Math Snap

STEP 1

1. The rocket's height as a function of time is given by a quadratic equation: $h(t) = -4.9t^2 + 40t + 185$.
2. The rocket splashes down when its height $h(t)$ is zero.
3. The peak height occurs at the vertex of the parabola described by the quadratic function.

STEP 2

1. Determine the time of splashdown by finding when $h(t) = 0$.
2. Determine the time at which the rocket reaches its peak height.
3. Calculate the peak height of the rocket.

STEP 3

Set the height function equal to zero to find the time of splashdown:
$$-4.9t^2 + 40t + 185 = 0$$

STEP 4

Use the quadratic formula to solve for $t$. The quadratic formula is given by:
$$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -4.9$, $b = 40$, and $c = 185$.

STEP 5

Calculate the discriminant $b^2 - 4ac$:
$$b^2 - 4ac = 40^2 - 4(-4.9)(185)$$ $$= 1600 + 3626$$ $$= 5226$$

STEP 6

Substitute the values into the quadratic formula:
$$t = \frac{-40 \pm \sqrt{5226}}{2(-4.9)}$$

STEP 7

Calculate the two possible values for $t$:
$$t = \frac{-40 + \sqrt{5226}}{-9.8} \quad \text{and} \quad t = \frac{-40 - \sqrt{5226}}{-9.8}$$

STEP 8

Since time cannot be negative, we only consider the positive value:
$$t \approx \frac{-40 + 72.31}{-9.8}$$ $$t \approx \frac{32.31}{9.8}$$ $$t \approx 3.30$$

STEP 9

To find the time at which the rocket reaches its peak height, use the vertex formula for a parabola $t = -\frac{b}{2a}$:
$$t = -\frac{40}{2(-4.9)}$$ $$t = \frac{40}{9.8}$$ $$t \approx 4.08$$

STEP 10

Substitute $t = 4.08$ back into the height function to find the peak height:
$$h(4.08) = -4.9(4.08)^2 + 40(4.08) + 185$$

Calculate the peak height:
$$h(4.08) \approx -4.9(16.6464) + 163.2 + 185$$ $$h(4.08) \approx -81.57 + 163.2 + 185$$ $$h(4.08) \approx 266.63$$ The rocket splashes down after $\boxed{3.30}$ seconds. The rocket peaks at $\boxed{266.63}$ meters above sea-level.