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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:
   \[
   d = \sqrt{(70 - 10)^2 + (80 - 10)^2}
   = \sqrt{60^2 + 70^2}
   = \sqrt{3600 + 4900}
   = \sqrt{8500} \approx 92.20 \, \text{mm}
   \]

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?

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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.

A CNC horizontal milling machine is tasked with cutting a straight slot of 30 mm width and 5 mm depth using a cutter with a diameter of 25 mm. The slot must be milled between points (10,10) and (70,80) on the xy plane (dimensions in mm). The feed rate used for milling is 60 mm/min. The milling time required to complete the slot in _____ seconds. (Hint: Assume that (10,10) and (70,80) are the distance between slot)

Learn how to calculate the milling time for a CNC machine tasked with cutting a straight slot between points (10,10) and (70,80) at a feed rate of 60 mm/min. This problem involves geometry and trigonometry concepts, including the distance formula and time calculation, and is suitable for Grades 10-12.

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