Question

a ball is in the shape of a sphere. a string is wrapped tightly around the ball at its widest point and goes around the ball exactly once. the length of the string is 37.7 centimeters. what is the volume, in cubic centimeters, of the ball? round your answer to the nearest hundredth.

Explanation:

Step1: Encontrar el radio

Sabemos que la longitud de la cuerda es la circunferencia $C = 2\pi r$. Dado $C=37.7$ cm, entonces $r=\frac{C}{2\pi}=\frac{37.7}{2\pi}$.

Step2: Calcular el volumen de la esfera

La fórmula del volumen de una esfera es $V = \frac{4}{3}\pi r^{3}$. Sustituimos $r=\frac{37.7}{2\pi}$ en la fórmula:
\[

$$\begin{align*} V&=\frac{4}{3}\pi(\frac{37.7}{2\pi})^{3}\\ &=\frac{4}{3}\pi\times\frac{37.7^{3}}{8\pi^{3}}\\ &=\frac{4\times37.7^{3}}{24\pi^{2}} \end{align*}$$

\]
Evaluando numéricamente con $\pi\approx3.14$:
\[

$$\begin{align*} V&=\frac{4\times37.7^{3}}{24\times3.14^{2}}\\ &=\frac{4\times53589.733}{24\times9.8596}\\ &=\frac{214358.932}{236.6304}\\ &\approx278.98 \end{align*}$$

\]

Answer:

278.98