Question

Johnny was able to drive an average of 27 miles per hour faster in his car after the traffic

cleared. He drove 25 miles in traffic before it cleared and then drove another 104 miles. If the
total trip took 3 hours, then what was his average speed in traffic?

Answer 1

Answer: 40 mph

Step-by-step explanation: let s = speed in traffic

(s+17) = out of traffic

:

Write a time equation; time = dist/speed

traffic time + normal speed time = 4 hrs

46%2Fs + 80%2F%28%28s%2B17%29%29 = 4

multiply by s(s+17) to cancel the denominators

46(s+17) + 80s = 4s(s+17)

46s + 782 + 80s = 4s^2 + 68s

126s + 782 = 4s^2 + 68s

Arrange as a quadratic equation on the right

2s^2 - 29s - 391 = 0

Use the quadratic formula a=2; b=-29; c=-391

I got a positive solution of

s = 23 mph in traffic

:

:

:

Check this by finding the actual times. Normal speed: 23 + 17 = 40 mph

46/23 = 2 hrs in traffic

80/40 = 2 hrs at normal speed

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

total time 4 hrs

0 = 4s^2 + 68s - 126s - 782

4s^2 - 58s - 782 = 0

simplify, divide by 2

so the answer is 40mph

Answer 2

Johnny's average speed in traffic was 25 miles per hour.

The Breakdown

Assuming that Johnny's average speed in traffic was "x" miles per hour. Since he drove 25 miles in traffic, the time spent in traffic can be calculated as 25/x hours.

After the traffic cleared, Johnny was able to drive 27 miles per hour faster than his speed in traffic. So, his speed after the traffic cleared would be (x + 27) miles per hour. He drove another 104 miles at this speed, which took him 104/(x + 27) hours.

The total time for the trip is given as 3 hours. Therefore, the sum of the time spent in traffic and the time spent after the traffic cleared should equal 3 hours:

25/x + 104/(x + 27) = 3

To solve this equation, we can multiply through by x(x + 27) to eliminate the denominators:

25(x + 27) + 104x = 3x(x + 27)

Simplifying the equation:

25x + 675 + 104x = 3x² + 81x

Combining like terms:

0 = 3x² + 81x - 129x - 675

0 = 3x² - 48x - 675

Now, we can solve this quadratic equation using quadratic formula.

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 3, b = -48, and c = -675. Plugging these values into the quadratic formula:

x = (-(-48) ± √((-48)² - 4 × 3 × -675)) / (2 × 3)

Simplifying further:

x = (48 ± √(2304 + 8100)) / 6

x = (48 ± √10404) / 6

x = (48 ± 102) / 6

Now, we have two possible solutions for x:

x1 = (48 + 102) / 6 = 150 / 6 = 25

x2 = (48 - 102) / 6 = -54 / 6 = -9

Since speed cannot be negative, we discard the negative solution. Therefore, Johnny's average speed in traffic was 25 miles per hour.