Exercise 47 Page 352

To find the average speed we can use the relationship between rate — the average speed — distance traveled, and the time needed for the trip. distance=rate* time We can use this equation to express the time needed for the trip in terms of the distance. To do this, let's introduce the variable d for the length of one leg of the journey.

Time Needed to get There

To find the time needed for the forward journey t_(forward), let's use the given rate, 30 miles per hour.

Distance=Rate* Time

SubstituteExpressions

Substitute expressions

d=30t_(forward)

DivEqn

.LHS /30.=.RHS /30.

d/30=t_(forward)

RearrangeEqn

Rearrange equation

t_(forward)=d/30

Time Needed to get Back

Using the given rate, we can also find the time needed for Mitsu to get back. Note that the distance back is the same as the distance forward.

Distance=Rate* Time

SubstituteExpressions

Substitute expressions

d=65t_(back)

DivEqn

.LHS /65.=.RHS /65.

d/65=t_(back)

RearrangeEqn

Rearrange equation

t_(back)=d/65

Total Time Needed

We can add these times to find the time needed for the round trip journey.

t_(total)=t_(forward)+t_(back)

SubstituteII

t_(forward)= d/30, t_(back)= d/65

t_(total)= d/30+ d/65

▼

Simplify right-hand side

MoveNumRight

a/b=1/b* a

t_(total)=1/30d+1/65d

FactorOut

Factor out d

t_(total)=(1/30+1/65)d

AddFrac

Add fractions

t_(total)=19/390d

Finding the Average Speed

Notice that the distance of the full journey is 2d, d on the way forward and d on the way back. Now that we know the time needed for the full journey, we can find the rate — the average speed.

Distance=Rate* Time

SubstituteExpressions

Substitute expressions

2d=rt_(total)

Substitute

t_(total)= 19/390d

2d=r( 19/390d)

▼

Solve for r

DivEqn

.LHS /d.=.RHS /d.

2=r*19/390