Math Problem Statement
the is length of the wire increased by 0.02% and its radius is decrease by 0.005% the percentage change in volume by
Solution
The volume of a cylindrical wire is given by:
where:
- is the radius of the wire,
- is the length of the wire.
Now, let the percentage change in length be and the percentage change in radius be .
Step 1: Percentage change in volume formula
The total change in volume can be approximated using the formula for small percentage changes:
This equation is derived from differentiating the volume formula with respect to and .
Step 2: Applying the values
Substitute these values into the formula:
Final Answer:
The percentage change in volume is 0.01%.
Would you like more details or further clarification?
Here are five related questions for further exploration:
- How does the volume change if the radius decreases by a larger percentage?
- What happens to the volume if both the radius and length increase?
- How does the volume change when only the radius changes?
- Can this method be used for other shapes, such as spheres?
- What is the significance of small percentage changes in engineering calculations?
Tip: When dealing with small changes, using approximations like linearization makes calculations easier and faster!
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Math Problem Analysis
Mathematical Concepts
Geometry
Calculus
Percentage Change
Formulas
Volume of a cylinder V = πr²l
Percentage change in volume: ΔV/V ≈ 2(Δr/r) + Δl/l
Theorems
Linear Approximation
Volume Formula for Cylinders
Suitable Grade Level
Grades 10-12
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