URGENT! MY MATHLETES COMP IS TOMORROW! This was a question in my mathletes packet, and similar ?s keep coming up, I would love a solution and explanation for it, thanks!

Question

Pilot Matt flew the first 340 miles of his 1190 mile flight at a constant speed. He checked his fuel consumption and realized that to be safe he should decrease his speed by 1.5 miles per minute. Matt landed his plane with not much fuel to spare, and he noticed that if he had flown his entire flight at the lower speed the trip would have only taken 6 minutes longer. What was the slower speed, in miles per hour? Express your answer to the nearest whole number. My mathletes competition is in the morning of 1/25/19

Answer 1

You have known for a long time that distance = rate * time (the rate is the speed).

If you solve this for time, you find that time = distance/rate

Let t = the time it took to fly there at two different speeds, and it will be in hours.

This way our rate (speed) can be in miles per hour.

Let r = faster rate, and r-90 = slower rate (Converted 1.5 mi/min * 60 min/hour = 90 miles/hour)

So t = 340/r + 850/(r-90)

Going only the slower rate (r-90), the time required would have been t (our original time) + 0.1 (6 min = 0.1 hours) and that would be the whole 1190/(r-90)

So we now have 2 equations that can both be simplified to = t (our original time)

t = 340/r + 850/(r-90)

t + 0.1 = 1190/(r-90), solving this for "t", our second equation becomes

t = 1190/(r-90) - 0.1

Since we have two expressions that both = t, they equal each other so we can set up this equation:

340/r + 850/(r-90) = 1190/(r-90) - 0.1

Getting a common denominator [ r(r-90) ] on the left, we have

(1190r -30600)

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

r ( r-90)

Getting a common denominator on the right we get

1280 - 0.1 r

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

r - 90

Need the right to have the same denominator as the right, so multiply top and bottom by "r"

Now the right side has

1280 r - 0.1 r²

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

r ( r - 90 )

Now that the left and right sides both have the same denominator, we can just deal with the numerators.

(or put another way, multiply both sides by "r ( r - 90 )" and all that will be left are the numerators.)

1190r - 30600 = 1280r - 0.1 r²

To solve a quadratic, put everything on one side so other side = 0

0.1 r² - 90 r - 30600 = 0

I multiplied everything by 10 so I did not have to deal with the 0.1.

Now I have r² - 900 r - 30600 = 0

This solves to:

r = 1163 miles per hour or r = -263 miles per hour. Generally we think of speed as being positive, so

the faster speed was 1163 miles per hour. Subtract 90 miles per hour to find that the slower speed was

1073 miles per hour.