Question

Naomi bikes the 40 miles to Hillsboro averaging a certain speed. The return trip is made at a speed 6 miles per hour slower. The total time for the round trip is 14 hours. Find Naomi's average speed on each part of the trip.

Answer 1

Let x be the initial speed.
x-6 be the return trip speed
y be the amount of initial time
14 - y be the the return trip time
Since Naomi traveled the same distance because she only had a return trip you can formulate d=d. Such that d = St. Where d is the distance, s is the seed and t is the time.
Can you do the next steps now? Or still needs help?

Answer 2

Could you do the next step please?

Answer 3

*i mean speed

Answer 4

Wait I think something is wrong here! hahahah!

Answer 5

speed there = x miles/hr
speed back = y miles per hour
note: the return speed is 6 miles per hour slower than the speed there
so y = x - 6

Answer 6

total time = 14hrs
total distance = 80 miles
average speed would be 80 miles / 14 hrs = about 5.7 miles per hours

but actual speed (x and y) ...

Answer 7

are 80 miles in 14 hrs
40 miles / x (miles per hour) there = time to get there
40 miles / y (miles per hour) back = time to get home
as y = x - 6
40 miles / (x - 6) = time to get home
time to get there + time to get home = 14 hrs
solve for x

Answer 8

so what are your 2 possible answers for x? @justine97 ?

Answer 9

I'm writing the problem right now, just give me a sec

Answer 10

no worries, sorry

Answer 11

So I got 40/x + 40/x-6 = 14

Answer 12

But I'm having trouble solving for x

Answer 13

\[\frac{ 40 }{ x } + \frac{ 40 }{x - 6 } = 14\]
cross multiply to get common denominator
\[\frac{ 40(x-6) }{ x(x - 6) } + \frac{ 40x }{ x(x-6) } = 14\]
common base, so simply add the numerators together (top line)
\[\frac{ 40(x-6) + 40x }{ x(x-6) } = 14\]
add like terms, expand brackets and simplify
\[\frac{ 40x - 240 + 40x }{ x^2 -6x } = 14\]
\[\frac{ 80x - 240 }{ x^2 - 6x } = 14\]
times both sides by (x^2 - 6x) to remove denominator

Answer 14

\[80x - 240 = 14(x^2 -6x)\]
\[80x - 240 = 14x^2 - 84x\]
\[0 = 14x^2 - 164x + 240\]
solve using quadratic formula

Answer 15

ur familiar with this one... yeah?

Answer 16

Okay, got it! :)

Answer 17

2(7x-12)(x-10) = 0

Answer 18

in this case: a = 14
b = -164
c = 240
so solve using the formula and u should get something close to:

x = 10
or
x = 1.7

now 1.7 is a crazy slow speed to be travelling as to get 40 miles at 1.7 miles per hour would take 23 ish hours, which is heaps more that the 14 hours total to travel 80 miles

... but at 10 miles per hour, it would take 4 hours to do the first leg... which is sensible

Answer 19

So we throw out 1.7 because the number doesn't make sense?

Answer 20

and given the return trip is 6 miles slower
10 - 6 = 4
so y = 4 miles per hour
to travel 40 miles would take... 10 hours
so 4 hrs there, 10 hours return: 14 hours total... sweet

Answer 21

we throw out the 1.7 as it dosent fit our equation, we always do a logic test on the answers as there's 2 possible answers to every sqr root (+ve and -ve)

Answer 22

ie 4^2 = 16
but (-4)^2 also = 16

Answer 23

WOW, this is amazing! Thank you so so so much! :)

Answer 24

nah all good dude, happy to help hey
it's all makin sense now tho?

Answer 25

Yes! This made a lot of sense. Again, you were super helpful and patient. I appreciate the help!

Answer 26

anytime hey, slaters ;D