If the surface area of a cell that is shaped like a cube increases 100 times, it volume increases about a. 5 times b. 10 timesc. 100 times d. 1000 times***
To find the correct answer, let's first understand the relationship between surface area and volume in a cube.
A cube has six square faces, each with side length "s". The formula for the surface area of a cube is: SA = 6s^2.
The volume of a cube is given by: V = s^3.
Now, let's analyze the situation. We have a cube whose surface area increased 100 times. Let's call the original surface area SA1 and the new surface area SA2.
SA2 = 100 * SA1
We are asked about the change in volume, so we need to find the relationship between the volumes, V1 and V2.
To do that, let's relate the surface area and the side length of the cube.
For the original cube:
SA1 = 6s1^2
For the new cube:
SA2 = 6s2^2
Since SA2 = 100 * SA1, we can write the relationship as:
6s2^2 = 100 * 6s1^2
Now, let's relate the volumes by substituting the side length of the new cube into the volume formula:
V1 = s1^3
V2 = s2^3
We need to find the relationship between V1 and V2. To do that, we need to find the relationship between s1 and s2 from the previously obtained equation:
6s2^2 = 100 * 6s1^2
Dividing both sides by 6, we get:
s2^2 = 100 * s1^2
Taking square root on both sides:
s2 = 10 * s1
Now, we can substitute this value into the volume equation:
V2 = s2^3 = (10 * s1)^3 = 1000 * (s1^3) = 1000 * V1
So, the volume of the cube, V, increases about 1000 times (d. 1000 times) when the surface area increases 100 times.
Hence, the correct answer is d. 1000 times.