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7nadin3 [17]
3 years ago
10

Se desea construir un depósito de agua con forma cilíndrica de radio r y de altura 2·r. A. Hallar la función que proporciona el

volumen del depósito en función de su radio. B. ¿Cuáles deben ser las dimensiones del depósito para que su capacidad sea 100 litros?
Mathematics
1 answer:
musickatia [10]3 years ago
5 0

Answer:

a) La función del volumen del depósito está dada por V = 2\pi\cdot r^{3}, b) Las dimensiones del depósito de 100 litros de capacidad son: Radio = 25.15 centímetros, Altura = 50.30 centímetros.

Step-by-step explanation:

a) Sabemos por Geometría que el volumen de un cilindro recto (V), medido en decímetros cúbicos o litros, es: (Nótese que 1 litro equivale a un decímetro cúbico)

V = \pi\cdot r^{2}\cdot h

Donde:

r - Radio del área transversal del cilindro, medido en decímetros.

h - Altura del cilindro, medida en decímetros.

Si h = 2\cdot r, entonces la fórmula de volumen es:

V = \pi\cdot r^{2}\cdot (2\cdot r)

V = 2\pi\cdot r^{3}

La función del volumen del depósito está dada por V = 2\pi\cdot r^{3}.

b) Si sabemos que V = 100\,L = 100\,dm^{3}, entonces el radio se obtiene al ser despejado de la función hallada en a).

r = \sqrt[3]{\frac{V}{2\pi} }

r = \sqrt[3]{\frac{100\,dm^{3}}{2\pi} }

r \approx 2.515\,dm\,(25.15\,cm)

Ahora, la altura, medida en decímetros, se obtiene a continuación:

h = 2\cdot (2.515\,dm)

h = 5.03\,dm\,(50.30\,cm)

Las dimensiones del depósito de 100 litros de capacidad son: Radio = 25.15 centímetros, Altura = 50.30 centímetros.

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