https://file.aiquickdraw.com/mathbot/file/1730420036648neee4mdk.jpg

Let's analyze and solve the problem presented in the image.

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### Problem Statement:
In the triangle shown, \( \theta \) is increasing at a constant rate of 3 radians per minute. At what rate is \( y \) decreasing in units per minute when \( x \) equals 3 units?

### Given Information:
1. The rate of change of \( \theta \) with respect to time, \( \frac{d\theta}{dt} \), is 3 radians per minute.
2. We need to find the rate of change of \( y \) with respect to time, \( \frac{dy}{dt} \), when \( x = 3 \) units.

### Step-by-Step Solution:

Since we are dealing with a right triangle, we can apply trigonometric relationships. In this case, the tangent function is useful because it involves both \( x \) and \( y \).

#### 1. Set up the tangent relationship:
\[
\tan(\theta) = \frac{y}{x}
\]

#### 2. Differentiate both sides with respect to \( t \):
Using implicit differentiation:
\[
\frac{d}{dt} \left( \tan(\theta) \right) = \frac{d}{dt} \left( \frac{y}{x} \right)
\]

Using the chain rule on the left side, we get:
\[
\sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{x} \cdot \frac{dy}{dt} - \frac{y}{x^2} \cdot \frac{dx}{dt}
\]

Since \( x \) is constant (it’s given as 3 units), \( \frac{dx}{dt} = 0 \). This simplifies the equation to:
\[
\sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{x} \cdot \frac{dy}{dt}
\]

#### 3. Solve for \( \frac{dy}{dt} \):
Rearrange to isolate \( \frac{dy}{dt} \):
\[
\frac{dy}{dt} = x \cdot \sec^2(\theta) \cdot \frac{d\theta}{dt}
\]

#### 4. Substitute known values:
We know that \( x = 3 \), \( \frac{d\theta}{dt} = 3 \) radians per minute, and we need to determine \( \sec^2(\theta) \).

Using the triangle, where \( \tan(\theta) = \frac{y}{x} \), let’s find \( \sec^2(\theta) \) when \( x = 3 \).

If we had the specific value of \( y \), we could calculate this directly. However, assuming the angle can be calculated from given values would provide an exact numeric answer.

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Let me know if you want a continuation of calculations.

In the triangle shown above, if θ increases at a constant rate of 3 radians per minute, at what rate is y decreasing in units per minute when x equals 3 units?

Solve a related rates problem involving trigonometry where θ increases at a rate of 3 radians per minute. Learn how to find the rate at which y decreases when x equals 3 units, using concepts like implicit differentiation and trigonometric functions.

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