Home » Maths for Planet Earth » Deep-Sea Bubbles

Deep-Sea Bubbles

Facebook
Twitter
Pinterest
Print

Deep sea vents can emit harmful gases, such as hydrogen sulphide. Since these bubbles are small, they shrink once they leave the vent, as the gases dissolve into the ocean.

An autonomous submarine takes a photograph of these bubbles that looks like this graph sketch. 

deep-sea bubbles

A scientist discovers that the surface of the bubbles visible in this photograph can be represented by the curve:

\[\sin\frac{y^{3}}{100} + e^{- x}y + x^{2} = 5\ ,\ y > 5\]

By first using implicit differentiation to show that

\[\frac{\text{dy}}{\text{dx}} = \frac{100(y – 2e^{x}x)}{3e^{x}y^{2}\cos\frac{y^{3}}{100} + 100}\]

Show that at the turning points on the curve, where \(\frac{\text{dy}}{\text{dx}} = 0\), \(y = 2xe^{x}\)

[8 marks]

Start exploring

Latest from blog

More Maths for Planet Earth

GCSE
Jean owns a trucking company (with ten trucks) which produces 1062.8 tonnes of carbon dioxide in the year 2015. The company’s carbon emissions increase
GCSE
As the atmosphere warms, the air holds more water vapour, and this could lead to more intense rainfall events, resulting in an increased flood
GCSE
The table below shows information about the annual CO2 emissions from 140 cities. a) Work out an estimate for the mean CO2 emissions across
A Level
The rate of CO2 emissions for the UK was measured every 5 years, from 1990 to 2015. The results are given in the table
BACK TO TOP