t(v)=253/v+70 +413/v+75

t(2)=

t(v)=253/v+70 +413/v+75

t(2)=

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Westford, MA

After reading Robert's answer, I would guess that the correct equation is:

t(v) = 253/(v+70) + 413/(v+75)

Then you drove 253 miles where the speed limit was 70 mph and 413 miles where the speed limit was 75.

Then t(2) = 253/72 + 413/77 = 253/72 + 59/11.

t(2) = (216 + 37)/72 + (55 + 4)/11

t(2) = 3 + 37/72 + 5 + 4/11

t(2) = 8 + 37/72 + 4/11

t(2) ≈ 8 + 0.5 + 0.4 ≈ 8.9 hours

How close is our estimate?

t(2) = 8 + 37/72 + 4/11

t(2) ≈ 8 + 0.51388888888889 + 0.36363636363636 ≈ 8.87752525252525 hours

Darn close!

And the estimate did not use a calculator (in case you forgot it, it's broken, the batteries are dead, you lent it to a friend, the dog chewed it up, it's a new whiz-bang graphing calculator your parent's bought you for your birthday and you don't know how to use it because you didn't read the user's manual and do the practice problems, etc.)

Boxford, MA

Hello Mondrea,

I will first assume that you've written the equation properly. I'm wondering if the expression "253/v + 70" is really "253/(v+70)," for example. The difference in the two is that the first gives the time to drive 235 miles at velocity v, and 70 is added time, perhaps for stops. The second expression gives the time required for travelling 253 miles at a speed of v+70.

Assuming the expression is written correctly, lets assume that v represents the speed limit. t(v) would then represent the time required to drive the distance as specified in the equation going the speed limit, v. The equation itself represents 4 different components to the driving time. There is time required to drive the distance 253, which is the result of 253/v, the time to drive 413 at v (=413/v), and a total added time of 145 (=70+75) that is independent of speed (stopping along the way for meals, gas, or speeding tickets).

The time to drive at the speed limit plus 2 (=v+2) is given by:

t(v+2)= (253/(v+2)) + (413/(v+2)) + 145

I hope this helps.

Bob

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