Solve for $t$.
$\begin{array}{l}
t^3=125 \\
t=\square
\end{array}$
We are given the equation \(t^3 = 125\). To find the value of \(t\), we need to find the cube root of 125. Since 125 can be written as \(5 \times 5 \times 5\), or \(5^3\), the equation becomes \(t^3 = 5^3\). Therefore, the real solution for \(t\) is 5.
To confirm the solution, we can rewrite the equation as \(t^3 - 125 = 0\). Using the difference of cubes formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) with \(a=t\) and \(b=5\), we have:
\((t - 5)(t^2 + 5t + 25) = 0\).
Setting the first factor to zero gives \(t = 5\). The second factor, \(t^2 + 5t + 25 = 0\), has a discriminant \(D = 5^2 - 4(1)(25) = 25 - 100 = -75\). Since the discriminant is negative, there are no other real solutions for \(t\). Therefore, the only real solution is \(t = 5\).
This concept of finding the root of a power is commonly used in architecture and engineering when calculating dimensions. For example, if you know the volume of a cubic water tank is 125 cubic meters, you can find the side length of the tank by taking the cube root of the volume: \(s = \sqrt[3]{125} = 5\) meters. Understanding roots helps professionals determine physical dimensions from capacity or volume measurements.