# Suppose you travel a distance d (in miles) at an average speed of 20 mph and travel back at x mph. Explain why your average speed is 2d/ (d/20) + (d/x). Write the expression from...

Suppose you travel a distance d (in miles) at an average speed of 20 mph and travel back at x mph.

- Explain why your average speed is 2d/ (d/20) + (d/x).
- Write the expression from part as a simple rational expression in lowest terms.
- Let f(x) equal your answer from part b. What is lim x→∞ f(x)? What real-world interpretation does the limit have?

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### 1 Answer

1)

The distance travelled d+d = 2d

The time taken for forward jpurney = d/20

The time taken for the return journey = d/x.

So the total time taken for up and down = d/20+d/x.

Therefore the average speed = total distance/ total time taken = 2d/{d/20+d/x} mph..

2)

2d/{d/20+d/x) mph = 2d / d{(x+20)/20x } mph.

2d/{d/20+d/x)mph = 2*20x/(x+20) mph.

2d/{d/20+d/x) mph= 40x/(20+x) mph.

So 40x/(20+x) is the saverage in simplified rational form in the lowest terms.

3)

If f(x) = 40x/(20+x) , then to find the limit as x-->infinity.

Since both numerator and denomonator become infinite, f(x) becomes indeterminate as x goes infinite. So we divide both numerator and denominator by x and then take the limits.

f(x) = (40x/x){20/x+x/x} = 40/{20/x+1}

Therefore, Lt x--> infinity f(x) = 40/(0+1} = 40 mph.

Practical interpretation: The time taken to forward journey = d/20. The time taken for return journey is d/x is zero as the speed x is very very high. So the up and down journey distance 2d took only d/20 hours of time. So the average speed is 2d/ (d/20) = 40d/d = 40 mph.