A certain cube has volume n cubic inches and surface area n square inches.
What is the value of n?
Let's start with the formula for the volume of a cube: the volume of a cube with side length s is given by s^3. Since we know that the volume of the cube is n cubic inches, we can write the equation as s^3 = n.
The formula for the surface area of a cube is given by 6s^2. Since we also know that the surface area of the cube is n square inches, we can write the equation as 6s^2 = n.
We can solve either equation for s and substitute it into the other equation. Let's solve the first equation for s:
s^3 = n
Taking the cube root of both sides, we get:
s = ∛n
Now we substitute this value of s into the second equation:
6s^2 = n
6(∛n)^2 = n
6∛n * ∛n = n
6∛n^2 = n
Taking the cube root of both sides, we get:
6∛n = ∛n
Divide both sides by ∛n:
6 = 1
Since 6 does not equal 1, this equation does not hold true for any values of n. Therefore, there is no value of n that satisfies the condition that the volume and surface area of the cube are equal.