Certainly! Let's solve for the slant height [tex]\( \ell \)[/tex] in the given formula for the surface area [tex]\( S \)[/tex] of a right pyramid:
The formula given for the surface area [tex]\( S \)[/tex] is:
[tex]\[ S = \frac{1}{2} P \ell + B, \][/tex]
where:
- [tex]\( S \)[/tex] is the surface area,
- [tex]\( P \)[/tex] is the perimeter of the base,
- [tex]\( \ell \)[/tex] is the slant height,
- [tex]\( B \)[/tex] is the area of the base.
We need to solve this formula for [tex]\( \ell \)[/tex].
Step-by-Step Solution:
1. Isolate the term with [tex]\( \ell \)[/tex]:
Start by isolating the term containing [tex]\( \ell \)[/tex] on one side of the equation.
[tex]\[ S - B = \frac{1}{2} P \ell. \][/tex]
2. Eliminate the fraction:
To eliminate the fraction, multiply both sides of the equation by 2.
[tex]\[ 2(S - B) = P \ell. \][/tex]
3. Solve for [tex]\( \ell \)[/tex]:
Finally, solve for [tex]\( \ell \)[/tex] by dividing both sides of the equation by [tex]\( P \)[/tex].
[tex]\[ \ell = \frac{2(S - B)}{P}. \][/tex]
So, the slant height [tex]\( \ell \)[/tex] in terms of the surface area [tex]\( S \)[/tex], the perimeter of the base [tex]\( P \)[/tex], and the area of the base [tex]\( B \)[/tex] is:
[tex]\[ \ell = \frac{2(S - B)}{P}. \][/tex]
This formula allows you to find the slant height [tex]\( \ell \)[/tex] if you know the values of the surface area [tex]\( S \)[/tex], the perimeter of the base [tex]\( P \)[/tex], and the area of the base [tex]\( B \)[/tex].
