Question

is the following statement true or false? i have a sphere whose volume is 300 cm². i also have a cube which has the same volume. the surface areas of the two objects are the same.

Explanation:

Step1: Analyze Sphere Volume and Surface Area

The volume of a sphere is given by \( V_{sphere}=\frac{4}{3}\pi r^{3} \), and the surface area is \( S_{sphere} = 4\pi r^{2}\). Given \( V_{sphere}=300\space cm^{3}\) (note: volume unit should be \( cm^{3}\), not \( cm^{2}\)), we first find the radius \( r\) of the sphere.
From \( \frac{4}{3}\pi r^{3}=300 \), we solve for \( r\):
\( r^{3}=\frac{300\times3}{4\pi}=\frac{225}{\pi}\)
\( r = \sqrt[3]{\frac{225}{\pi}}\approx\sqrt[3]{\frac{225}{3.14}}\approx\sqrt[3]{71.66}\approx4.15\space cm\)
Then the surface area of the sphere:
\( S_{sphere}=4\pi r^{2}\approx4\times3.14\times(4.15)^{2}\approx4\times3.14\times17.22\approx216.5\space cm^{2}\)

Step2: Analyze Cube Volume and Surface Area

The volume of a cube is \( V_{cube}=a^{3}\), where \( a\) is the side length. Given \( V_{cube} = 300\space cm^{3}\), we find \( a\):
\( a=\sqrt[3]{300}\approx6.69\space cm\)
The surface area of a cube is \( S_{cube}=6a^{2}\), so:
\( S_{cube}=6\times(6.69)^{2}\approx6\times44.76\approx268.6\space cm^{2}\)

Step3: Compare Surface Areas

We see that \( S_{sphere}\approx216.5\space cm^{2}\) and \( S_{cube}\approx268.6\space cm^{2}\), so they are not the same.

Answer:

False