Ten boxes are packed tightly in a crate. From above, the packed crate looks like this: [asy] size(4cm); draw((0,10)--(12,10)--(12,0)--(0,0) --cycle, black+3); draw((0,7)--(12,7)); draw((0,4)--(12,4)); draw((4,10)--(4,4)); draw((8,10)--(8,4)); draw((3,4)--(3,0)); draw((6,4)--(6,0)); draw((9,4)--(9,0)); label("$\ell$",(6,10),N); draw((5.3,10+.9)--(0,10+.9),EndArro w(TeXHead)); draw((6.7,10+.9)--(12,10+.9),EndArr ow(TeXHead)); label("$w$",(12,5),E); draw((12.9,5.7)--(12.9,10),EndArrow (TeXHead)); draw((12.9,4.3)--(12.9,0),EndArrow( TeXHead)); [/asy] The visible faces of the boxes are all congruent rectangles (that means the ten small rectangles in the diagram are the same shape and size). The top of the crate is $\ell$ inches long by $w$ inches wide, where $\ell\ge w$ (as shown above). What is the ratio $\ell:w$ in simplified form?

For this option, you will work individually. the pythagorean theorem can be used in many real-world scenarios. part 1 write your own real-world scenario where the pythagorean theorem can be applied to find a missing piece. you may choose to write a problem that is two- or three-dimensional in nature. be sure that you will be able to draw a diagram of your scenario. write out your problem and submit it for part 1. be sure to end your scenario with a question. part 2 draw a diagram of the scenario you created in part 1. you may draw by hand and scan and upload your drawing or create a computer-generated drawing for submission. be sure to label all parts and dimensions of the drawing. part 3 solve the question that you posed in part 1. show all of your steps in answering the question. for this option, you will need to submit all three parts for full credit—your real-world problem and question, the diagram that you created, and your work solving the problem, showing all steps. * note that your instructor is looking for your own original idea. while it is acceptable to use the internet for research and inspiration, academic integrity policies apply.