To solve this problem, we need to calculate the angle between the line $TB$ and the base of the prism $ABCD$.

Key details from the problem:

• The base $ABCD$ is a rectangle with:
  • $AB = 10$ cm,
  • $BC = 7$ cm.
• $M$ is the midpoint of $AD$, so:
  • $AM = MD = \frac{10}{2} = 5$ cm.
• $T$ is directly above $M$ with $TM = 6$ cm.

To find the angle between $TB$ and the base $ABCD$, we can use vector geometry or trigonometry, focusing on the right triangle formed by:

• $TM$ (height),
• $MB$ (horizontal distance in the base),
• $TB$ (the slant height we need to calculate).

Steps:

1. Find the length of $MB$: Since $M$ is the midpoint of $AD$, we can apply the Pythagorean theorem to find $MB$ within the rectangle $ABCD$.

   • $MB^2 = AB^2 + AM^2$.
   • Substituting values: $MB^2 = 10^2 + 5^2 = 100 + 25 = 125$, so $MB = \sqrt{125} = 5\sqrt{5}$ cm.

2. Calculate the length of $TB$: Now apply the 3D Pythagorean theorem: $TB^2 = TM^2 + MB^2 = 6^2 + (5\sqrt{5})^2 = 36 + 125 = 161$ So, $TB = \sqrt{161} \approx 12.7$ cm.

3. Calculate the angle $\theta$ between $TB$ and the base $ABCD$: The angle can be calculated using the cosine rule for right triangles: $\cos(\theta) = \frac{MB}{TB} = \frac{5\sqrt{5}}{\sqrt{161}}.$ Now let's calculate the value of $\theta$. I will compute this.The angle between the line $TB$ and the base $ABCD$ is approximately $28.2^\circ$.

Would you like a detailed explanation or have any further questions? Here are some related questions that may help deepen your understanding:

1. How can we calculate angles in 3D geometry more efficiently?
2. Why did we use the Pythagorean theorem for distances in this problem?
3. What is the significance of vectors when calculating angles between lines and planes?
4. Can this method be applied to other shapes, like pyramids or cones?
5. How does understanding the midpoint of a line help in solving 3D problems?

Tip: When calculating angles in 3D, it's often useful to break the problem into simpler 2D triangles using right-angle triangles or coordinate geometry.