Problem 60 - Calculus Early Transcendentals

Question

Show that the function \(T(x)=60 D(60+x)^{-1}\) gives the time in minutes required to drive \(D\) miles at \(60+x\) miles per hour.

Step-by-Step Solution

Answer

Question: Show that the function \(T(x) = 60D(60+x)^{-1}\) represents the time in minutes required to travel a distance \(D\) miles at a speed of \(60+x\) mph. Answer: We have derived an equation \(d = 60D\) using the function \(T(x) = \frac{60D}{60+x}\). This equation shows that when we use the given function for time, we obtain the original distance traveled as \(D\) miles. Therefore, the function represents the time required to travel \(D\) miles at a speed of \(60+x\) mph.

Step 1: Write down the given function

The given function is \(T(x) = 60D(60+x)^{-1}\), where \(D\) is the distance in miles and \(60+x\) is the speed in miles per hour.

Step 2: Rewrite the function using exponents

We can rewrite \(T(x) = 60D(60+x)^{-1}\) as \(T(x) = \frac{60D}{60+x}\).

Step 3: Calculate the distance traveled

We know that \(t = \frac{d}{s}\). Therefore, we can calculate the distance traveled by multiplying both sides by the speed (\(60+x\)): \(d = t(60+x)\).

Step 4: Substitute the given function for time

Now, we can substitute the given function \(T(x) = \frac{60D}{60+x}\) for time \(t\) in the equation \(d = t(60+x)\): \(d = \frac{60D}{60+x}(60+x)\).

Step 5: Simplify the equation

Simplify the equation by canceling out terms in the numerator and denominator: \(d = \frac{60D(60+x)}{60+x}\). Since \(60+x\) appears both in the numerator and the denominator, we can cancel them out: \(d = 60D\).

Step 6: Conclusion

We've shown that the function \(T(x) = 60D(60+x)^{-1}\) gives the time in minutes required to drive \(D\) miles at \(60+x\) miles per hour. This is because when we use this function in the time formula, we obtain the distance traveled as \(D\) miles, which is the desired outcome of the problem.