Question

I am supposed to find when there will be 40 berry bushes. I am given the function p(t)=(16t+50t^3)^1/3. P is the population of a certain type of bush in a conservation area and t is time in years.

I know that i am supposed to make p(t) = 40, but i don't know how to solve for t.

Please Help!

Answer 1

40=cubrt (16t+50t^3)

cube both sides.

40^3=16t+50t^3

50t^3+16t-40^3=0

The easy what is to graph the function
y=50t^3+16t-40^3

do that on the graphing calc. Where it crosses zero, read off the t value.

Another way is Newton's solution. I will outline it here:
try t=4
50*64+64-64000= negative number, so increase t to 10

50*1000+160-64000 still negative, t=12
50*1728+192-64000 postive, reduce to t=11
50*1331+171-64000 positive, reduce to 10.5
57881+168-64000, negative, increase to t=10.7
anser slightly negative, try 10.8
answer slightly negative, try 10.85
very close. Close enought t=10.85 years.

with a calc, this back and forth goes really fast.

Answer 2

Thank you, but is there an easier way to do this? This is grade 12 calculus and I don't think I can use newtons solution for my answer if I have never heard of it.