Name: A person is standing 30 m30~\text{m} away from a lighthouse. A balloon
Uploaded: 2022-07-23T00:00:00.000Z
Duration: 6 min 31 s
Description: Watch the video explanation for the concept: A person is standing 30 m30~\text{m} away from a lighthouse. A balloon

Question

Watching an elevator An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer’s line of sight makes with the horizontal (it may be positive or negative). Assuming the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground? When the elevator is 40 m above the ground?

Accepted Answer

Step 1: Define the variables. Let $$ 	heta  be $$the angle of elevation, $$ y  be $$the height of the elevator above the ground, and $$ x  be $$the horizontal distance from the observer to the elevator shaft, which is constant at 20 m. Step 2: Use trigonometry to relate $$ 	heta $$ and $$ y . $$The tangent of the angle $$ 	heta  is $$given by $$ 	an(	heta) = \frac{y}{x} . $$Since $$ x = 20  m$$, we have $$ 	an(	heta) = \frac{y}{20} . $$Step 3: Differentiate both sides of the equation $$ 	an(	heta) = \frac{y}{20} $$ with respect to time $$ t . $$Use the chain rule: $$ \sec^2(	heta) \frac{d	heta}{dt} = \frac{1}{20} \frac{dy}{dt} . $$Step 4: Substitute the given rate of change of the elevator's height, $$ \frac{dy}{dt} = 5  m/s$$, into the differentiated equation: $$ \sec^2(	heta) \frac{d	heta}{dt} = \frac{1}{20} 	imes 5 . $$Step 5: Solve for $$ \frac{d	heta}{dt} $$ when $$ y = 10  m $$and $$ y = 40  m. $$First, find $$ 	heta $$ using $$ 	an(	heta) = \frac{y}{20} $$ for each case, then calculate $$ \sec^2(	heta) $$, and finally find $$ \frac{d	heta}{dt} $$ using the equation from Step 4.

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주요 개념

Related Rates

Trigonometric Functions

Implicit Differentiation