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!
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.
To find when there will be 40 berry bushes, you need to solve the equation p(t) = 40 for t. Given the function p(t) = (16t + 50t^3)^(1/3), we can substitute p(t) with 40 to get:
(16t + 50t^3)^(1/3) = 40
Now, let's solve this equation step by step:
1. Cube both sides of the equation to eliminate the cube root:
[(16t + 50t^3)^(1/3)]^3 = 40^3
This simplifies to:
16t + 50t^3 = 64,000
2. Re-arrange the equation in the standard form of a cubic equation:
50t^3 + 16t - 64,000 = 0
3. Now, you need to solve this cubic equation. Unfortunately, cubic equations do not have a general formula for solving, but you can use numerical methods or calculators to find an approximation of the roots. One popular numerical method is the Newton-Raphson method or you can use online calculators specifically designed for solving cubic equations.
Once the equation is solved, you will obtain the values of t when there will be 40 berry bushes.
To solve for t when p(t) = 40, you need to substitute 40 for p(t) in the given function and solve for t. Let's go step by step:
1. Start with the given function: p(t) = (16t+50t^3)^(1/3).
2. Replace p(t) with 40: 40 = (16t+50t^3)^(1/3).
3. Cube both sides of the equation to eliminate the cube root: (40)^3 = (16t+50t^3).
4. Simplify the left side of the equation: 64000 = 16t + 50t^3.
5. Rearrange the equation to isolate the cubic term: 50t^3 + 16t - 64000 = 0.
6. This equation is in cubic form. Unfortunately, solving cubic equations does not have a straightforward algebraic solution. However, you can use numerical methods or calculators to find an approximate solution.
To find an approximate solution:
- You can use graphing calculators or online tools that can solve equations numerically.
- Alternatively, you can use numerical methods like Newton's method, which involves iterations and might require some programming or a calculator.
Remember that the solution will give you an approximate value for t, and it might not be an exact value due to the nature of solving cubic equations.
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.