Certainly! Let's solve the formula \( V = \frac{1}{5} s^2 h \) for \( s \).
1. Start with the given formula:
[tex]\[
V = \frac{1}{5} s^2 h
\][/tex]
2. Isolate \( s^2 \) on one side of the equation:
To eliminate the fraction, multiply both sides by 5:
[tex]\[
5V = s^2 h
\][/tex]
3. Solve for \( s^2 \):
Divide both sides of the equation by \( h \) to solve for \( s^2 \):
[tex]\[
s^2 = \frac{5V}{h}
\][/tex]
4. Take the square root of both sides to solve for \( s \):
[tex]\[
s = \sqrt{\frac{5V}{h}}
\][/tex]
Now that we have the formula for \( s \), let's substitute hypothetical values for \( V \) and \( h \) to find \( s \). Assume the volume \( V \) of each piece is 100 cubic units and the height \( h \) is 10 units.
5. Substitute \( V = 100 \) and \( h = 10 \) into the equation:
[tex]\[
s = \sqrt{\frac{5 \times 100}{10}}
\][/tex]
6. Simplify the expression inside the square root:
[tex]\[
s = \sqrt{\frac{500}{10}}
\][/tex]
7. Further simplify the division:
[tex]\[
s = \sqrt{50}
\][/tex]
Thus, the value of \( s \) is approximately:
[tex]\[
s = \sqrt{50} \approx 7.0710678118654755
\][/tex]
Hence, the value of [tex]\( s \)[/tex] when [tex]\( V = 100 \)[/tex] and [tex]\( h = 10 \)[/tex] is approximately 7.0710678118654755 units.
