To solve this problem, we need to find the angle between the line $QT$ and the plane $PQRS$ of the rectangular-based pyramid $PQRST$.

Step-by-Step Solution:

1. Understand the Geometry of the Pyramid:

   • $PQRS$ is the base of the pyramid, and it is a rectangle.
   • The height of the pyramid $T$ is 24 cm above the center of the rectangle $PQRS$.
   • The dimensions of the rectangle are $PQ = 22$ cm and $PS = 25$ cm.

2. Find the Position of $Q$:

   • $Q$ is the midpoint of $PR$. To find $Q$, we first need the coordinates of $P$ and $R$.
   • Assume $P$ is at the origin $(0, 0)$, $Q$ will be at $\left(\frac{22}{2}, 0\right) = (11, 0)$.

3. Coordinates of Points:

   • $P(0, 0, 0)$
   • $Q(11, 0, 0)$
   • $R(22, 0, 0)$
   • $S(0, 25, 0)$
   • The point $T$, being directly above the center of the base rectangle, has coordinates $T(11, 12.5, 24)$.

4. Find the Line $QT$:

   • $QT$ connects the point $Q(11, 0, 0)$ to $T(11, 12.5, 24)$.
   • The direction vector of $QT$ is $T - Q = (11 - 11, 12.5 - 0, 24 - 0) = (0, 12.5, 24)$.

5. Normal to Plane $PQRS$:

   • The plane $PQRS$ is horizontal, so its normal vector is along the $z$-axis, which is $(0, 0, 1)$.

6. Find the Cosine of the Angle:

   • The angle $\theta$ between line $QT$ and the plane $PQRS$ can be found using the dot product: $\cos \theta = \frac{\text{Vector } QT \cdot \text{Normal to } PQRS}{\|QT\| \cdot \|Normal\|}$
   • The dot product $(0, 12.5, 24) \cdot (0, 0, 1) = 24$.
   • Magnitude of $QT$, $\|QT\| = \sqrt{0^2 + 12.5^2 + 24^2} = \sqrt{0 + 156.25 + 576} = \sqrt{732.25}$.
   • Magnitude of the normal vector $\|Normal\| = 1$.

   $\cos \theta = \frac{24}{\sqrt{732.25}}$

7. Calculate the Angle: $\cos \theta = \frac{24}{\sqrt{732.25}}$

   Let's calculate the angle $\theta$ in degrees:

$\theta = \cos^{-1}\left(\frac{24}{\sqrt{732.25}}\right)$

Let's compute this.The angle between the line $QT$ and the plane $PQRS$ is approximately 27.5 degrees when rounded to one decimal place.

Would you like more details on this calculation or have any questions?

Here are some related questions you might find interesting:

1. How do you calculate the dot product between two vectors?
2. What is the significance of the normal vector to a plane?
3. How can you find the height of a pyramid given the dimensions of its base and its slant height?
4. What are some other methods to find the angle between a line and a plane?
5. How can you determine if a line is perpendicular or parallel to a plane?

Tip: Always double-check your geometric interpretations, especially when working with 3D objects, to ensure the correct coordinates and vectors are used.