MHB How Can I Correctly Derive the Quadratic Equation from a Train Speed Problem?

AI Thread Summary
The discussion focuses on deriving the quadratic equation \(v^2 - 40v - 9600 = 0\) from a train speed problem. The problem states that a train traveling 240 km at speed \(v\) takes an hour longer when its speed is reduced by 40 km/h due to adverse weather. Participants clarify the kinematic relationships and how to manipulate equations to eliminate time, leading to the correct formulation of the quadratic equation. They also discuss the factorization of the equation and identify the positive solution for \(v\) as 120 km/h. The conversation emphasizes the importance of correctly applying algebraic principles to arrive at the solution.
Simonio
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I'm having difficulty deriving a quadratic from the info in this question:

A train usually covers a journey of 240 km at a steady speed of \(v \text{ kmh}^{-1}\). One day, due to adverse weather conditions, it reduces its speed by 40 \text{kmh}^{-1}\) and the journey takes an hour longer.

Derive the equation \(v^2 - 40v - 9600 =0\), and solve it to find the value of \(v\).

All I could come up with was: \(\frac{240}{v^2} = \frac{240}{v^2 - 40} -1\)

But i don't think this is right! Any help to get me on the right track appreciated! Thanks.
 
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Well, let's begin with the kinematic relationship between distance $d$, speed $v$, and time $t$, when speed is constant:

$$d=vt$$

Let's let the distance be measured in km, speed in kph and time in hrs.

We know in both cases that $d=240$. Under usual conditions, we are given:

$$240=vt$$

And when the weather was bad, we are given:

$$240=(v-40)(t+1)$$

Now, we want to get rid of $t$ so that we have an equation in $v$ only. Solving the first equation for $t$, we obtain:

$$t=\frac{240}{v}$$

Now, substitute for $t$ into the second equation:

$$240=(v-40)\left(\frac{240}{v}+1\right)$$

Can you continue?
 
MarkFL said:
Well, let's begin with the kinematic relationship between distance $d$, speed $v$, and time $t$, when speed is constant:

$$d=vt$$

Let's let the distance be measured in km, speed in kph and time in hrs.

We know in both cases that $d=240$. Under usual conditions, we are given:

$$240=vt$$

And when the weather was bad, we are given:

$$240=(v-40)(t+1)$$

Now, we want to get rid of $t$ so that we have an equation in $v$ only. Solving the first equation for $t$, we obtain:

$$t=\frac{240}{v}$$

Now, substitute for $t$ into the second equation:

$$240=(v-40)\left(\frac{240}{v}+1\right)$$

Can you continue?

Do we have to multiply throughout by \(v\)? Would that look like this:

\(v(v-40)v\) \((\frac{240}{v}+1)\)

Hmm...think I've gone wrong somewhere...
 
Simonio said:
Do we have to multiply throughout by \(v\)? Would that look like this:

\(v(v-40)v\) \((\frac{240}{v}+1)\)

Hmm...think I've gone wrong somewhere...

Yes, multiplying through by $v$ is a good next step, and you would have:

$$240v=(v-40)(240+v)$$

Do you see what I did there?
 
MarkFL said:
Yes, multiplying through by $v$ is a good next step, and you would have:

$$240v=(v-40)(240+v)$$

Do you see what I did there?
Thanks. Could you show the working..so I know how you got there? Thanks.
 
We start with:

$240 = (v - 40)\left(\dfrac{240}{v} + 1\right)$

If we multiply one side by $v$, we must do the same to the other side.

The left-hand side is easy: we wind up with $240v$.

The right-hand side is a bit more involved:

$(v - 40)\left(\dfrac{240}{v} + 1\right)v$

$ = (v - 40)\left(\dfrac{240}{v}\cdot v + v\right)$ (using the DISTRIBUTIVE rule)

$ = (v - 40)(240 + v)$ ("cancelling the $v$'s")

$ = v(240 + v) - 40(240 + v)$ (distributive rule again, expanding out this time)

$ = 240 v + v^2 - 9600 - 40v$ ( another expansion using the distributive rule)

$ = v^2 + 200v - 9600$ (collecting "like terms", and ordering by the power of $v$).

So that's what we wind up with the right-hand side, and now we equate the two sides:

$240v = v^2 + 200v - 9600$

Do you see what to do next?
 
So:

\(v^2 + 200v - 9600 =240v\)

Then: \(v^2 - 40v -9600 = 0\)

next, find factors of \(ac\) whose sum is \(c\) = 120, -80.

\(v2 + 120v -80v - 9600\)

Then: \(v(v-80) + 120(v-80)\)

So either \(v+120 = 0\) or \(v-80 = 0\)

Then \(v = -120\) or \(v = 80\)

hmm... the answer in my book gives v as +120 only-so I've gone wrong somewhere(Doh)
 
Since we have $v^2-40v-9600=0$:

Simonio said:
\(v2 + 120v -80v - 9600\)

The above should be $v^2-120v+80v-9600=0$...:) and try it again, Simonio, as I'm fairly certain you will end up getting $v=120$ as the positive solution to the problem.
 
Simonio said:
So:

\(v^2 + 200v - 9600 =240v\)

Then: \(v^2 - 40v -9600 = 0\)

next, find factors of \(ac\) whose sum is \(c\) = 120, -80.

\(v2 + 120v -80v - 9600\)

Then: \(v(v-80) + 120(v-80)\)

So either \(v+120 = 0\) or \(v-80 = 0\)

Then \(v = -120\) or \(v = 80\)

hmm... the answer in my book gives v as +120 only-so I've gone wrong somewhere(Doh)

What you have written is tantamount to:

$v^2 - 40v - 9600 = (v - 80)(v + 120)$.

However, in truth:

$(v - 80)(v + 120) = v^2 + 40v - 9600$

as -80 and 120 sum to +40, not -40, as required.

In general, if $ab < 0$ (and $a,b$ are integers), then exactly ONE of $a$ or $b$ must be negative. If $a + b < 0$ as well, the negative one has to be the one with the largest absolute value (or magnitude).

If we write:

$v^2 - 40v - 9600 = (v + a)(v + b) = v^2 + (a+b)v + ab$

we have $ab = -9600 < 0$ and $a+b = -40 < 0$.
 
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