Question

Can someone answer this entire page please thank you.

Answer 1

`18.\ Speed\ of\ current=3\ miles/hour\\\\19.\ 6\ hours\\\\20.\ Speed\ of\ Freight\ train=92\ mi/h,\ Speed\ of\ passenger\ train=112\ mi/h\\\\21.\ I\ can\ paint\ the\ dence\ 9\ hours\ and\ my\ sister\ can\ paint\ the\ fence\ in\ 18\ hours.\\\\22.\ Speed\ of\ James=(35)/(3)\ mi/h,\ Speed\ of\ Michael=(14)/(3)\ mi/h\\\\23.\ Speed\ of\ current=5\ mi/h\\\\24.\ Speed\ of\ dirt\ bike=35\ mi/h,\ Speed\ of\ motorcycle=43\ mi/h`

Explanation:

18.

`Let\ speed\ of\ river's\ current=x\ miles/hour\\\\Speed\ of\ boat=9\ miles/hour`

Downstream:

`Speed\ in\ the\ downstream=speed\ of\ boat+speed\ of\ current\\\\Speed\ in\ the\ downstream=x+9\\\\distance=24\ miles\\\\time=(distance)/(speed)\\\\Time(T_1)=(24)/(x+9)`

Upstream:

`Speed\ in\ the\ upstream=speed\ of\ boat-speed\ of\ current\\\\Speed\ in\ the\ upstream=9-x\\\\distance=24\ miles\\\\time=(distance)/(speed)\\\\Time(T_2)=(24)/(9-x)`

`Total\ time=T_1+T_2\\\\Total\ time=6\\\\T_1+T_2=6\\\\(24)/(x+9)+(24)/(9-x)=6\\\\24((x+9-x+9)/(81-x^2))=6\\\\24(18)=6(81-x^2)\\\\divide\ by\ 6\ both\ sides\\\\4(18)=81-x^2\\\\x^2-81+72=0\\\\x^2=9\\\\x=\pm 3\\\\speed\ can\ not\ be\ negative\\\\x=3\ miles/hour`

19.

Since they are working together but time taken by each driver to complete the route will be same so minimum time to cover the distance 6 hours.

20.

`Let\ speed\ of\ freight\ train=x\ mi/h\\\\Speed\ of\ passenger\ train=speed\ of\ freight\ train+20\\\\Speed\ of\ passenger\ train=x+20`

Passenger Train:

`Speed=x+20\\\\distance=392\ mi\\\\Time=(distance)/(speed)\\\\Time=(392)/(x+20)`

Freight Train:

`Speed=x\\\\distance=322\ mi\\\\Time=(distance)/(speed)\\\\Time=(322)/(x)`

`Time\ taken\ by\ both\ trains\ is\ same.\\\\(392)/(20+x)=(322)/(x)\\\\Cross\ multiplication\\\\392x=322(20+x)\\\\divide\ by\ 322\\\\(392x)/(322)=x+20\\\\(28)/(23)x=x+20\\\\(28)/(23)x-x=20\\\\(5x)/(23)=20\\\\x=(20* 23)/(5)\\\\x=4* 23\\\\x=92\ mi/h\\\\Speed\ of\ Freight\ train=92\ mi/h\\\\Speed\ of\ passenger\ train=92+20=112\ mi/h`

21.

`Let\ I\ can\ paint\ the\ fence\ in\ x\ hours\\\\My\ sister\ can\ paint\ the\ fence\ in\ 2x\ hours\ (as\ I'm\ two\ times\ faster\ than\ her)\\\\`

In 1 hour:

`Per\ hour\ painiting=(1)/(hours\ taken\ to\ paint\ whole\ fance)\\\\I\ can\ paint\ per\ hour=(1)/(x)\\\\My\ sister\ can\ paint\ per\ hour=(1)/(2x)\\\\Together\ we\ can\ paint\ per\ hour=(1)/(x)+(1)/(2x)\\\\it\ takes\ 6\ hours\ to\ both\ of\ us\ to\ paint\ the\ fence \\\\Per\ hour\ painting\ =(1)/(6)\\\\(1)/(x)+(1)/(2x)=(1)/(6)\\\\(3)/(2x)=(1)/(6)\\\\2x=18\\\\x=9\\\\I\ can\ paint\ the\ fence\ in\ 9\ hours`

`My\ sister\ can\ paint\ the\ fence\ in\ 9*2=18\ hours`

22.

`Let\ speed\ of\ Micheal\ = x\ mi/hr\\\\Speed\ of\ James\ =speed\ of\ James\ +7\\\\Speed of James =x+7\\\\Time\ of\ Michael =(distance)/(speed)=(8)/(x)\\\\Time\ of\ james\ =(distance)/(Speed)=(20)/(x+7)\\\\Both\ time\ are\ same\\\\(8)/(x)=(20)/(x+7)\\\\8x+56=20x\\\\20x-8x=56\\\\12x=56\\\\x=(56)/(12)\\\\x=(14)/(3)\ mi/h\\\\x+7=(14)/(3)+7=(35)/(3)\ mi/h\\\\Speed\ of\ Michael\ =(14)/(3)\ mi/h\\\\ Speed\ of\ James\ =(35)/(3)\ mi/h`

23.

`Let\ speed\ of\ current\ =x\ mi/hr\\\\ Upstream\ speed\ =speed\ of\ boot-speed\ of\ river\\\\Upstream\ speed\ =15-x\\\\Downstream\ speed=sped\ of\ boat\ +speed\ of\ river=15+x\\\\Upstream\ time=(distance)/(speed)=(10)/(15-x)\\\\Downstream=(distance)/(speed)=(10)/(15+x)\\\\Downstream\ =1.5\ hours\\\\(10)/(15-x)+(10)/(15+x)=1.5\\\\10\left((15+x+15-x)/(225-x^2)\right)=(3)/(2)\\\\(10*30)/(225-x^2)=(3)/(2)\\\\225-x^2=10*30*(2)/(3)`

`225-x^2=200\\\\x^2=225-200\\\\x^2=25\\\\x=\pm5\\\\Speed\ can\ not\ be\ negative\\\\x=5\\\\Hence\ speed\ of\ current\ =5\ mi/hr`

24.

`Let\ speed\ of\ dirt\ bike=x\ mi/h\\\\Speed\ of\ motorcycle\ =speed\ of\ dirt\ bike\ +8\\\\Speed\ of\ motorcycle\ =x+8\\\\Time\ taken\ on\ dirt\ bike=(distance)/(speed)=(52.5)/(x)\\\\Time\ taken\ on\ motorcycle=(distance)/(speed)=(64.5)/(x+8)\\\\Both\ the\ times\ are\ equal\\\\(52.5)/(x)=(64.5)/(x+8)\\\\Cross\ multiplication\\\\52.5(x+8)=64.5x\\\\(64.5)/(52.5)x=x+8\\\\(43)/(35)x=x+8\\\\(43)/(35)x-x=8\\\\(8x)/(35)=8\\\\x=35`

`Speed\ of\ dirt\ bike=35\ mi/hr\\\\Speed\ of\ motorcycle=35+8=43\ mi/hr`