crecimiento de arboles
un vivero de plantas verdes suele vender cierto arbusto despues de 6 años de crecimiento y cuidado. La velocidad de crecimiento durante esos 6 años es aproximadamente dh/dt = 1.5t + 5, donde t es el tiempo en años y h es la altura en centimetros. Las plantas de semillero miden 12 centimetros de altura cuando se plantan (t=0).
determinar la altura despues de t años
que altura tiene los arbustos cuando se venden

Answers

Answer 1

Queremos encontrar la altura de las plantas luego de t años, dado que conocemos la altura inicial y la velocidad de crecimiento.

La ecuación de la altura es:

[tex]h(t) = 0.75*t^2 + 5*t + 12[/tex]

Con ella veremos que los arbustos miden 69cm cuando son vendidos.

Veamos como resolver esto.

h(t) es la función que modela la altura de las plantas.

Sabemos que:

[tex]\frac{dh(t)}{dt} = 1.5*t + 5[/tex]

integramos, obtendremos:

[tex]h(t) = (1/2)*1.5*t^2 + 5*t + c[/tex]

donde C es una constante de integración, la cual representa la altura inicial (para t = 0) de las plantas, notar que:

[tex]h(0) = c[/tex]

Y sabemos que las plantas miden 12 cm cuando son plantadas, entonces c = 12.

La ecuación de la altura es entonces:

[tex]h(t) = 0.75*t^2 + 5*t + 12[/tex]

Esta ecuación nos da la altura en centimetros a tiempo t.

Sabemos que los arbustos se venden luego de 6 años, entonces la altura que tendran cuando son vendidos es:

[tex]h(6) = 0.75*6^2 + 5*6 + 12 = 69[/tex]

Es decir, los arbustos miden 69cm cuando son vendidos.

Sí quieres aprender más, puedes leer.

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