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03.05.2021 •
Mathematics
Dos máquinas tuneladoras horadarán una montaña desde puntos opuestos para hacer un túnel de 24 km de longitud. La tuneladora A, desde la cara norte de la montaña, avanza a un ritmo de 200 m por día, y la B, algo más lenta, horada 150 m cada día, desde la cara sur. 1. ¿En qué punto del túnel se encontrarán ambas y cuánto tiempo emplearán en hacerlo? 2. La empresa de la tuneladora A cobra 1,5 millones de euros por día trabajado y 0,2 millones de euros por cada 100 metros avanzados. La empresa de la B cobra 1 millón de euros por día trabajado y 0,3 millones por cada 100 metros. La fracción de día se cobra como un día completo, y cada fracción de 100 metros, también como 100 metros completos. ¿Cuánto cobrará cada empresa por la obra? 3. Si hubiera que elegir la misma empresa para horadar ambos lados con dos máquinas iguales, ¿cuál sería el presupuesto total de la obra en cada caso? ¿Cuál habría que elegir si interesase la más barata?
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Ответ:
I think how you approach this depends on your knowledge of calculus.
If you don't know how to compute definite integrals yet, but you do know that they represent signed areas under curves, then you can plot both curves |x - 4| and √(36 - x ²), then recognize that the areas represented by these integrals are areas of geometric shapes. (See attached images)
First integral: if you plot |x - 4| on the interval [3, 6], you'll see that the integral corresponds to the area of two triangles. One of them has base = height = 1, and the other has base = height = 2. Then
Second integral: if
, then
, meaning this curve is the upper half of a circle with radius 6. On the interval [-6, 0], the area amounts to 1/4 of the total area of such a circle, so that
* * *
If you already know a few things about calculus and integration, you can compute these areas directly.
First integral:
Second integral:
Substitute x = 6 sin(t ) and dx = 6 cos(t ) dt, then
For t ∈ [-π/2, 0], cos(t ) > 0, so |cos(t )| = cos(t ) :
Recall the half-angle identity,
cos²(t ) = (1 + cos(2t )) / 2