On Monday, Dev goes on a 3.6 km run. His average speed for the first 1.2 km is $x$ km/h. Simplify and write down an expression, in terms of $x$, for the time taken for the first 1.2 km. Answer ... minutes His average speed for the last 2.4 km of the run is 2 km/h slower than the first 1.2 km. Simplify and write down an expression, in terms of $x$, for the time taken for the final 2.4 km. Answer ... minutes Dev takes 25 minutes to complete the full 3.6 km run. Form an equation in $x$ and show that it simplifies to $25x^{2} - 266x + 144 = 0.$ Solve the equation $25x^{2} - 266x + 144 = 0$, leaving your answers correct to 3 decimal places. Answer x = ... On Friday, Dev completed a 4 km run on the same average speed that he ran for the last 2.4 km of the 3.6 km run on Monday. Calculate the time Dev took to run 4 km on Friday.
Give your answer in minutes and seconds, correct to the nearest seconds. Answer ...

1. Find the time for the first 1.2 km and convert it to minutes.

$\text{Time}=\frac{\text{distance}}{\text{speed}}=\frac{1.2}{x}\text{ hours}$
Since $1$ hour $=60$ minutes,
$\frac{1.2}{x}\times 60=\frac{72}{x}\text{ minutes}$

1. Express the time for the final 2.4 km in minutes using the slower speed.

Speed for the final 2.4 km is $(x-2)$ km/h.

$\text{Time}=\frac{2.4}{x-2}\text{ hours}$

Convert to minutes:

$\frac{2.4}{x-2}\times 60=\frac{144}{x-2}\text{ minutes}$

1. Form the equation by adding the two running times to get the total time.

From parts (a)(i) and (a)(ii),

Time for first $1.2$ km $=\dfrac{72}{x}$ minutes,

Time for final $2.4$ km $=\dfrac{144}{x-2}$ minutes.

Since the total time is $25$ minutes,
$\frac{72}{x}+\frac{144}{x-2}=25.$

2. Clear fractions and simplify to the required quadratic equation.

Multiply throughout by $x(x-2)$:
$72(x-2)+144x=25x(x-2).$

Expand and collect terms:
$72x-144+144x=25x^2-50x$
$216x-144=25x^2-50x$
$0=25x^2-266x+144.$

Therefore,
$25x^2-266x+144=0.$

1. Identify the coefficients and apply the quadratic formula.

For $25x^{2}-266x+144=0$,
$a=25,\quad b=-266,\quad c=144.$
Using
$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},$
$x=\frac{-(-266)\pm\sqrt{(-266)^2-4(25)(144)}}{2(25)}.$

2. Evaluate the discriminant and simplify the expression for the roots.

$x=\frac{266\pm\sqrt{70756-14400}}{50}$
$x=\frac{266\pm\sqrt{56356}}{50}.$

3. Calculate both roots and round to 3 decimal places.

$x=\frac{266+\sqrt{56356}}{50}=10.070\ldots$
$x=\frac{266-\sqrt{56356}}{50}=0.569\ldots$
Correct to 3 decimal places,
$x=10.070\text{ or }0.570.$

1. Choose the physically valid value of $x$ and find the Friday running speed.

From part (a)(iv), $x=0.570$ or $x=10.070$.

Since the speed for the last $2.4$ km was $x-2$, $x=0.570$ would give a negative speed, so it is rejected.

Hence
$x=10.070$
Friday speed $=10.070-2=8.070$ km/h.

2. Calculate the time for 4 km and convert it to minutes and seconds.

$\text{Time}=\frac{4}{8.070}\text{ h}$
$=\frac{4}{8.070}\times 60\text{ min}$
$=29.739\ldots\text{ min}$

$0.739\ldots\times 60=44.3\ldots$ seconds.

So the time is approximately $29$ minutes $44$ seconds, to the nearest second.