Hannah and Jenna are both travelling to a volleyball tournament in Grande Prairie, and leave at the same time. Hannah’s parents drive her from Edmonton to Grande Prairie, a distance of 460 km. Jenna’s team takes a bus from Dawson Creek, BC to Grande Prairie, a distance of 130 km. Hannah’s parents’ vehicle travels 10 km/h faster than Jenna’s, and Jenna arrives at the tournament 3 hours earlier than Hannah. Determine how fast Hannah’s parents are driving
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time = distance / rate
460 / x = [130 / (x - 10)] + 3
460 x - 4600 = 130 x + 3 x^2 - 30 x
3 x^2 - 360 x + 4600 = 0
solve for x ... use the quadratic formula
the larger solution is the correct one
To solve this problem, we can start by setting up equations based on the given information. Let's assume Jenna's speed is "x" km/h, so Hannah's speed would be "x + 10" km/h.
We know that distance = speed × time. Therefore, Jenna's time can be calculated as:
130 km = x km/h × (Hannah's time - 3 hours)
Similarly, Hannah's time can be calculated as:
460 km = (x + 10) km/h × Hannah's time
Now, we have two equations with two unknowns, which we can solve to find the value of "x" (Jenna's speed) and then calculate Hannah's parents' speed.
Let's solve the equations step by step:
From the first equation, we can rearrange it to solve for Hannah's time:
130 km = x km/h × (Hannah's time - 3 hours)
130 km = x × (Hannah's time - 3)
130/x = Hannah's time - 3
Hannah's time = 130/x + 3
Now, substitute the value of Hannah's time in the second equation:
460 km = (x + 10) km/h × (130/x + 3)
460 km = (x + 10)(130/x + 3)
Simplify the equation by multiplying both sides by the common denominator "x":
460x = (x + 10)(130 + 3x)
Expand and simplify:
460x = 130(x + 10) + 3x(x + 10)
460x = 130x + 1300 + 3x^2 + 30x
Rearrange the equation in standard form:
0 = 3x^2 + 90x + 1300
Divide both sides by 3 to simplify the equation:
0 = x^2 + 30x + 433.33
At this point, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, since the equation doesn't factor easily, we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the formula, we can find the value of "x" (Jenna's speed) by substituting the values of a, b, and c from the quadratic equation:
x = (-30 ± √(30^2 - 4(1)(433.33))) / (2(1))
x = (-30 ± √(900 - 1733.32)) / 2
x = (-30 ± √(-833.32)) / 2
Since the value inside the square root (√) is negative, it means the equation doesn't have real solutions. Therefore, there is no real value for Jenna's speed.
Based on the information given, it seems there might be an error or inconsistency in the problem statement. Please double-check the information provided or clarify any additional details to find a valid solution.