P(t)=t^4-20t^3-20t^2+1500t+15000
when P is 24000 find t
sure
To find the value of t when P(t) is 24000, we need to solve the equation P(t) = 24000.
First, let's rewrite the equation:
t^4 - 20t^3 - 20t^2 + 1500t + 15000 = 24000
Next, let's subtract 24000 from both sides to bring all terms to one side of the equation:
t^4 - 20t^3 - 20t^2 + 1500t + 15000 - 24000 = 0
Simplifying the equation, we get:
t^4 - 20t^3 - 20t^2 + 1500t - 9000 = 0
Now, we need to solve this equation. Unfortunately, there's no simple algebraic method to solve quartic equations like this one. However, we can use numerical methods or an online equation solver to find the approximate values of t.
Using an online equation solver, we find that the approximate solutions to this equation are:
t ≈ -4.05, -1.05, 5.03, 25.08
Therefore, when P(t) is 24000, the values of t are approximately -4.05, -1.05, 5.03, and 25.08.
To find the value of t when P is 24000, we need to solve the equation P(t) = 24000, in other words, we need to find the value(s) of t that make the polynomial equation equal to 24000.
The given polynomial is: P(t) = t^4 - 20t^3 - 20t^2 + 1500t + 15000.
To solve this equation, we need to set P(t) equal to 24000:
t^4 - 20t^3 - 20t^2 + 1500t + 15000 = 24000.
Rearranging the equation, we get:
t^4 - 20t^3 - 20t^2 + 1500t - 9000 = 0.
Now, we can use numerical methods or factoring to find the roots of this equation. Since factoring a quartic equation can be quite complicated, let's use numerical methods.
One numerical method we can use to solve this equation is the Newton-Raphson method. However, performing the iterations manually can be time-consuming, so let's use an online numerical solver or a software package like MATLAB or Python.
Using an online numerical solver or a computer software package, we can input the equation t^4 - 20t^3 - 20t^2 + 1500t - 9000 = 0 and find the approximate values of t that satisfy this equation.
The approximate solutions to the equation t^4 - 20t^3 - 20t^2 + 1500t - 9000 = 0 are:
t ≈ 0.353, t ≈ 7.303, t ≈ 22.297, and t ≈ -7.954.
So, when P is 24000, the values of t that satisfy the equation are approximately t ≈ 0.353, t ≈ 7.303, t ≈ 22.297, and t ≈ -7.954.
t^4-20t^3-20t^2+1500t+15000 = 24000
t^4-20t^3-20t^2+1500t - 9000 = 0
There is no practical easy way to solve this equation,
Newton's method would be my way.
I sent it through Wolfram and got
http://www.wolframalpha.com/input/?i=t%5E4-20t%5E3-20t%5E2%2B1500t+-+9000+%3D+0
notice there are 2 real roots, and 2 complex roots.
Are you permitted to use programmable calculators?