first she had to go 65 miles per hour for a certain amount of time, but then she was able to go 75 miles per hour for the rest of the trip and in total she drove 485 miles. Assuming she made no stops, how long did she drive 65 miles per hour?

Answer 1

Putting that in equation form:

\[t_1+t_2 = 7\]\[t_1*65+t_2*75=485\]
Solve the simultaneous equations

Answer 2

\[v_1=\frac{x_1-0}{t_1}\\ \\ \frac{x_1}{t_1}=65......(1)\]
\[v_2=\frac{v_2-v_1}{t_2} \\ \\ \frac{v_2-v_1}{t_2}=75......(2)\]
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Total time needed is \(t_1+t_2=T\) , so, \(t_2=7-t_1\)
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\[x_2-x_1=75t_2 \\ \\ x_1=x_2-75t_2\]
also, \(x_2=485\) so,
\[x_1=485-75t_2\]
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Using back (1),\[ \frac{x_1}{t_1}=65\]
\[\frac{(485-75(7-t_1)}{t_1}=65 \\ \\ t_1=4s\]