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The volume of a spherical bubble is increasing at a constant rate - AQA - A-Level Maths Pure - Question 10 - 2019 - Paper 1
Question 10
The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, r, of the bubble is inversely proportional to r². ... show full transcript
Worked Solution & Example Answer:The volume of a spherical bubble is increasing at a constant rate - AQA - A-Level Maths Pure - Question 10 - 2019 - Paper 1
Step 1
Differentiate the Volume with Respect to Time
Answer
We start with the formula for the volume of a sphere:
[ V = \frac{4}{3} \pi r^3 ]
To find the rate of change of the volume with respect to time, we differentiate with respect to time (t):
[ \frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) ]
Applying the chain rule, we get:
[ \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt} ]
Step 2
Express \( \frac{dr}{dt} \)
Answer
Now, rearranging the differentiated equation gives us:
[ \frac{dr}{dt} = \frac{1}{4 \pi r^2} \frac{dV}{dt} ]
Since the volume (V) is increasing at a constant rate, ( \frac{dV}{dt} ) is a constant.
Thus, we can express the rate of increase of the radius as:
[ \frac{dr}{dt} \propto \frac{1}{r^2} ]
This confirms that the rate of increase of the radius, ( r ), is inversely proportional to ( r^2 ).