Lesson 10

Edge Lengths, Volumes, and Cube Roots

Let’s explore the relationship between volume and edge lengths of cubes.

Problem 1

  1. What is the volume of a cube with a side length of
    1. 4 centimeters?
    2. \sqrt[3]{11} feet?
    3. s units?
  2. What is the side length of a cube with a volume of
    1. 1,000 cubic centimeters?
    2. 23 cubic inches?
    3. v cubic units?

Problem 2

Write an equivalent expression that doesn’t use a cube root symbol.

  1. \sqrt[3]{1}
  2. \sqrt[3]{216}
  3. \sqrt[3]{8000}
  4. \sqrt[3]{\frac{1}{64}}
  5. \sqrt[3]{\frac{27}{125}}
  6. \sqrt[3]{0.027}
  7. \sqrt[3]{0.000125}

Problem 3

Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.

  1. t^3=216

  2. a^2=15

  3. m^3=8

  4. c^3=343

  5. f^3=181

Problem 4

For each cube root, find the two whole numbers that it lies between.

  1. \sqrt[3]{11}
  2. \sqrt[3]{80}
  3. \sqrt[3]{120}
  4. \sqrt[3]{250}

Problem 5

Order the following values from least to greatest:

\displaystyle \sqrt[3]{530},\;\sqrt{48},\;\pi,\;\sqrt{121},\;\sqrt[3]{27},\;\frac{19}{2}

 

Problem 6

The equation x^2=25 has two solutions. This is because both 5 \boldcdot 5 = 25, and also \text-5 \boldcdot \text-5 = 25. So, 5 is a solution, and also -5 is a solution. But! The equation x^3=125 only has one solution, which is 5. This is because 5 \boldcdot 5 \boldcdot 5 = 125, and there are no other numbers you can cube to make 125. (Think about why -5 is not a solution!)

Find all the solutions to each equation.

  1. x^3=8
  2. \sqrt[3]x=3
  3. x^2=49
  4. x^3=\frac{64}{125}