Answer :
To find the slant height of a right square pyramid with an altitude of 10 and each side of the base being 6, follow these steps:
1. Calculate the half-length of the base diagonal:
- The base of the pyramid is a square with each side of length 6.
- For a square, the length of the diagonal can be found using the Pythagorean theorem.
- The diagonal [tex]\(d\)[/tex] of the square is given by [tex]\(d = \sqrt{6^2 + 6^2}\)[/tex].
- Simplifying, [tex]\(d = \sqrt{36 + 36} = \sqrt{72} \approx 8.485\)[/tex].
- Half of this diagonal is [tex]\(\frac{8.485}{2} \approx 4.242\)[/tex].
2. Find the slant height of the pyramid:
- The slant height forms the hypotenuse of a right triangle where the altitude of the pyramid (10) and half the base diagonal (4.242) are the legs.
- Use the Pythagorean theorem to find the slant height [tex]\(s\)[/tex]:
[tex]\[ s = \sqrt{10^2 + 4.242^2} \][/tex]
- Simplifying, [tex]\( s = \sqrt{100 + 18} \approx \sqrt{118} \approx 10.9 \)[/tex].
Thus, the slant height of the pyramid, to the nearest tenth, is:
[tex]\[ \boxed{10.9} \][/tex]
1. Calculate the half-length of the base diagonal:
- The base of the pyramid is a square with each side of length 6.
- For a square, the length of the diagonal can be found using the Pythagorean theorem.
- The diagonal [tex]\(d\)[/tex] of the square is given by [tex]\(d = \sqrt{6^2 + 6^2}\)[/tex].
- Simplifying, [tex]\(d = \sqrt{36 + 36} = \sqrt{72} \approx 8.485\)[/tex].
- Half of this diagonal is [tex]\(\frac{8.485}{2} \approx 4.242\)[/tex].
2. Find the slant height of the pyramid:
- The slant height forms the hypotenuse of a right triangle where the altitude of the pyramid (10) and half the base diagonal (4.242) are the legs.
- Use the Pythagorean theorem to find the slant height [tex]\(s\)[/tex]:
[tex]\[ s = \sqrt{10^2 + 4.242^2} \][/tex]
- Simplifying, [tex]\( s = \sqrt{100 + 18} \approx \sqrt{118} \approx 10.9 \)[/tex].
Thus, the slant height of the pyramid, to the nearest tenth, is:
[tex]\[ \boxed{10.9} \][/tex]
