QuestionPecro left home at noon and cycled 72 kim to his Jamily coltage. His sister, Alexandra, left home on her bike at 1 PM and arrived at the coltage 12 minutes auter peoro. If she cycles, on average $3 \mathrm{~km} t \mathrm{~h}$ Vaster than Pecro, how long dit it take Peoro to make the trip and what was his aerage speed?

Studdy Solution

STEP 1

1. Pecro left home at noon.
2. Pecro cycled 72 km to his family cottage.
3. Alexandra left home at 1 PM.
4. Alexandra arrived at the cottage 12 minutes after Pecro.
5. Alexandra cycles, on average, 3 km/h faster than Pecro.
6. We need to find how long it took Pecro to make the trip and his average speed.

STEP 2

1. Define variables for Pecro's time and speed.
2. Write equations based on the problem description.
3. Solve the equations to find Pecro's time and speed.

STEP 3

Define variables for Pecro's time and speed.
Let $t$ be the time it took Pecro to make the trip in hours. Let $v$ be Pecro's average speed in km/h.

STEP 4

Write equations based on the problem description.
Since Pecro cycled 72 km, we can write the equation for his trip as: $v \times t = 72$
Alexandra left an hour later and arrived 12 minutes after Pecro, so her travel time is $t - 1 + \frac{12}{60}$ hours. Her speed is $v + 3$ km/h.
The equation for Alexandra's trip is: $(v + 3) \times \left(t - 1 + \frac{12}{60}\right) = 72$

STEP 5

Solve the equations to find Pecro's time and speed.
First, simplify Alexandra's travel time: $t - 1 + \frac{12}{60} = t - 1 + 0.2 = t - 0.8$
Substitute into Alexandra's equation: $(v + 3) \times (t - 0.8) = 72$
Now, solve the system of equations:
1. $v \times t = 72$
2. $(v + 3) \times (t - 0.8) = 72$
From the first equation, express $v$ in terms of $t$ : $v = \frac{72}{t}$
Substitute $v$ into the second equation: $\left(\frac{72}{t} + 3\right) \times (t - 0.8) = 72$
Distribute and solve for $t$ : $\left(\frac{72}{t} + 3\right) \times t - 0.8 \left(\frac{72}{t} + 3\right) = 72$
$72 + 3t - \frac{57.6}{t} - 2.4 = 72$
$3t - \frac{57.6}{t} = 2.4$
Multiply through by $t$ to clear the fraction: $3t^2 - 57.6 = 2.4t$
Rearrange into a standard quadratic form: $3t^2 - 2.4t - 57.6 = 0$
Solve this quadratic equation for $t$ .

STEP 6

Use the quadratic formula to solve for $t$ : $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Where $a = 3$ , $b = -2.4$ , and $c = -57.6$ .
$t = \frac{-(-2.4) \pm \sqrt{(-2.4)^2 - 4 \times 3 \times (-57.6)}}{2 \times 3}$
$t = \frac{2.4 \pm \sqrt{5.76 + 691.2}}{6}$
$t = \frac{2.4 \pm \sqrt{696.96}}{6}$
Calculate the discriminant: $\sqrt{696.96} \approx 26.4$
So, $t = \frac{2.4 \pm 26.4}{6}$
Choose the positive root: $t = \frac{2.4 + 26.4}{6}$
$t = \frac{28.8}{6}$
$t = 4.8$
Now, find Pecro's average speed $v$ : $v = \frac{72}{t} = \frac{72}{4.8} = 15$
Pecro's time is $4.8$ hours and his average speed is $15$ km/h.

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