exponential growth and decay (1 Viewer)

[talulahbay]

1. The half life of radium is 1600 years.
a) find the percentage of the radium that will be decyaed after 500 years
b) find the number of years that it will take for 75% of the radium to decay.
For a i got 80.5% as an answer but for the correct answer you need to minus this from 100% and i dont really understand why

2. The population of a city is P(t) at any one time. The rate of decline in population is proportional to the population P(t) that is dP(t)/dt=-kP(t).
a) show that P(t) =P(to)e^-kt is a solution of the differential equation dP(t)/dt=-kP(t)
b) What percentage decline in population will there be after 10 years, given a 10% decline in 4 years

if any can help me with these then thankyouu!!

 

[braintic]

1 (a) The usual calculation gives the amount of radium present at any time, ie. the amount of radium left. But the question asks for the amount of radium decayed (ie. gone).
(b) Don't need a serious calculation for this one. Half-Life is 1600 years. So after 1600 years, 50% is left. After another 1600 years, half of that is left. That is 25% left, or 75% decayed after 3200 years.

2. (a) Differentiating P=P0.e^(-kt): P'=-kP0.e^(-kt) = -kP (subbing in the original eqn).
(b) If you can do Q1, you can do this one: Just let P=0.9P0 when t=4 (10% decline means 90% remaining), cancel the P0s on either side of the eqn , then solve for k.
Then let k=10. If you get say P=0.53P0 (I've made that up), then there has been a 47% decline.

 

[talulahbay]

thankyouu I have some more
1. Find the equation of the tangent to the curve y=4^x+1 at the point (0,4)
2. Find the equation of the normal to the curve y=log3x at the point where x=3
3. find the exact volume of the solid formed when the curve y=logex is rotated about the y axis from y=1 to y=3
4. Find the area between the curve y=lnx, the y axis and the lines y=2 and y=4, correct to 3 significant figures

 

[Sy123]

thankyouu I have some more
1. Find the equation of the tangent to the curve y=4^x+1 at the point (0,4)
2. Find the equation of the normal to the curve y=log3x at the point where x=3
3. find the exact volume of the solid formed when the curve y=logex is rotated about the y axis from y=1 to y=3
4. Find the area between the curve y=lnx, the y axis and the lines y=2 and y=4, correct to 3 significant figures

You can do the rest, using



The rest is just generic like last question







Then you can integrate normally.







Then you can do the rest.

Always make sure to know why I did the steps that I did.

 

[talulahbay]

for the first question i dont really understand why the +1 isn't a power any more, in the question i didn't really write this clearly but its y=4^(x+1) if that changes anything?
and with the second question i had the first bit you wrote its just after that im struggling with the differentation if you could write that bit out please
thankyouu