IM 8.8.12 Lesson: Edge Lengths and Volumes
Let [math]a[/math],[math]b[/math], [math]c[/math], [math]d[/math], [math]e[/math], and [math]f[/math] be positive numbers.[br][br]Given these equations, arrange [math]a[/math],[math]b[/math], [math]c[/math], [math]d[/math], [math]e[/math], and [math]f[/math] from least to greatest. Explain your reasoning.[br][list][*][math]a^2=9[/math][/*][*][math]b^3=8[/math][/*][*][math]c^2=10[/math][/*][*][math]d^3=9[/math][/*][*][math]e^2=8[/math][/*][*][math]f^3=7[/math][/*][/list]
A cube has a volume of 8 cubic centimeters. A square has the same value for its area as the value for the surface area of the cube. How long is each side of the square?
For each card with a letter and value, find the two other cards that match. One shows the location on a number line where the value exists, and the other shows an equation that the value satisfies.
Explain your reasoning.[br][br]
IM 8.8.12 Practice: Edge Lengths and Volumes
What is the volume of a cube with a side length of 4 centimeters?
What is the volume of a cube with a side length of [math]\sqrt[3]{11}[/math] feet?
What is the volume of a cube with a side length of [math]s[/math] units?
What is the side length of a cube with a volume of 1,000 cubic centimeters?
What is the side length of a cube with a volume of [math]v[/math] cubic units?
Write an equivalent expression that doesn’t use a cube root symbol.
[math]\sqrt[3]{1}[/math][br]
[math]\sqrt[3]{216}[/math]
[math]\sqrt[3]{8000}[/math][br]
[math]\sqrt[3]{\frac{1}{64}}[/math]
[math]\sqrt[3]{\frac{27}{125}}[/math]
[math]\sqrt[3]{0.027}[/math]
[math]\sqrt[3]{0.000125}[/math]
Find the distance between each pair of points. If you get stuck, use the applet below.
[math]X=(5,0)[/math] and [math]Y=(-4,0)[/math]
[math]K=(-21,-29)[/math] and [math]L=(0,0)[/math]
Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.
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[/img]