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lutik1710 [3]
2 years ago
8

You are looking at the New York ball drop on New Year’s Eve at a distance of 100 m away from the base of the structure. If the b

all drops at a constant rate of 2 m/s, what is the rate of change of the
angel between you and the ball when the angle is ( ?
Mathematics
1 answer:
Fofino [41]2 years ago
8 0

The question is an illustration of related rates.

The rate of change between you and the ball is 0.01 rad per second

I added an attachment to illustrate the given parameters.

The representations on the attachment are:

\mathbf{x = 100\ m}

\mathbf{\frac{dy}{dt} = 2\ ms^{-1}} ---- the rate

\mathbf{\theta = \frac{\pi}{4}}

First, we calculate the vertical distance (y) using tangent ratio

\mathbf{\tan(\theta) = \frac{y}{x}}

Substitute 100 for x

\mathbf{y = 100\tan(\theta) }

\mathbf{\tan(\theta) = \frac{y}{100}}

Differentiate both sides with respect to time (t)

\mathbf{ \sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{100} \cdot \frac{dy}{dt}}

Substitute values for the rates and \mathbf{\theta }

\mathbf{ \sec^2(\pi/4) \cdot \frac{d\theta}{dt} = \frac{1}{100} \cdot 2}

This gives

\mathbf{ (\sqrt 2)^2 \cdot \frac{d\theta}{dt} = \frac{1}{100} \cdot 2}

\mathbf{ 2 \cdot \frac{d\theta}{dt} = \frac{1}{100} \cdot 2}

Divide both sides by 2

\mathbf{ \frac{d\theta}{dt} = \frac{1}{100} }

\mathbf{ \frac{d\theta}{dt} = 0.01 }

Hence, the rate of change between you and the ball is 0.01 rad per second

Read more about related rates at:

brainly.com/question/16981791

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