¹/₂ cm/s
Further explanation
Given:
The area of a circle increases at a rate of 1 cm²/s.
Question:
How fast is the radius changing when the circumference is 2 cm?
The Process:
Step-1: differentiate the area equation of a circle
The area of the circle is [tex]\boxed{ \ A = \pi r^2 \ }[/tex].
Let us differentiate this function with respect to r.
[tex]\boxed{ \ \frac{dA}{dr} = 2 \pi r \ }[/tex]
Step-2: using the chain rule
Let us use the chain rule for composite functions.
[tex]\boxed{ \ \frac{dA}{dr} = 2 \pi r \ }[/tex]
Hence [tex]\boxed{ \ \frac{dA}{dt} \cdot \frac{dt}{dr} = 2 \pi r \ }[/tex]
Step-3: find out how fast is the radius changing when the circumference is 2 cm
We knew that,
- [tex]\boxed{ \ \frac{dA}{dt} = 1 \ cm^2/s \ }[/tex]
- the circumference is 2πr = 2 cm
Let us substitute in the equation from the previous chain rule.
[tex]\boxed{ \ 1 \cdot \frac{dt}{dr} = 2 \ }[/tex]
Multiply both sides by dr/dt and ¹/₂.
[tex]\boxed{ \ 1 = 2 \cdot \frac{dr}{dt} \ }[/tex]
Multiply both sides by ¹/₂.
[tex]\boxed{ \ \frac{dr}{dt} = \frac{1}{2} \ }[/tex]
Thus, we get how fast the radius changing when the circumference is 2 cm, which is ¹/₂ cm/s.
Learn more
- Using the product rule https://brainly.com/question/1578252
- The derivatives of the composite function https://brainly.com/question/6013189
- What is the general form of the equation of the given circle with center A(-3,12) and the radius is 5? https://brainly.com/question/1506955
