IM Geo.5.7 Lesson: The Root of the Problem
A cube whose side lengths measure 1 unit has been dilated by several scale factors to make new cubes.
For what scale factor will the volume of the dilated cube be 27 cubic units?[br]
For what scale factor will the volume of the dilated cube be 1,000 cubic units?[br]
Estimate the scale factor that would be needed to make a cube with volume of 1,001 cubic units.[br]
Estimate the scale factor that would be needed to make a cube with volume of 7 cubic units[br]
A shipping company makes cube-shaped boxes. Their basic box measures 1 foot per side. They want to know how to scale the basic box to build new boxes of various volumes.
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[/img][br][br]If the company wants a box with a volume of 8 cubic feet, by what scale factor do they need to dilate the box?[br]
If they want a box with a volume of 10 cubic feet, by approximately what scale factor do they need to dilate the box?[br]
The company decides to create a graph to help analyze the relationship between volume (x) and scale factor (y). Complete the table, rounding values to the nearest hundredth if needed. Use the applet below the table to help, if you choose.
Plot the points and connect them with a smooth curve.
The graph shows the relationship between the volume of the dilated box and the scale factor. Write an equation that describes this relationship.[br]
Suppose the company builds a box with volume 21 cubic feet, then decides to build another box with volume 25 cubic feet. Use your graph to estimate how much the scale factor changes between these two dilated boxes.[br]
Use your graph to estimate how the scale factor changes between a box with volume of 1 cubic foot and one with volume of 5 cubic feet.
[size=150]A government agency is redesigning a satellite, or an object that goes in orbit around Earth. The surface of the satellite is covered with solar panels that supply the satellite with energy. The interior of the satellite is filled with scientific instruments. In the current design, the satellite has a surface area of 5.4 square feet and a volume of 1.2 cubic feet.[/size][br][br]If the agency wants to increase the surface area to 21.6 square feet so the satellite can generate more energy, by what scale factor do they need to dilate the satellite?
If the agency instead wants to increase the volume to 4.05 cubic feet to fit in more scientific instruments, by what scale factor do they need to dilate the satellite?[br]
Is it possible to dilate the satellite in the activity so that the number of square feet of surface area is equal to the number of cubic feet of volume? Explain or show your reasoning.
IM Geo.5.7 Practice: The Root of the Problem
[size=150]A solid with volume 8 cubic units is dilated by a scale factor of [math]k[/math] to obtain a solid with volume [math]V[/math] cubic units. Find the value of [math]k[/math] which results in an image with each given volume.[br][br][/size]216 cubic units
1 cubic unit
1,000 cubic units
[size=150]A solid has volume 7 cubic units. The equation [math]k=\sqrt[3]{\frac{V}{7}}[/math] represents the scale factor of [math]k[/math] by which the solid must be dilated to obtain an image with volume [math]V[/math] cubic units. Select [b]all [/b]points which are on the graph representing this equation.[/size]
[size=150]A solid with surface area 8 square units is dilated by a scale factor of [math]k[/math] to obtain a solid with surface area [math]A[/math] square units. Find the value of [math]k[/math] which leads to an image with each given surface area.[br][/size][br]512 square units
[math]\frac{1}{2}[/math] square unit
8 square units
[size=150]It takes [math]\frac{1}{8}[/math] of a roll of wrapping paper to completely cover all 6 sides of a small box that is shaped like a rectangular prism. The box has a volume of 10 cubic inches. [/size][br][size=150][br]Suppose the dimensions of the box are tripled.[/size][br][list=1][br][/list]How many rolls of wrapping paper will it take to cover all 6 sides of the new box?
What is the volume of the new box?
[size=150]A solid with volume 8 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}{2}[/math]
[math]k=0.6[/math]
[math]k=1[/math]
[math]k=1.5[/math]
[size=150]A figure has an area of 9 square units. The equation [math]y=\sqrt{\frac{x}{9}}[/math] represents the scale factor of [math]y[/math] by which the solid must be dilated to obtain an image with area of [math]x[/math] square units. [/size][size=150]Select [b]all [/b]points which are on the graph representing this equation.[br][/size]
[size=150]Noah edits the school newspaper. He is planning to print a photograph of a flyer for the upcoming school play. The original flyer has an area of 576 square inches. The picture Noah prints will be a dilation of the flyer using a scale factor of [math]\frac{1}{4}[/math]. [/size][br][br]What will be the area of the picture of the flyer in the newspaper?[br]
[size=150]Angle [math]S[/math] is 90 degrees and angle [math]T[/math] is 45 degrees. Side [math]ST[/math] is 3 feet.[/size][size=150] How long is side [math]SU[/math]?[br][/size]