You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are pointing the hose at the same angle and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you blocked?
To solve this problem, we can use the concept of conservation of flow rate. When you put your finger over the hose opening and increase the distance the water reaches, the flow rate of the water remains constant.
Let's assume the original flow rate of the water without blocking any part of the hose opening is Q.
Now, when you block a fraction of the hose opening, let's say f (where 0 < f < 1), the effective area through which the water can flow decreases by a factor of (1 - f).
Therefore, the effective flow rate, Q', can be calculated as:
Q' = Q * (1 - f)
Given that the distance the water reaches increases by a factor of 4, we can imply that the effective flow rate also increases by a factor of 4. So, we have:
Q' = 4 * Q
Now, we can substitute Q' with 4Q in the equation above:
4Q = Q * (1 - f)
Simplifying this equation, we get:
4 = 1 - f
Rearranging the equation, we find:
f = 1 - 4
f = -3
But, in this context, the fraction cannot be negative, so we discard that solution.
Therefore, there is no fraction of the hose opening blocked.