Question

a toy has various shaped objects that a child can push through matching holes. the volume of a cube - shaped block is 343 cubic centimeters. a. which edge length can be found? explain. b. will the block fit in the square hole? explain. a. since 343 is a perfect cube, the length of each edge of the block can be found. each edge is 7 centimeters long. b. the area of one face of the block is square centimeters. since this is than the area of the hole, the block fit in the hole.

Explanation:

Step1: Recall cube - volume formula

The volume formula for a cube is $V = s^{3}$, where $s$ is the length of an edge. Given $V = 343$ cubic centimeters, we need to solve for $s$.

Step2: Solve for edge - length

We set up the equation $s^{3}=343$. Taking the cube - root of both sides, $\sqrt[3]{s^{3}}=\sqrt[3]{343}$. Since $7\times7\times7 = 343$, $s = 7$ centimeters.

Step3: Compare area of cube - face and hole area

The area of one face of the cube is $A_{cube}=s^{2}$. Substituting $s = 7$ centimeters, we get $A_{cube}=7^{2}=49$ square centimeters. The area of the square hole is $A_{hole}=14$ square centimeters. Since $49>14$, the block will not fit in the hole.

Answer:

a. The edge - length of the block is 7 centimeters because the volume of a cube $V=s^{3}$ and when $V = 343$, $s=\sqrt[3]{343}=7$.
b. The block will not fit in the square hole. The area of one face of the block is $7\times7 = 49$ square centimeters, which is greater than the area of the hole (14 square centimeters).