Respuesta :
Answer:
1/10 per sec
Step-by-step explanation:
When he's walked x feet in the eastward direction, the angle Θ that the search light makes has tangent
tanΘ = x/18
Taking the derivative with respect to time
sec²Θ dΘ/dt = 1/18 dx/dt.
He's walking at a rate of 18 ft/sec, so dx/dt = 18.
After 3seconds,
Speed = distance/time
18ft/sec =distance/3secs
x = 18 ft/sec (3 sec)
= 54ft. At this moment
tanΘ = 54/18
= 3
sec²Θ = 1 + tan²Θ
1 + 3² = 1+9
= 10
So at this moment
10 dΘ/dt = (1/18ft) 18 ft/sec = 1
10dΘ/dt = 1
dΘ/dt = 1/10 per sec
At 1/10 per second is the rate when the light turns 3 seconds after he started walking east.
It is given that a man starts at point A and walks 18 feet north. He then turns and walks due east at 18 feet per second.
It is required to find at what rate is the light turning 3 seconds after he started walking east.
What is the trigonometric ratio?
The trigonometric ratio is defined as the ratio of the pair of a right-angled triangle.
The [tex]\rm tan\theta[/tex] is the ratio of the perpendicular to the base.
When he started walking eastward direction, the searchlight makes an angle [tex]\theta[/tex]
Let the distance is x
And the [tex]\rm tan\theta = \frac{x}{18}[/tex]
After performing the derivate with respect to time, we get:
[tex]\rm sec^2\theta\frac{d\theta}{dt} = \frac{1}{18} \frac{dx}{dt}[/tex] ...(1) ( the differentiation of [tex]\rm tan\theta \ is \ sec^2\theta\\[/tex])
He walking at the rate of 18 ft per second ie.
[tex]\rm \frac{dx}{dt} = 18[/tex]
After 3 seconds [tex]\rm Speed=\frac{Distance}{Time}[/tex]
By using speed- time formula we can calculate the distance:
Distance x = 18×3 ⇒ 54 ft.
[tex]\rm tan\theta = \frac{54}{18}[/tex] ⇒3
We know that:
[tex]\rm sec^2\theta = 1+tan^2\theta[/tex]
[tex]\rm sec^2\theta = 1+3^2\\\\\rm sec^2\theta = 10\\[/tex]
Put this value in the (1) equation, we get:
[tex]\rm 10\frac{d\theta}{dt} = \frac{1}{18} \times18[/tex] ∵ [tex]\rm (\frac{dx}{dt} =18)[/tex]
[tex]\rm 10\frac{d\theta}{dt} = 1\\\\\rm \frac{d\theta}{dt} = \frac{1}{10} \\\\[/tex]per second.
Thus, at 1/10 per second what rate is the light turning 3 seconds after he started walking east.
Know more about trigonometry here:
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