To solve this problem, we need to calculate the time required to mill a slot between the points $(10,10)$ and $(70,80)$ at a feed rate of $60 \, \text{mm/min}$.

Steps to Solution

1. Calculate the Distance Between Points $(10,10)$ and $(70,80)$: Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates:

   = \sqrt{60^2 + 70^2} = \sqrt{3600 + 4900} = \sqrt{8500} \approx 92.20 \, \text{mm}$$

2. Determine the Milling Time: Given the feed rate is $60 \, \text{mm/min}$, we can find the time required using: $\text{Time} = \frac{\text{Distance}}{\text{Feed Rate}}$ Substitute the values: $\text{Time} = \frac{92.20}{60} \approx 1.54 \, \text{minutes}$

3. Convert to Seconds: Since we want the time in seconds: $1.54 \, \text{minutes} \times 60 = 92.40 \, \text{seconds}$

Answer

The milling time required to complete the slot is approximately 92.40 seconds.

Would you like further details on any step or have any additional questions?

Here are some related questions for further practice:

1. How would the milling time change if the feed rate was increased to $100 \, \text{mm/min}$?
2. What if the distance between the points was doubled—how would that affect the milling time?
3. How would you calculate the distance if the points were given in 3D coordinates?
4. If the width of the slot changes, does it affect the milling time under the same feed rate?
5. How would you convert the time to hours if the feed rate was very slow?

Tip: Remember, the feed rate is essential for time calculations in milling, as it directly impacts the speed at which the cutter moves across the material.