Final answer:
To find the new surface area of the dilated sphere, we can solve for the radius of the original sphere and use the formula for the surface area of a sphere. After finding the radius, we can calculate the new surface area by multiplying the original surface area by the square of the scale factor.
Step-by-step explanation:
To find the new surface area of the dilated sphere, we need to use the formula for the surface area of a sphere, which is A = 4πr², where r is the radius of the sphere.
Given that the original surface area is 48π cm², we can set up the equation 48π = 4πr² and solve for r. Dividing both sides of the equation by 4π, we get r² = 12. At this point, we can take the square root of both sides to find that r = √12, which simplifies to 2√3.
Now, to find the new surface area after dilation with a scale factor of k = 3/4, we need to multiply the original surface area by the square of the scale factor. Since the original surface area is 48π, the new surface area will be (48π)(3/4)² = 27π cm².