Balloon or expanding shape

Given a balloon or similar expanding shape with volume increasing at a given rate, find the rate of change of radius or other dimension.

4 questions · Standard +0.1

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OCR C3 2010 January Q7
7 marks Standard +0.3
  1. Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures. [4]
  2. The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, \(m\) grams, at a time \(t\) years from now is given by $$m = 150e^{-kt},$$ where \(k\) is a positive constant. Find, in terms of \(k\), the number of years from now at which the mass will be decreasing at a rate of 3 grams per year. [3]
OCR MEI C3 2011 January Q3
5 marks Standard +0.3
The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]
OCR MEI C3 2012 January Q6
8 marks Standard +0.3
Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20(1 - e^{-0.2t}).$$
  1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time. [4]
  2. Find the rate at which the area of the slick is increasing when \(t = 2\). [4]
OCR MEI C3 Q7
6 marks Moderate -0.3
An oil slick is circular with radius \(r\) km and area \(A\) km\(^2\). The radius increases with time at a rate given by \(\frac{dr}{dt} = 0.5\), in kilometres per hour.
  1. Show that \(\frac{dA}{dt} = \pi r\). [4]
  2. Find the rate of increase of the area of the slick at a time when the radius is 6 km. [2]