Home / V_2 = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi \cdot 27r^3 = 27 \cdot \frac{4}{3} \pi r^3 = 27V_1

V_2 = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi \cdot 27r^3 = 27 \cdot \frac{4}{3} \pi r^3 = 27V_1

February 22, 2026

$V_2 = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi \cdot 27r^3 = 27 \cdot \frac{4}{3} \pi r^3 = 27V_1$

Let the original radius be $r$, so original volume is:
V_1 = \frac{4}{3} \pi r^3
After tripling the radius, new radius is $3r$, so new volume $V_2$ is:
Thus, the volume increases by a factor of $\boxed{27}$.
Question:** In a right triangle, one leg is 6 cm and the hypotenuse is 10 cm. Find the length of the shortest altitude.
Let the triangle have legs $a = 6$ cm, hypotenuse $c = 10$ cm. Use the Pythagorean theorem to find the other leg $b$: