MHB How do I derive and solve the equation for the athlete's running speed?

AI Thread Summary
The discussion revolves around deriving and solving the equation for an athlete's running speed based on a scenario where he reduces his speed and experiences a time increase. The key equation derived is \(\frac{80}{v} + \frac{8}{3} = \frac{160}{2v-5}\), which relates the time taken at normal speed and reduced speed. Participants discuss how to manipulate the equation to eliminate variables and simplify the calculations. The conversation emphasizes understanding the relationships between speed, time, and distance to arrive at the correct solution for \(v\). Ultimately, the focus is on guiding through the algebraic steps necessary to solve for the athlete's speed.
Simonio
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I'm having problems getting going on the following question, any help appreciated:

As part of his training an athlete usually runs 80 km at a steady speed of \(v\) km h. One day he decided to reduce his speed by 2.5 km h and his run takes him an extra 2h 40 mins.

Derive the equation \(\frac{80}{v}+ \frac{8}{3} = \frac{160}{2v-5}\) and solve it to find the value of \(v\).
 
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Simonio said:
I'm having problems getting going on the following question, any help appreciated:

As part of his training an athlete usually runs 80 km at a steady speed of \(v\) km h. One day he decided to reduce his speed by 2.5 km h and his run takes him an extra 2h 40 mins.

Derive the equation \(\frac{80}{v}+ \frac{8}{3} = \frac{160}{2v-5}\) and solve it to find the value of \(v\).

where is the problem deriving or solving
 
Simonio said:
I'm having problems getting going on the following question, any help appreciated:

As part of his training an athlete usually runs 80 km at a steady speed of \(v\) km h. One day he decided to reduce his speed by 2.5 km h and his run takes him an extra 2h 40 mins.

Derive the equation \(\frac{80}{v}+ \frac{8}{3} = \frac{160}{2v-5}\) and solve it to find the value of \(v\).

Hi Simonio!

Suppose we define $t$ to be the time it takes the athlete to run 80 km at a speed of $v$ km/h.
Can you set up the equations that correspond to the run times?
 
I like Serena said:
Hi Simonio!

Suppose we define $t$ to be the time it takes the athlete to run 80 km at a speed of $v$ km/h.
Can you set up the equations that correspond to the run times?

Well I think I can say that \(t\)= \(\frac{80}{v}\) and with the slower speed \(t\) = \(\frac{80}{v-2.5}\) not sure of next step
 
Simonio said:
Well I think I can say that \(t\)= \(\frac{80}{v}\)

Good! :)

and with the slower speed \(t\) = \(\frac{80}{v-2.5}\)

With the slower speed it takes 2h40 longer, which is $2\frac 2 3$ hours.
So it should be:
$$t + 2\frac 2 3 = \frac{80}{v-2.5}$$

not sure of next step

Can you combine the 2 equations and eliminate $t$?
 
I like Serena said:
Good! :)
With the slower speed it takes 2h40 longer, which is $2\frac 2 3$ hours.
So it should be:
$$t + 2\frac 2 3 = \frac{80}{v-2.5}$$
Can you combine the 2 equations and eliminate $t$?

OK-you've given me a good hint there!

so: \(\frac{80}{v} + \frac{8}{3} = \frac{160}{2v-5}\)
 
Simonio said:
OK-you've given me a good hint there!

so: \(\frac{80}{v} + \frac{8}{3} = \frac{160}{2v-5}\)

Not sure about the next bit: do I multiply throughout by \(2v-5\)?

That would make: \((2v-5) \frac{80}{v} + (2v-5) \frac{8}{3} = 160\)

Think I'm on the wrong track
 
Simonio said:
OK-you've given me a good hint there!

so: \(\frac{80}{v} + \frac{8}{3} = \frac{160}{2v-5}\)

Good!

Simonio said:
Not sure about the next bit: do I multiply throughout by \(2v-5\)?

That would make: \((2v-5) \frac{80}{v} + (2v-5) \frac{8}{3} = 160\)

Think I'm on the wrong track

You're right on track.
Continue by multiplying throughout by $v$.
 
I like Serena said:
Good!
You're right on track.
Continue by multiplying throughout by $v$.
Do you mean multiply by \(2v-5\)?
 
  • #10
Simonio said:
Do you mean multiply by \(2v-5\)?

You already did that.
But you're still left with $v$ in a denominator.
 
  • #11
I like Serena said:
You already did that.
But you're still left with $v$ in a denominator.

Then I get: \(v(2v-5)\frac{80}{v}\ +v(2v-5)\frac{8}{3} = 160v\)

= \(80(2v^2-5) + (2v^2-5)\frac{8}{3}\ = 160v\)

= \(160v^2 - 400v + \frac{16v^2-40v}{3}\ = 160v\)

Not sure whether I've lost the plot here!
 
  • #12
Simonio said:
Then I get: \(v(2v-5)\frac{80}{v}\ +v(2v-5)\frac{8}{3} = 160v\)

= \(80(2v^2-5) + (2v^2-5)\frac{8}{3}\ = 160v\)

= \(160v^2 - 400v + \frac{16v^2-40v}{3}\ = 160v\)

Not sure whether I've lost the plot here!

When you multiply $v(2v-5)$, you should get $(2v^2-5v)$ instead of $(2v^2-5)$.
That is because generally $a(b+c)=ab+ac$.
You can check this with for instance $4(2+3)$.

Furthermore, you can write:
$$v(2v-5)\frac{80}{v} = (2v-5)\frac{80}{v}v = (2v-5)80$$
That is because if you first divide by $v$ and then multiply by $v$ the effects cancel.
 
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