Let's solve the problem step by step.
1. Identify the given values:
- Altitude of the pyramid (distance from the apex to the center of the base): 10 units
- Side length of the base square: 6 units
2. Calculate the half-diagonal of the base square:
- A square's diagonal can be calculated using the Pythagorean theorem. For a square with side length [tex]\(a\)[/tex]:
[tex]\[
\text{Diagonal} = a \sqrt{2}
\][/tex]
- Therefore, for a side length of 6 units, the diagonal is:
[tex]\[
\text{Diagonal} = 6 \sqrt{2}
\][/tex]
- We're interested in the half-diagonal, so:
[tex]\[
\text{Half-diagonal} = \frac{6 \sqrt{2}}{2} = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} \approx 4.2426 \text{ units}
\][/tex]
3. Calculate the slant height:
- The slant height is the distance from the apex to the midpoint of the base's side, but we use the half-diagonal obtained above.
- Using the Pythagorean theorem in the triangle formed by the apex, the center of the base, and the midpoint of one side of the base:
[tex]\[
\text{Slant height} = \sqrt{(\text{altitude})^2 + (\text{half-diagonal})^2}
\][/tex]
- Substitute the known values:
[tex]\[
\text{Slant height} = \sqrt{10^2 + (3\sqrt{2})^2} = \sqrt{100 + 18} \approx 10.8628 \text{ units}
\][/tex]
4. Round to the nearest tenth:
- The slant height rounded to the nearest tenth is:
[tex]\[
\text{Slant height} \approx 10.9 \text{ units}
\][/tex]
Therefore, the distance from the apex (top) of the pyramid to each vertex of the base, rounded to the nearest tenth, is 10.9 units.
