IM Geo.5.6 Lesson: Scaling Solids
Find the volume of each cube mentally.
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Use the applet to build cubes that result from dilating a unit cube by each scale factor shown in the table. Then, complete the table with the surface area and volume of each dilated cube.
Suppose a unit cube is dilated by some scale factor k.
Write an expression for the surface area of the dilated cube.[br]
Write an expression for the volume of the dilated cube.[br]
Compare and contrast the expression for surface area and the expression for volume.[br]
[size=150]Clare says, “We know that if we dilate a cube by a factor of [math]k[/math], the cube’s volume is multiplied by [math]k^3[/math]. It seems like that must apply to [i]all[/i] solids, but I’m not sure how to prove it.”[br][br]Elena says, “Earlier in the unit, we showed that we can cover any two-dimensional shape with rectangles, so the property that area changes by [math]k^2[/math] when we dilate a figure by [math]k[/math] applies to all shapes, not just rectangles. Can we do something similar here?”[/size][br][br]Use Elena’s line of reasoning to argue that for [i]any[/i] solid, if it’s dilated by a factor of [math]k[/math], the volume is multiplied by [math]k^3[/math].
Suppose a triangular prism has surface area 84 square centimeters and volume 36 cubic centimeters. The prism is dilated by scale factor [math]k=4[/math]. Calculate the surface area and volume of the dilated prism.
The image shows a figure called Sierpinski’s triangle.
[size=150]It’s formed by starting with an equilateral triangle, then repeatedly removing equilateral triangles created by joining the midpoints of the existing triangle’s sides. The first few stages are shown.[br][br][/size][table][tr][td]stage 1[/td][td]stage 2[/td][td]stage 3[/td][/tr][tr][td][img]data:image/png;base64,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[/img][/td][td][img]data:image/png;base64,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[/img][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br][br][size=150]If we continue this process of removal forever, we are left with some points that never get removed from the triangle. The remaining points are what we call Sierpinski’s triangle. At any given stage, the triangle at the top of the figure is a scaled copy of the triangle at the previous stage.[br][/size][br][table][tr][td]stage 4[/td][td]top triangle[/td][/tr][tr][td][img]data:image/png;base64,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[/img][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br][br][size=150]For the completed Sierpinski’s triangle, though, the top triangle is a scaled copy not of the previous stage, but of the full Sierpinski’s triangle.[/size][br][br]For the completed figure, what scale factor takes Sierpinski’s triangle to its scaled copy at the top?[br]
Based on the scale factor, what fraction of the original shaded region should be contained in the scaled copy at the top?[br]
The scaled copy at the top actually contains [math]\frac{1}{3}[/math] of the shading of the original. Provide reasoning that shows that this is true.[br]
IM Geo.5.6 Practice: Scaling Solids
[size=100][size=150]It takes 2 ounces of paint to completely cover all 6 sides of a rectangular prism box which holds 15 cups of sugar. Double the dimensions of the box.[br][/size][br]Approximately how much paint would the new box need? [/size]
How much sugar would it hold?
[size=150]A solid with volume 12 cubic units is dilated by a scale factor of [math]k[/math]. Find the volume of the image for each given value of [math]k[/math].[/size][br][br][math]k=\frac{1}{4}[/math]
[math]k=0.4[/math]
[math]k=1[/math]
[math]k=1.2[/math]
[math]k=\frac{5}{3}[/math]
[size=150]A solid’s volume is 10 cubic inches. The solid is dilated by a scale factor of 3.5. Kiran says, “I calculated the volume of the image as 35 cubic inches, but I don’t think that’s right.”[/size][br][br][size=100]What might Kiran have done wrong?[/size][br]
[size=100]What is the volume of the image?[/size][br]
Plot the points in the table on coordinate axes and connect them to create a smooth curve.
A parallelogram has an area of 10 square feet. Complete the table that shows the relationship between the dilated area (x) and the scale factor (y).
[size=150]A figure has an area of 4 square units. The equation [math]y=\sqrt{\frac{x}{4}}[/math] represents the scale factor of [math]y[/math] by which the solid must be dilated to obtain an image with area [math]x[/math] square units.[/size][br][size=100][br]Select [b]all [/b]points which are on the graph representing this equation.[/size]
[size=150]Tyler is designing a banner that will welcome people to a festival. The design for the banner has an area of 1.5 square feet. The actual banner will be a dilation of the design by a factor of 5. W[/size][size=150]hat will the area of the actual banner be?[/size]
[size=150]The horizontal cross sections of this figure are dilations of the bottom rectangle using a point above the rectangle as a center and scale factors from [math]\frac{1}{2}[/math] t[/size][size=150]o 1.Sketch an example of a cross section that is created from using a scale factor of [math]\frac{3}{4}[/math]. Label the dimensions of the cross section that you sketch.[/size]
A regular hexagon is inscribed in a circle of radius 1 inch.
[size=100]What is the area of the shaded region?[br][/size]
[size=100][size=150]Two distinct lines, [math]\ell[/math] and [math]m[/math], are each perpendicular to the same line [math]n[/math]. Explain why [math]\ell[/math] and [math]m[/math] are parallel lines.[/size][/size]