SOLUTION: A car leaves a town 20 minutes after a truck leaves. The speed of the car is 10 miles per hour greater than the truck. After traveling 100 miles the car overtakes the truck. Find t

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Question 732945: A car leaves a town 20 minutes after a truck leaves. The speed of the car is 10 miles per hour greater than the truck. After traveling 100 miles the car overtakes the truck. Find the average speed of both vehicles.
I can not figure out this one. Help!
Found 2 solutions by lynnlo, josmiceli:
Answer by lynnlo(4176) About Me  (Show Source):
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Average speed is ALWAYS ( change in total distance ) / ( change in time )
Let the speed of the truck = +s mi/hr
+s+%2B+10+ = the speed of the car
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Note that 20 min = +20%2F60+=+1%2F3+ hr
The truck gets a head start of +d%5B1%5D+=+s%2A%281%2F3%29+
Start a stopwatch when the car leaves. Let +t+ =
the time in hrs on the stopwatch when the car catches the truck.
-----------------------------
Equation for the car:
(1) +100+=+%28+s+%2B+10+%29%2At+
Equation for the truck:
(2) +100+-+%281%2F3%29%2As+=+s%2At+
------------------------
(1) +100+=+s%2At+%2B+10t+
(1) +s%2At+=+100+-+10t+
(1) +s+=+100%2Ft+-+10+
and
(2) +300+-+s+=+3s%2At+
(2) +3s%2At+%2B+s+=+300+
(2) +s%2A%28+3t+%2B+1+%29+=+300+
Substitute (1) into (2)
(2) +%28+100%2Ft+-+10+%29%2A%28+3t+%2B+1+%29+=+300+
(2) +10%2A%28+10%2Ft+-+1+%29%2A%28+3t+%2B+1+%29+=+300+
(2) +%28+10%2Ft+-+1+%29%2A%28+3t+%2B+1+%29+=+30+
(2) +30+-+3t+%2B+10%2Ft+-+1+=+30+
(2) +-3t+-+1+=+-10%2Ft+
Multiply both sides by +t+
(2) +-3t%5E2+-+t+=+-10+
(2) +3t%5E2+%2B+t+-+10+=+0+
Using the quadratic formula:
+t+=+%28+-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+3+
+b+=+1+
+c+=+-10+
+t+=+%28+-1+%2B-+sqrt%28+1%5E2+-+4%2A3%2A%28-10%29+%29%29+%2F+%282%2A3%29+
+t+=+%28+-1+%2B-+sqrt%28+1+%2B+120+%29%29+%2F+6+
+t+=+%28+-1+%2B-+sqrt%28+121+%29%29+%2F+6+
+t+=+%28+-1+%2B+11+%29+%2F+6+
+t+=+5%2F3+ hrs
------------
The car goes 100 mi in +5%2F3+ hr
Ave speed = +100+%2F+%285%2F3%29++=+300%2F5+
+300%2F5+=+60+
The car averages 60 mi/hr
-------------
The truck goes +100+ mi in +1%2F3+%2B+5%2F3+=+2+ hrs
( Note that the 100 mi has the head start included )
Equation for truck:
+100+=+s%2A2+
+s+=+50+
The truck averages 50 mi/hr
-------------------------
check:
Equation for the car:
(1) +100+=+%28+s+%2B+10+%29%2At+
(1) +100+=+%28+50+%2B+10%29%2At+
(1) +t+=+100%2F60+
(1) +t+=+5%2F3+
Equation for the truck:
(2) +100+-+%281%2F3%29%2As+=+s%2At+
(2) +100+-+%281%2F3%29%2A50+=+50%2At+
(2) +300+-+50+=+150t+
(2) +150t+=+250+
(2) +t+=+250%2F150+
(2) +t+=+5%2F3+
OK