3.3. Diseños notables

Formas que se repiten: el álgebra detrás del diseño

El cofre

Las marcas de automóviles diseñan cofres de techo para aumentar la capacidad de carga.

Antes de fabricarlos, trabajan con modelos geométricos simplificados, que permiten analizar el espacio que ocupa el cofre y la cantidad de material necesaria.

En una primera fase, el equipo de diseño parte de una forma base y van añadiendo o ajustando capas para mejorar la capacidad, la aerodinámica y la resistencia del cofre.

Para el estudio se utilizan expresiones algebraicas en lugar de medidas fijas.

El área del cofre se puede aproximar por un rectángulo cuyo lado mide \( x±a \) unidades.

Áreas

Los resultados finales del área de los cofres se basan en los siguientes patrones:

Cofre cuadrado aumentando el lado: \( (x+a)^2=x^2+2ax+a^2\)

Cofre cuadrado disminuyendo el lado: \( (x-a)^2=x^2-2ax+a^2\)

Cofre rectangular, un lado aumenta y el otro disminuye: \((x+a)\cdot (x-a)=x^2-a^2\)

¿Los recuerdas? Son las "Identidades o productos notables"

Ej. \((x+a)^2\)

\((x+1)^2=x^2+2\cdot x\cdot 1+1^2=x^2+2x+1\)
\((2x+3)^2=(2x)^2+2\cdot 2x\cdot 3+3^2=4x^2+12x+9\)
\((x^2+5)^2=(x^2)^2+2\cdot x^2\cdot 5+5^2=x^4+10x^2+25\)

Ej. \((x-a)^2\)

\((x-4)^2=x^2-2\cdot x\cdot 4+4^2=x^2-8x+16\)
\((3x-1)^2=(3x)^2-2\cdot 3x\cdot 1+1^2=9x^2-6x+1\)
\((2x^2-1)^2=(2x^2)^2-2\cdot 2x^2\cdot 1+1^2=4x^4-4x^2+1\)

Ej. \((x+a)(x-a)\)

\((x-2)(x+2)=x^2-2^2=x^2-4\)
\((4x+3)(4x-3)=(4x)^2-3^2=16x^2-9\)
\((3x^2-4)(3x^2+4)=(3x^2)^2-4^2=9x^4-16\)

Clasifica fórmulas de diseños

Antes de comenzar el trabajo, demuestra que eres capaz de clasificar las fórmulas según el tipo de diseños (si tienes alguna duda siempre puedes consultar los ejemplos anteriores).

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%E0%F6%F3%E0%B2%E7%FC%F3%B2%E4%F7%E8%B0%BE%B0%FF%E1%F5%DB%FC%F4%FD%E0%FF%F3%E6%FB%FD%FC%B0%A8%B0%DB%FC%F4%FD%E0%FF%F3%F1%FBa%FC%B0%BE%B0%FF%E1%F5%CB%FD%E7%C1%F1%FD%E0%F7%B0%A8%B0%C6%E7%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B0%BE%B0%FF%E1%F5%D3%E7%E6%FA%FD%E0%B0%A8%B0%D3%E7%E6%FD%E0%7F%F3%B0%BE%B0%FF%E1%F5%DD%FC%FE%EB%C1%F3%E4%F7%D3%E7%E6%FD%B0%A8%B0%C6%E7%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B2%E1%F7%B2%F5%E7%F3%E0%F6%F3%E0s%B2%F6%F7%E1%E2%E7%7B%E1%B2%F6%F7%B2%F1%F3%F6%F3%B2%E2%E0%F7%F5%E7%FC%E6%F3%BC%B2%C1a%FE%FD%B2%E2%E7%F7%F6%F7%E1%B2%F8%E7%F5%F3%E0%B2%E7%FC%F3%B2%E4%F7%E8%BC%B0%BE%B0%FF%E1%F5%C1%F3%E4%F7%D3%E7%E6%FD%B0%A8%B0%C6%E7%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B2%E1%F7%B2%F5%E7%F3%E0%F6%F3%E0s%B2%F3%E7%E6%FD%FFs%E6%FB%F1%F3%FF%F7%FC%E6%F7%B2%F6%F7%E1%E2%E7%7B%E1%B2%F6%F7%B2%F1%F3%F6%F3%B2%E2%E0%F7%F5%E7%FC%E6%F3%BC%B0%BE%B0%FF%E1%F5%C1%F7%E4%F7%E0%F3%FE%C1%F1%FD%E0%F7%B0%A8%B0%C2%E7%F7%F6%F7%E1%B2%F5%E7%F3%E0%F6%F3%E0%B2%FE%F3%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B2%E6%F3%FC%E6%F3%E1%B2%E4%F7%F1%F7%E1%B2%F1%FD%FF%FD%B2%E3%E7%FB%F7%E0%F3%B0%BE%B0%FF%E1%F5%CB%FD%E7%DE%F3%E1%E6%C1%F1%FD%E0%F7%B0%A8%B0%DE%F3%B2h%FE%E6%FB%FF%F3%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B2%F5%E7%F3%E0%F6%F3%F6%F3%B2%F7%E1%B0%BE%B0%FF%E1%F5%D3%F1%E6%FB%E6%EB%D1%FD%FF%E2%FE%EB%B0%A8%B0%CB%F3%B2%FA%F3%E1%B2%E0%F7%F3%FE%FB%E8%F3%F6%FD%B2%F7%E1%E6%F3%B2%F3%F1%E6%FB%E4%FB%F6%F3%F6%BC%B0%BE%B0%FF%E1%F5%C2%FE%F3%EB%C1%F7%E4%F7%E0%F3%FE%C6%FB%FF%F7%E1%B0%A8%B0%C2%E7%F7%F6%F7%E1%B2%E0%F7%F3%FE%FB%E8%F3%E0%B2%F7%E1%E6%F3%B2%F3%F1%E6%FB%E4%FB%F6%F3%F6%B2%F1%E7%F3%FC%E6%F3%E1%B2%E4%F7%F1%F7%E1%B2%E3%E7%FB%F7%E0%F3%E1%B0%BE%B0%FF%E1%F5%D1%FE%FD%E1%F7%B0%A8%B0%D1%F7%E0%E0%F3%E0%B0%BE%B0%FF%E1%F5%D3%E7%F6%FB%FD%B0%A8%B0%D3%E7%F6%FB%FD%B0%BE%B0%FF%E1%F5%CB%F7%E1%B0%A8%B0%C1%7F%B0%BE%B0%FF%E1%F5%DC%FD%B0%A8%B0%DC%FD%B0%BE%B0%FF%E1%F5%C6%FB%FF%F7%DD%E4%F7%E0%B0%A8%B0%C6%E7%B2%E6%FB%F7%FF%E2%FD%B2%FA%F3%B2%F4%FB%FC%F3%FE%FB%E8%F3%F6%FD%BC%B2%DB%FC%E6%7B%FC%E6%F3%FE%FD%B2%F6%F7%B2%FC%E7%F7%E4%FD%B0%BE%B0%FF%F5%E1%D5%F3%FF%F7%C1%E6%F3%E0%E6%B0%A8%B03%D7%FE%B2%F8%E7%F7%F5%FD%B2%FA%F3%B2%F1%FD%FF%F7%FC%E8%F3%F6%FD%B3%B2%D3%E0%E0%F3%E1%E6%E0%F3%B2%F1%F3%F6%F3%B2%E6%F3%E0%F8%F7%E6%F3%B2%FA%F3%E1%E6%F3%B2%E1%E7%B2%F1%FD%FC%E6%F7%FC%F7%F6%FD%E0%B0%BE%B0%FF%E1%F5%C1%F7%FE%F7%F1%E6%D1%F3%E0%F6%B0%A8%B0%C1%F7%FE%F7%F1%F1%FB%FD%FC%F3%B2%FD%E6%E0%F3%B2%E6%F3%E0%F8%F7%E6%F3%B0%BE%B0%FF%E1%F5%C1%F7%FE%F7%F1%E6%D1%F3%E0%F6%DD%FC%F7%B0%A8%B0%C1%F7%FE%F7%F1%F1%FB%FD%FC%F3%B2%E7%FC%F3%B2%E6%F3%E0%F8%F7%E6%F3%B0%BE%B0%FF%E1%F5%C0%F7%F0%FD%FD%E6%B0%A8%B0%C0%F7%FB%FC%FB%F1%FB%F3%E0%B0%BE%B0%FF%E1%F5%C6%F7%E1%E6%C2%F3%E1%E1%F7%F6%B0%A8%B03%D5%F7%FC%FB%F3%FE%B3%B23%C2%E0%E7%F7%F0%F3%B2%E1%E7%E2%F7%E0%F3%F6%F3%B3%B0%BE%B0%FF%E1%F5%C6%F7%E1%E6%D4%F3%FB%FE%F7%F6%B0%A8%B0%DC%FD%B2%FA%F3%E1%B2%E1%E7%E2%F7%E0%F3%F6%FD%B2%FE%F3%B2%E2%E0%E7%F7%F0%F3%BC%B2%DB%FC%E6%7B%FC%E6%F3%FE%FD%B2%F6%F7%B2%FC%E7%F7%E4%FD%B0%BE%B0%FF%E1%F5%C0%F7%F0%FD%FD%E6%D5%F3%FF%F7%B0%A8%B0-%D6%F7%E1%F7%F3%E1%B2%E0%F7%FB%FC%FB%F1%FB%F3%E0%B2%F7%E1%E6%F3%B2%E2%F3%E0%E6%FB%F6%F3%AD%B0%BE%B0%FF%E1%F5%D1%FD%FC%E6%FB%FC%E7%F7%B0%A8%B0%D1%FD%FC%E6%FB%FC%E7%F3%E0%B0%BE%B0%FF%E1%F5%C1%FA%FD%E5%D3%FC%E1%E5%F7%E0%E1%B0%A8%B0%D1%FD%FF%E2%E0%E7%F7%F0%F3%B2%E0%F7%E1%E7%FE%E6%F3%F6%FD%E1%B0%BE%B0%FF%E1%F5%C7%FC%F3%FC%E1%E5%F7%E0%F7%F6%C3%E7%F7%E1%E6%FB%FD%FC%E1%B0%A8%B0%C6%F7%B2%FA%F3%FC%B2%E3%E7%F7%F6%F3%F6%FD%B2%B7%E1%B2%E6%F3%E0%F8%F7%E6%F3%E1%B2%E2%FD%E0%B2%F1%FD%FE%FD%F1%F3%E0%B2%F1%FD%E0%E0%F7%F1%E6%F3%FF%F7%FC%E6%F7%BC%B2-%C3%E7%FB%F7%E0%F7%E1%B2%FB%FC%E6%F7%FC%E6%F3%E0%FE%FD%B2%F6%F7%B2%FC%E7%F7%E4%FD%AD%B0%BE%B0%FF%E1%F5%D3%FE%FE%D1%FD%E0%E0%F7%F1%E6%B0%A8%B03%D5%F7%FC%FB%F3%FE%B3%B23%C6%FD%F6%FD%B2%E2%F7%E0%F4%F7%F1%E6%FD%B3%B0%BE%B0%FF%E1%F5%C6%FD%FD%DF%F3%FC%EB%C6%E0%FB%F7%E1%B0%A8%B03%D7%E1%E6%E7%E2%F7%FC%F6%FD%B3%B2%DA%F3%E1%B2%E0%F7%E1%E7%F7%FE%E6%FD%B2%F7%E1%E6%F3%B2%F3%F1%E6%FB%E4%FB%F6%F3%F6%B2%F7%FC%B2%B7%E1%B2%FB%FC%E6%F7%FC%E6%FD%E1%BC%B23%C1%F7%F5%E7%E0%FD%B2%E3%E7%F7%B2%E2%E7%F7%F6%F7%E1%B2%FA%F3%F1%F7%E0%FE%FD%B2%FFs%E1%B2%E0s%E2%FB%F6%FD%B3%B0%BE%B0%FF%E1%F5%C3%A7%B0%A8%B0%DC%FD%B2%FA%F3%E1%B2%F1%FD%FE%FD%F1%F3%F6%FD%B2%B7%E1%B2%E6%F3%E0%F8%F7%E6%F3%E1%B2%F7%FC%B2%E1%E7%B2%FE%E7%F5%F3%E0%BC%B23%DB%FC%E6%7B%FC%E6%F3%FE%FD%B2%F6%F7%B2%FC%E7%F7%E4%FD%B3%B0%BE%B0%FF%E1%F5%C3%A5%B0%A8%B03%D7%E1%E6%E7%E2%F7%FC%F6%FD%B3%B2%DA%F3%E1%B2%F1%FE%F3%E1%FB%F4%FB%F1%F3%F6%FD%B2%F1%FD%E0%E0%F7%F1%E6%F3%FF%F7%FC%E6%F7%B2%FE%F3%B2%FF%F3%EB%FD%E0%B2%E2%F3%E0%E6%F7%B2%F6%F7%B2%FE%F3%E1%B2%E6%F3%E0%F8%F7%E6%F3%E1%BE%B2%B7%E1%BE%B2%E2%F7%E0%FD%B2%F3h%FC%B2%E2%E7%F7%F6%F7%E1%B2%FF%F7%F8%FD%E0%F3%E0%B0%BE%B0%FF%E1%F5%C3%AB%B0%A8%B03%DF%F3%F5%FC%7F%F4%FB%F1%FD%B3%B2%C6%F3%FC%B2%E1%FD%FE%FD%B2%E6%F7%B2%FA%F3%FC%B2%E3%E7%F7%F6%F3%F6%FD%B2%B7%E1%B2%E6%F3%E0%F8%F7%E6%F3%E1%B2%E2%FD%E0%B2%F1%FD%FE%FD%F1%F3%E0%B2%F1%FD%E0%E0%F7%F1%E6%F3%FF%F7%FC%E6%F7%BC%B2%D0%E7%E1%F1%F3%B2%FE%F3%B2%E2%F7%E0%F4%F7%F1%F1%FBa%FC%B0%BE%B0%FF%E1%F5%C1%F3%E4%F7%D5%F3%FF%F7%D3%E7%E6%FD%B0%A8%B0%C1%E7%B2%E2%E7%FC%E6%E7%F3%F1%FBa%FC%B2%E1%F7%B2%F5%E7%F3%E0%F6%F3%E0s%B2%F3%E7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Su navegador no es compatible con esta herramienta.

Analiza cambios en las fórmulas

Analiza cómo cambia la superficie del cofre cuando se modifican las medidas de su diseño.

(x+4)² = Rellenar huecos (1):JXUwMDIw ^{Rellenar huecos (2):JXUwMDZh} + Rellenar huecos (3):JXUwMDYwJXUwMDQw + Rellenar huecos (4):JXUwMDY5JXUwMDA3
(3x - 1)² = Rellenar huecos (5):JXUwMDYxJXUwMDQx ^{Rellenar huecos (6):JXUwMDZh} - Rellenar huecos (7):JXUwMDZlJXUwMDRl + Rellenar huecos (8):JXUwMDY5
(x+6)² = Rellenar huecos (9):JXUwMDIw ^{Rellenar huecos (10):JXUwMDZh}+ Rellenar huecos (11):JXUwMDY5JXUwMDAzJXUwMDRh + Rellenar huecos (12):JXUwMDZiJXUwMDA1
(x+1)(x-1) = Rellenar huecos (13):JXUwMDIw ^{Rellenar huecos (14):JXUwMDZh} - Rellenar huecos (15):JXUwMDY5
(2x - 5)² = Rellenar huecos (16):JXUwMDZjJXUwMDRj ^{Rellenar huecos (17):JXUwMDZh}- Rellenar huecos (18):JXUwMDZhJXUwMDAyJXUwMDQ4 + Rellenar huecos (19):JXUwMDZhJXUwMDA3
(x+3)(x-3) = Rellenar huecos (20):JXUwMDIw ^{Rellenar huecos (21):JXUwMDZh} - Rellenar huecos (22):JXUwMDYx

Habilitar JavaScript

Adaptándose a los requisitos de la marca

En el equipo de ingeniería se hicieron modificaciones del diseño para ver qué opción se adapta mejor a los requisitos de la marca.

Después de un largo día de trabajo, descubren que sus notas están medio borradas.

¿Puedes ayudarles a descubrir los términos que faltan?

Une factores y diseños

Relaciona cada tarjeta con su pareja.

{"typeGame":"Relaciona","author":"","randomCards":true,"instructions":"

Relaciona cada tarjeta con su pareja.

","showMinimize":false,"itinerary":{"showClue":false,"clueGame":"","percentageClue":40,"showCodeAccess":false,"codeAccess":"","messageCodeAccess":""},"cardsGame":[{"url":"","x":0,"y":0,"author":"","alt":"","audio":"","color":"#000000","backcolor":"#ffffff","eText":"49x%C2%B2%20-%2016","urlBk":"","xBk":0,"yBk":0,"authorBk":"","altBk":"","audioBk":"","colorBk":"#000000","backcolorBk":"#ffffff","eTextBk":"(7x%20%2B%204)%20(7x%20-%204)"},{"url":"","x":0,"y":0,"author":"","alt":"","audio":"","color":"#000000","backcolor":"#ffffff","eText":"x%C2%B2%20-%2018%20x%20%2B%2081","urlBk":"","xBk":0,"yBk":0,"authorBk":"","altBk":"","audioBk":"","colorBk":"#000000","backcolorBk":"#ffffff","eTextBk":"(x%20-%209)%C2%B2%20"},{"url":"","x":0,"y":0,"author":"","alt":"","audio":"","color":"#000000","backcolor":"#ffffff","eText":"4x%C2%B2%20%2B%2016x%20%2B%2016","urlBk":"","xBk":0,"yBk":0,"authorBk":"","altBk":"","audioBk":"","colorBk":"#000000","backcolorBk":"#ffffff","eTextBk":"(2x%20%2B%204)%C2%B2%20"},{"url":"","x":0,"y":0,"author":"","alt":"","audio":"","color":"#000000","backcolor":"#ffffff","eText":"25%20-%20100y%C2%B2%20","urlBk":"","xBk":0,"yBk":0,"authorBk":"","altBk":"","audioBk":"","colorBk":"#000000","backcolorBk":"#ffffff","eTextBk":"(5%20%2B%2010y)%20(5%20-%2010y)"},{"url":"","x":0,"y":0,"author":"","alt":"","audio":"","color":"#000000","backcolor":"#ffffff","eText":"(3x%20-%204)%C2%B2%20","urlBk":"","xBk":0,"yBk":0,"authorBk":"","altBk":"","audioBk":"","colorBk":"#000000","backcolorBk":"#ffffff","eTextBk":"9x%C2%B2%20-%2024x%20%2B%2016"}],"isScorm":1,"textButtonScorm":"Guardar la puntuación","repeatActivity":true,"textAfter":"","version":1.3,"percentajeCards":100,"type":2,"showSolution":true,"timeShowSolution":3,"time":3,"evaluation":false,"evaluationID":"","id":"20251229203733-60","msgs":{"msgSubmit":"Enviar","msgClue":"¡Genial! La pista es:","msgCodeAccess":"Código de acceso","msgPlayStart":"Pulsa aquí para jugar","msgScore":"Puntuación","msgErrors":"Errores","msgHits":"Aciertos","msgMinimize":"Minimizar","msgMaximize":"Maximizar","msgFullScreen":"Pantalla Completa","msgExitFullScreen":"Salir del modo pantalla completa","msgNoImage":"Pregunta sin imágenes","msgEndGameScore":"Antes de guardar la puntuación comience la partida.","msgScoreScorm":"La puntuación no se puede guardar porque esta página no forma parte de un paquete SCORM.","msgOnlySaveScore":"¡Sólo puedes guardar la puntuación una vez!","msgOnlySave":"Sólo puedes guardar una vez","msgInformation":"Información","msgYouScore":"Tu puntuación","msgAuthor":"Autoría","msgOnlySaveAuto":"Tu puntuación se guardará después de cada pregunta. Sólo puedes jugar una vez.","msgSaveAuto":"Tu puntuación se guardará automáticamente después de cada pregunta.","msgSeveralScore":"Puedes guardar la puntuación tantas veces como quiera","msgYouLastScore":"La última puntuación guardada es","msgActityComply":"Ya ha realizado esta actividad.","msgPlaySeveralTimes":"Puedes realizar esta actividad cuantas veces quieras","msgClose":"Cerrar","msgAudio":"Audio","msgNumQuestions":"Número de tarjetas","msgTryAgain":"Necesitas al menos un %s% de respuestas correctas para conseguir la información. Vuelva a intentarlo.","msgEndGameM":"Has completado el juego. Tu puntuación es %s.","msgUncompletedActivity":"Actividad no completada","msgSuccessfulActivity":"Actividad superada. Puntuación: %s","msgUnsuccessfulActivity":"Actividad no superada. Puntuación: %s","msgTypeGame":"Relaciona","msgCheck":"Comprueba","msgRestart":"Reiniciar"}}

Su navegador no es compatible con esta herramienta.

Revisa la labor del equipo

Demuestra lo que has aprendido en el equipo de diseño.

Calentamiento inicial

(4 + a)² = 16 + Rellenar huecos (1):JXUwMDYw Rellenar huecos (2):JXUwMDM5 + Rellenar huecos (3):JXUwMDM5 ^{Rellenar huecos (4):JXUwMDZh}
(5y - 3)² = Rellenar huecos (5):JXUwMDZhJXUwMDA3 Rellenar huecos (6):JXUwMDIx ^{Rellenar huecos (7):JXUwMDZh} - 30y + Rellenar huecos (8):JXUwMDYx
(x+7)(x+7) = x ^{Rellenar huecos (9):JXUwMDZh} - Rellenar huecos (10):JXUwMDZjJXUwMDBk

Entrenamiento

(2 - 2x²)² = Rellenar huecos (11):JXUwMDZj - Rellenar huecos (12):JXUwMDYw x²+ 4x^{Rellenar huecos (13):JXUwMDZj}
(4 - 3x²)(4 + 3x²) = Rellenar huecos (14):JXUwMDY5JXUwMDA3 - Rellenar huecos (15):JXUwMDYx Rellenar huecos (16):JXUwMDIw ^{Rellenar huecos (17):JXUwMDZj}
(5y + 2x)² = Rellenar huecos (18):JXUwMDZhJXUwMDA3 y ^{Rellenar huecos (19):JXUwMDZh} + Rellenar huecos (20):JXUwMDZhJXUwMDAy y Rellenar huecos (21):JXUwMDIw + 4 Rellenar huecos (22):JXUwMDIw ^{Rellenar huecos (23):JXUwMDZh}

Competición

(6a + 5ab²)² = 36 Rellenar huecos (24):JXUwMDM5 ^{Rellenar huecos (25):JXUwMDZh} + 60a²b²+ 25 a^{Rellenar huecos (26):JXUwMDZh} Rellenar huecos (27):JXUwMDNh ^{Rellenar huecos (28):JXUwMDZj}
(\( \dfrac{3x}{2} - 1 \))² = \(\dfrac{9x^2}{4}\) - Rellenar huecos (29):JXUwMDZi Rellenar huecos (30):JXUwMDIw + Rellenar huecos (31):JXUwMDY5
(6xy - \( \dfrac{5}{2} x^4 \))(6xy + \( \dfrac{5}{2}x^4 \)) = Rellenar huecos (32):JXUwMDZiJXUwMDA1 Rellenar huecos (33):JXUwMDIw ² y ^{Rellenar huecos (34):JXUwMDZh} - 25 Rellenar huecos (35):JXUwMDIw ^{Rellenar huecos (36):JXUwMDYw}

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