- A 200 hp car can go from 0 to 100 mph in \(t_1\) seconds.
- A 250 hp car can go from 0 to 100 mph in \(t_2\) seconds.
Relationship between horsepower and time:
Since the time to reach 100 mph varies inversely with horsepower, we can write the relationship as:
$$t = k \cdot \frac{1}{hp}$$
where \(t\) is the time in seconds, \(hp\) is the horsepower of the car, and \(k\) is a constant.
Finding the constant \(k\):
We are given that a 200 hp car can go from 0 to 100 mph in \(t_1\) seconds. Substituting these values into the equation above, we get:
$$t_1 = k \cdot \frac{1}{200}$$
Solving for \(k\), we get:
$$k = t_1 \cdot 200$$
Calculating \(t_2\):
Now we can use the value of \(k\) to find \(t_2\), the time it takes for a 250 hp car to go from 0 to 100 mph. Substituting \(hp = 250\) into the equation \(t = k \cdot \frac{1}{hp}\), we get:
$$t_2 = k \cdot \frac{1}{250}$$
Substituting the value of \(k\) that we found earlier, we get:
$$t_2 = (t_1 \cdot 200) \cdot \frac{1}{250}$$
Simplifying, we get:
$$t_2 = \frac{4}{5} \cdot t_1$$
Conclusion:
Therefore, if a 200 hp car can go from 0 to 100 mph in \(t_1\) seconds, then a 250 hp car will take \(\frac{4}{5} \cdot t_1\) seconds to go from 0 to 100 mph.
