Lesson 12

Edge Lengths and Volumes

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 distance between each pair of points. If you get stuck, try plotting the points on graph paper.

  1. \(X=(5,0)\) and \(Y=(\text-4,0)\)

  2. \(K=(\text-21,\text-29)\) and \(L=(0,0)\)

(From Unit 8, Lesson 11.)

Problem 4

Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.

A rectangle with a point on the bottom side. Two line segments are drawn from the point to the top left vertex and from the point to the top right vertex of the rectangle creating 3 triangles.

 

(From Unit 8, Lesson 9.)

Problem 5

Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.

Equilateral triangle. Length of each side = 2 units. A height is drawn. 

  1. Find the exact height.
  2. Find the area of the equilateral triangle.
  3. (Challenge) Using \(x\) for the length of each side in an equilateral triangle, express its area in terms of \(x\).
(From Unit 8, Lesson 10.)