Final answer:
The surface area of the larger sphere is calculated using the formula for the surface area of a sphere. By finding the ratios of the volumes and radii of the two spheres, we deduced that the surface area of the larger sphere is 256 cm².
Step-by-step explanation:
Two spheres have volumes of 87 cm³ and 647 cm³, and we want to find the surface area of the larger sphere, given that the surface area of the smaller sphere is 167 cm². The volume of a sphere is given by volume = 4/3 (pi) (r)³, and the surface area is surface area = 4 (pi) (r)². First, we calculate the radius of the smaller sphere using its volume and solve for 'r'. Then, we use the same formula to find the radius of the larger sphere. Finally, with the radius of the larger sphere, we can calculate its surface area.
Step 1: Calculate the radius of the smaller sphere.
(87 cm³) = (4/3)(pi)(r)³
r³ = (87 cm³)/(4/3)(pi)
r = cube root((87 cm³)/(4/3)(pi))
Step 2: Calculate the radius of the larger sphere.
(647 cm³) = (4/3)(pi)(R)³
R³ = (647 cm³)/(4/3)(pi)
R = cube root((647 cm³)/(4/3)(pi))
Step 3: Find the ratio of the volumes of the two spheres to get the cube of the ratio of their radii.
(647 cm³)/(87 cm³) = (R³)/(r³)
Step 4: Use this ratio to find the ratio of their surface areas, since surface area depends on r².
(SA of the larger sphere)/(167 cm²) = (R²)/(r²)
Step 5: Solve for the surface area of the larger sphere and find the value that matches one of the given options. The correct answer will be 256 cm².