The speed of the tortoise is given by the sinusoidal function T(t) = 1/2 - 1/2 cos(2πt) whereas the speed of the hare is H(t) = √t. If the race is over in 1 hour, use integration t... The speed of the tortoise is given by the sinusoidal function T(t) = 1/2 - 1/2 cos(2πt) whereas the speed of the hare is H(t) = √t. If the race is over in 1 hour, use integration to find who won the race and by how much? Read more

Steps to Solve

1. Define the functions for distance
   To find the distance traveled by each participant, we need to integrate their speed functions over the time interval from $t = 0$ to $t = 1$ hour.

2. Integrate the tortoise's speed function
   The tortoise's speed is given by ( T(t) = \frac{1}{2} - \frac{1}{2} \cos(2\pi t) ).
   We need to compute the integral: $$ D_T = \int_0^1 T(t) , dt = \int_0^1 \left( \frac{1}{2} - \frac{1}{2} \cos(2\pi t) \right) dt $$

3. Calculate the integral for the tortoise
   Calculating the integral, we first break it into two parts: $$ D_T = \int_0^1 \frac{1}{2} , dt - \int_0^1 \frac{1}{2} \cos(2\pi t) , dt $$
   The first integral is: $$ \int_0^1 \frac{1}{2} , dt = \frac{1}{2} $$ The second integral can be computed using the antiderivative of cosine: $$ \int \cos(2\pi t) dt = \frac{1}{2\pi} \sin(2\pi t) + C $$ Thus, $$ \int_0^1 \frac{1}{2} \cos(2\pi t) , dt = \left[ \frac{1}{2} \cdot \frac{1}{2\pi} \sin(2\pi t) \right]_0^1 = 0 $$

Combining these results, we have: $$ D_T = \frac{1}{2} - 0 = \frac{1}{2} $$

4. Integrate the hare's speed function
   The hare's speed is given by ( H(t) = \sqrt{t} ).
   We need to compute the integral: $$ D_H = \int_0^1 H(t) , dt = \int_0^1 \sqrt{t} , dt $$

5. Calculate the integral for the hare
   The integral for ( \sqrt{t} ) is: $$ \int_0^1 \sqrt{t} , dt = \left[ \frac{2}{3} t^{3/2} \right]_0^1 = \frac{2}{3} $$

6. Compare the distances
   Now we compare the distances:

• Distance traveled by the tortoise: ( D_T = \frac{1}{2} ) km
• Distance traveled by the hare: ( D_H = \frac{2}{3} ) km

Since ( \frac{2}{3} > \frac{1}{2} ), the hare won.

7. Calculate the margin
   To find the margin by which the hare won: $$ \text{Margin} = D_H - D_T = \frac{2}{3} - \frac{1}{2} $$
   Finding a common denominator (which is 6), we have: $$ \frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6} $$
   So, $$ \text{Margin} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} $$