Average Speed and Average Rate of Change

Question

A train travels from city A to city B and then to city C. The distance from A to B is 97.5 miles and the distance from B to C is 70 miles. The average speed from A to B was 65 miles per hour, and the average speed from B to C was 35 mph. What was the average speed from A to C (that is for the entire trip) in miles per hour?

 

A train travels from city A to city B to city C. The distance from A to B is 28 miles. The distance from B to C is 45 miles. The train took 1 hour for the trip from A to B and then went from B to C at an average speed of 30 miles per hour. What was the average speed of the train for the entire trip (in miles per hour)?

 

Let f(x)=8x2.
Find a value A such that the average rate of change of f(x) from 1 to A equals 104

Answer 1

1) d=r*t

A to B is 97.5 miles at 65 mph

97.5=65*t

97.5/65=t

t=1.5 hours

B to C is 70 miles at 35 mph

70=35*t

70/35=t

t=2 hours

total distance=rate*total time

(97.5+70)=rate*(1.5+2)

167.5=rate*(3.5)

1675=rate*35

1675/35=rate

simplify the fraction

335/7=rate

47 6/7 mph is the average speed

 

2) d=r*t

A to B is 28 miles in 1 hour (rate=28 mph)

B to C is 45 miles at 30 mph

d=r*t

45=30*t

45/30=t

1.5=t

total distance=rate*total time

73=rate*2.5  (28+45=73 and 1+1.5=2.5)

730=rate*25

730/25=rate

29.2 mph=rate

rate=29.2 mph is the average speed of the train

 

3) f(x)=8x^2

[8(x+h)^2-8x^2]/h

[8(x^2+2xh+h^2)-8x^2]/h

(16xh+8h^2)/h

16x+8h

1 to A using x=1 and h=A-1

16(1)+8(A-1)=104

16+8A-8=104

8+8A=104

8A=104-8

8A=96

A=12

 

Answer 2

Here is another approach for solving problem 3:

 

Average rate of change of f from 1 to A = [f(A) - f(1)]/[A - 1]

 

                                                       = [8A² - 8]/[A - 1]

 

                                                       = 8(A² - 1)/(A - 1)

 

                                                       = 8(A+1)(A-1)/(A-1)

 

                                                       = 8(A + 1)

 

So, we are looking for a value of A such that 8(A+1) = 104

 

                                                                 A + 1 = 13

 

                                                                       A = 12