Word problem troubles.

Question

When a tortoise travels across a highway, his speed is 3ft/hr faster than normal due to the incentive of making it across safely. If he can cross a 36 foot wide road in one hour less than the time he can travel the same distance at his normal speed, then what is his normal speed?

Answer 1

Hi Kaleb, this is a standard D(istance) = R(ate) * T(ime) problem. Set up a grid with the known information.

 

Object                 Distance        Rate        Time

-----------------------------------------------------

Normal tortoise         36             x              t

 

Turbo tortoise           36           x + 3       t - 1

 

The common measure is the distance, in both cases the tortoise crosses 36 feet.

 

Normal tortoise: D = x * t

 

Solve for t gives: D / x

 

Turbo tortoise: D = (x + 3) * (t - 1)

 

Solve for t gives: D / (x + 3) + 1

 

Equating both equations to t gives::

 

D / x = D / (x + 3) + 1

 

Removing the fractions, substituting for D = 36:

 

36 * (x + 3) =  36 * x + x * (x + 3)

 

Combining terms:

 

36x + 108 = 36x + x² + 3x

 

The 36x terms cancel out leaving

 

x² + 3x - 108 = 0

 

We can factor this trinomial or use the binomial theorem to solve.

 

Factoring gives: (x - 9) * (x + 12) = 0

 

Applying the zero product rule gives:

 

x - 9 = 0, or x = 9

 

and 

 

x + 12 = 0, x = -12. We can discard the -12 answer since the tortoise has positive speed.

 

So, the tortoise's normal rate of speed is 9 feet per hour. It would take him 4 hours to cross the 36 foot road.

 

On the other hand, turbo tortoise would travel at a rate of speed of 9 + 3, or 12 feet per hour and be able to cross the road in 3 hours.

 

Questions?