More differentiation application questions
1. A child’s height h cm at age years can be modelled by the equation
for ages . Find the child’s annual growth rate at age 12 and at age 15.
2. The temperature C measured at a distance cm from a candle flame is given by for distances
a) Find the temperature gradient (i.e the rate at which temperature decreases with distance) 10 cm from the flame.
b) Find the rate of change of temperature gradient with respect to distance 5cm from the flame.
3. Some water is gradually evaporating from an open vessel, so that the volume V, measured in cm3 after t days is given by where is a constant
a) Given that the water evaporates completely after exactly 27 days, find and hence the initial volume of water in the vessel.
b) Find the rate of evaporation after 8 days.
4. A circular pipe has outer diameter 4cm and thickness tcm.
a) Show that the area of cross section cm2 is given by .
b) Find the rate of increase of with respect to t when and when , leaving in the answer.
5. At time hours after the beginning of a storm, mm of rain have fallen where for
a) Find the rate of rainfall one and two hours after the beginning of the storm.
b) Find the rate at which the rate of rainfall is changing after 3 hours.
1. 9.176,2 2a) -2.13 ° per cm b) 6.04 °per cm per cm 3a) 12,288cm3 b) 10cm3/day 4b)3.5π,3π 5a) 3, 0 b)6
6. A region of high pressure is centred at a point C. The pressure in millibars a distance km from C obeys the equation
a. Find the pressure 20km from C correct to the nearest millibar
b. Find the pressure gradient (the rate of change of pressure with respect to position), correct to 2 s.f., 40 km from C.
c. Find the rate of change of pressure gradient with respect to position 60km from C, giving your answer to 2 s.f.
6 a) 1004 mb b)0.25mb/km c).0073mb/km2