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. \(x^3=\frac{64}{125}\)