Math Problem Statement
The base of a right triangle is fixed at 4 meters. The height y is increasing at a rate of 2m/s when y = 3 meters. Determine the rate at which theta is increasing, in radians per second, at this time.
Solution
To find the rate at which the angle is increasing in a right triangle with a fixed base and a variable height, we can use trigonometry and related rates.
Step-by-Step Solution:
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Identify Variables and Known Rates:
- Base of the triangle () is fixed at meters.
- Height () is variable and is increasing at .
- At the instant we are considering, meters.
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Set Up the Trigonometric Relation:
- In a right triangle, we can express in terms of and :
- Since is fixed, this relationship simplifies to being a function of alone.
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Differentiate with Respect to Time:
- We differentiate with respect to time :
- Solving for :
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Calculate :
- Since at the given time, we can find using the tangent inverse:
- Now, :
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Substitute Values:
- Substitute , , and into the differentiated equation: Simplifying:
Answer:
The rate at which is increasing at this time is .
Would you like a breakdown of any specific step, or do you have additional questions?
Here are some related questions for further exploration:
- How would the answer change if the base length was variable instead of fixed?
- What if the rate of height increase was a function of time?
- Can this problem be solved using a different trigonometric function?
- How does the result change if the triangle’s base is extended to a larger fixed length?
- What implications would this rate of angle change have in real-world applications like tilting structures?
Tip: When dealing with related rates, always differentiate with respect to time, even if the initial relationship doesn’t directly involve time.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Related Rates
Calculus
Formulas
tan(θ) = y / x
sec^2(θ) = 1 + tan^2(θ)
dθ/dt = (1 / (x * sec^2(θ))) * (dy/dt)
Theorems
Trigonometric identity
Related rates theorem
Suitable Grade Level
Grades 11-12
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