Question

Alfonso runs 10 km at an average speed of x km/h.

The next day he runs 12 km at an average speed of (x - 1) km/h.
The time taken for the 10km run is 30 minutes less than the time taken for the 12 km run.
(a) (i)
Write down an equation in x and show that it simplifies to x? – 5x – 20 = 0.​

Answer 1

The solution involves establishing time equations for Alfonso's runs, converting minutes to hours for consistency, and solving for the average speed. Equations representing distance over speed for each day's run are set up, revealing a quadratic equation when we express the time difference of 30 minutes.

Step-by-step explanation:

To solve part (a) of the student's question, we need to establish two time equations based on the information given about Alfonso's runs. The first equation is for the 10 km run at an average speed of x km/h, and the second is for the 12 km run at (x - 1) km/h. We know from the problem that the time taken to run 10 km is 30 minutes less than the time taken to run 12 km.

First, we must remember that average speed is defined as the total distance traveled divided by the time taken. Therefore, the time for each run can be found by dividing the distance by the speed. We convert 30 minutes to hours to use consistent units, so 30 minutes is 0.5 hours.

Time for 10 km run = Distance/Speed = 10/x hours

Time for 12 km run = Distance/Speed = 12/(x - 1) hours

We are given that 10/x is equal to 12/(x - 1) - 0.5. Multiplying throughout by x(x - 1) to clear the fractions, we get 10(x - 1) = 12x - 0.5x(x - 1). Expanding the expressions and simplifying leads to the quadratic equation x² - 5x - 20 = 0.

Answer 2

Step-by-step explanation:

Note that t = d/r where t is time, d is distance, and r is rate/speed.

We can come up with two equations with the information given and the equation:

t_1 hr = (10 km)/(x km/hr)

t_2 hr = (12 km)/(x - 1 km/hr)

where t_1 is the time taken to run the 10km the first day and t_2 is the time taken to run the 12km the second day.

We know that 30 minutes is 1/2 of an hour and that t_1 is 30 minutes less than t_2 (as stated in the question). Therefore, we can write:

t_1 = t_2 - 1/2

Substituting the values we derived:

(10 km)/(x km/hr) = (12 km)/(x - 1 km/hr) -1/2

Then we can evaluate by multiplying by 2x(x-1) on both sides:

20(x-1) = 24x - (x)(x-1)

20x - 20 = 24x - x^2 + x

x^2 -5x -20 = 0

And we are done.

I hope this helps! :)