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Aright square pyramid has an altitude of 10 and each side of the base is 6. To the nearest tenth
or top of the pyramid, to each vertex of the base?



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]