Math Problem Statement
The sides of a triangle are changing in a way that the base is increasing at the rate of 13insec, and the altitude to that base is decreasing at a rate of 14insec. At what rate is the area of this triangle changing when the base is 18 inches and the altitude to that base is 6 inches?
Solution
We are given a triangle where the base is increasing and the altitude is decreasing, and we need to find the rate at which the area of the triangle is changing at a specific moment.
Step 1: Formula for the Area of a Triangle
The area of a triangle is given by: Let:
- = base of the triangle (in inches),
- = height or altitude to the base (in inches),
- = area of the triangle (in square inches).
Thus, the area at any time is:
Step 2: Differentiate the Area with Respect to Time
To find how fast the area is changing, we take the derivative of the area with respect to time : Where:
- is the rate of change of the base,
- is the rate of change of the height.
Step 3: Plug in the Given Values
From the problem, we are given:
- inches per second (the base is increasing),
- inches per second (the height is decreasing),
- inches,
- inches.
Now, substitute these values into the differentiated equation: Simplifying:
Step 4: Conclusion
The area of the triangle is decreasing at a rate of when the base is 18 inches and the height is 6 inches.
Would you like further clarification or details on any of the steps?
Here are 5 related questions to expand on this concept:
- How would the rate of change of the area be affected if the altitude were also increasing?
- What happens to the rate of area change when the base and altitude change at the same rate?
- How would you find the rate of change of the perimeter of the triangle?
- Can the rate of change of the area ever become positive in this scenario?
- How does the chain rule apply in solving related rates problems like this?
Tip: In related rates problems, always differentiate the relevant equation with respect to time before plugging in values.
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Math Problem Analysis
Mathematical Concepts
Geometry
Related Rates
Differentiation
Formulas
Area of a triangle A = (1/2) × base × height
Related rates formula for area change: dA/dt = (1/2) × (db/dt × height + base × dh/dt)
Theorems
Chain Rule of Differentiation
Suitable Grade Level
Grades 10-12
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