Find the lateral area of a cube given that its volume is 1,728 cubic centimeters.
In this question, we are given the volume of a cube. And we need to use this information to find the lateral area of the cube.
To answer this question, we can begin by recalling that the lateral area of a prism or cylinder is the surface area of the shape where we exclude the area of the base and top. We can use this definition to find a formula for the lateral area of a cube. Let’s say that we have a cube with sides of length 𝐿. Then, the lateral area of the cube is the surface area of the cube where we ignore the base and top of the cube. This leaves us the surface area of the four faces of the cube, which is equal to four 𝐿 squared. This means that we can find the lateral area of the cube from its side length.
We can also recall that the volume of a cube with sides of length 𝐿 is 𝐿 cubed. We can substitute the given volume of the cube into the formula to obtain 1,728 is equal to 𝐿 cubed.
To find the value of 𝐿 using this equation, we need to take cube roots of both sides of the equation. We can do this by recalling that for any real number 𝑎, the cube root of 𝑎 cubed is equal to 𝑎. This means that the cube root of 𝐿 cubed is equal to 𝐿. And we can use this to evaluate the cube root of 1,728 if we can write the radicand as a perfect cube.
To check if 1,728 is a perfect cube, we can look for perfect cube factors of 1,728. Since it is even, we can start by checking if eight is a factor. We see that 1,728 is equal to eight times 216. We can apply this process again to see that 216 is equal to eight times 27. We can then note that 27 is equal to three cubed. This allows us to rewrite the equation as 𝐿 is equal to the cube root of two cubed times two cubed times three cubed.
Then, either by expanding the exponents or by using the laws of exponents, we can rewrite this equation as 𝐿 is equal to the cube root of two times two times three cubed. Therefore, 1,728 is equal to 12 cubed, and we can see that 𝐿 must be equal to 12 centimeters. We can substitute this value into our formula for the lateral area. This gives us a lateral area of four times 12 squared, which we can evaluate is equal to 576 square centimeters.