Question

1. the population of a species is modeled by the equation $p(t) = -t^4 + 56t^2 + 240$, where $t$ is the number of years. find the approximate number of years until the species is extinct.

$-t^4 + 56t^2 + 240=0$

Explanation:

Step1: Set population to 0

$$-t^4 + 56t^2 + 240 = 0$$

Step2: Substitute $u=t^2$

$$-u^2 + 56u + 240 = 0$$
Rewrite as:
$$u^2 - 56u - 240 = 0$$

Step3: Apply quadratic formula

Quadratic formula: $u=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $au^2+bu+c=0$. Here $a=1, b=-56, c=-240$
$$u=\frac{56\pm\sqrt{(-56)^2-4(1)(-240)}}{2(1)}$$
$$u=\frac{56\pm\sqrt{3136+960}}{2}$$
$$u=\frac{56\pm\sqrt{4096}}{2}$$
$$u=\frac{56\pm64}{2}$$

Step4: Solve for valid $u$

Positive solution only (since $u=t^2\geq0$):
$$u=\frac{56+64}{2}=\frac{120}{2}=60$$

Step5: Solve for $t$

$t=\sqrt{u}=\sqrt{60}\approx7.75$

Answer:

Approximately 8 years (or $\sqrt{60} \approx 7.75$ years)