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Diano4ka-milaya [45]
3 years ago
7

You stand a known distance from the base of the tree, measure the angle of elevation the top of the tree to be 15â—¦ , and then

compute the height of the tree above eye level. Use the appropriate linear approximation to estimate the maximum possible error in your measurement of the angle (measuered in degrees) to be sure that your computation of the height has a relative error of at most ±p%. Give an exact answer, simplified as much as possible. Do not use a calculator. Assume p ∼ 0.
Mathematics
1 answer:
gogolik [260]3 years ago
4 0

Answer:

The maximum possible error of in measurement of the angle is  d\theta_1  =(14.36p)^o

Step-by-step explanation:

From the question we are told that

    The angle of elevation  is  \theta_1  =  15 ^o =  \frac{\pi}{12}

     The height of the tree is  h

      The distance from the base is  D

h is mathematically represented as

            h  = D tan \theta       Note : this evaluated using SOHCAHTOA i,e

                                               tan\theta  =  \frac{h}{D}

Generally for small angles the series approximation of  tan \theta \  is

          tan \theta  =  \theta  + \frac{\theta ^3 }{3}

So given that \theta =  15 \ which \ is \ small

       h = D (\theta + \frac{\theta^3}{3} )

       dh = D (1 + \theta^2) d\theta

=>        \frac{dh}{h} =  \frac{1 + \theta ^2}{\theta + \frac{\theta^3}{3} } d \theta

Now from the question the relative error of height should be at  most

        \pm  p%

=>    \frac{dh}{h} =   \pm p

=>    \frac{1 + \theta ^2}{\theta + \frac{\theta^3}{3} } d \theta  = \pm p

=>      d\theta  =  \pm  \frac{\theta +  \frac{\theta^3}{3} }{1+ \theta ^2} *    \ p

 So  for   \theta_1

            d\theta_1  =  \pm  \frac{\theta_1 +  \frac{\theta^3_1 }{3} }{1+ \theta_1 ^2} *    \ p

substituting values  

          d [\frac{\pi}{12} ]  =  \pm  \frac{[\frac{\pi}{12} ] +  \frac{[\frac{\pi}{12} ]^3 }{3} }{1+ [\frac{\pi}{12} ] ^2} *    \ p

 =>       d\theta_1  = 0.25 p

Converting to degree

           d\theta_1  = (0.25* 57.29) p

            d\theta_1  =(14.36p)^o

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Se tienen 16 maquinas cuyo rendimiento es del 90% y produce 4800 articulos en 6 dias trabajando 10horas diarias. Si se desea pro
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Answer:

5 máquinas.

Step-by-step explanation:

Sabemos que con 16 máquinas, cada una con un rendimiento del 90%, se producen 4800 artículos en 6 días trabajando 10 horas diarias.

Primero extraigamos toda la información útil de esto.

Sea R la velocidad con la que trabaja una máquina con un rendimiento del 100%.

Entonces nuestras máquinas van a trabajar a:

(90%/100%)*R = 0.9*R

16 máquinas juntas trabajaran en total a 16 veces esa cantidad, o:

16*0.9*R

Luego también sabemos que las máquinas trabajan 10 horas al día por 6 días, es decir, trabajan un total de:

6*10h = 60h

60 horas.

Entonces podemos plantear la ecuación:

(16*0.9*R)*60h = 4800 artículos.

Ahora podemos despejar el valor de R.

R = (4800 artículos)/(16*0.9*60 horas) = 5.556 artículos/hora

Ahora:

Se desea producir 1200 artículos.

En 8 días trabajando 9 horas al día, es decir en:

8*9h = 72 horas

Con N máquinas al 60%.

Cada máquina al 60% trabajara con una velocidad de:

0.6*R

N máquinas entonces trabajaran en conjunto a:

N*(0.6*R)

Con la información que se nos da, podemos plantear la ecuación:

N*(0.6*R)*72horas = 1200 artículos.

Queremos resolver esto para N, el número de máquinas, entonces aislamos N

N = (1200 artículos)/((0.6*R)*72horas)

Reemplazando el valor de R, que es 5.556 artículos/hora

N = (1200 artículos)/((0.6*5.556 artículos/hora)*72horas) = 5

Se necesitaran 5 máquinas.

5 0
3 years ago
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