Final answer:
To find how long it takes for the helicopter to rise to 32 feet, solve the equation 4t^2 + 8t = 32. Using the quadratic formula, the two possible solutions are t = 2 and t = -4. Since time cannot be negative, the helicopter will take 2 seconds to rise to a height of 32 feet.
Step-by-step explanation:
To find how long it takes for the helicopter to rise to 32 feet, we need to solve the equation s(t) = 32, where s(t) represents the distance from the ground at time t. Substituting s(t) = 32 into the given equation s(t) = 4t^2 + 8t, we get 4t^2 + 8t = 32. Rearranging the equation, we have 4t^2 + 8t - 32 = 0. This is a quadratic equation that we can solve using the quadratic formula.
Using the quadratic formula, we get t = (-b ± √(b^2 - 4ac)) / (2a), where a = 4, b = 8, and c = -32. Plugging in these values, we have t = (-8 ± √(8^2 - 4(4)(-32))) / (2(4)). Simplifying the expression under the square root, we get t = (-8 ± √(64 + 512)) / 8. This further simplifies to t = (-8 ± √576) / 8. Taking the square root of 576, we get t = (-8 ± 24) / 8. This gives us two possible solutions: t = (-8 + 24) / 8 = 2 and t = (-8 - 24) / 8 = -4.
Since time cannot be negative in this context, we discard the negative solution. Therefore, it will take the helicopter 2 seconds to rise to a height of 32 feet.