To find the distance from the top of a right square pyramid to each vertex of the base, we need to use the Pythagorean theorem. Here's a detailed step-by-step solution:
1. Identify given values:
- The altitude (height) of the pyramid: [tex]\( 10 \)[/tex] units
- Each side of the base: [tex]\( 6 \)[/tex] units
2. Calculate the half diagonal of the base:
Since each side of the square base is [tex]\( 6 \)[/tex] units, half of the side of the base (which is needed for the calculation) is:
[tex]\[
\text{Half base side} = \frac{6}{2} = 3 \text{ units}
\][/tex]
3. Use the Pythagorean theorem to find the distance from the top to a base vertex:
In a right square pyramid, the distance from the top to a vertex of the base forms a right triangle with the altitude and half of the diagonal of the base. The Pythagorean theorem is used here to find this slant distance.
[tex]\[
\text{Distance} = \sqrt{(\text{Altitude})^2 + (\text{Half base side})^2}
\][/tex]
Substituting the values:
[tex]\[
\text{Distance} = \sqrt{10^2 + 3^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{100 + 9}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{109}
\][/tex]
[tex]\[
\text{Distance} \approx 10.44030650891055
\][/tex]
4. Rounding to the nearest tenth:
The approximate distance to the nearest tenth is:
[tex]\[
\text{Distance} \approx 10.4 \text{ units}
\][/tex]
So, the distance from the top of the pyramid to each vertex of the base, rounded to the nearest tenth, is:
[tex]\[
x = 10.4 \text{ units}
\][/tex]
