. On the reality TV show Last Man Standing, a package is thrown from a plane to the ocean below. The contestants must swim to the package to receive the “free pass” located inside the package. The path that the package follows can be modelled by the quadratic function -4.9t^2+10t+1200 , where d represents the distance, in metres, that the package travels, and t is the time, in seconds.
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To determine the distance the package travels and the time it takes to reach the water, we can use the quadratic function -4.9t^2 + 10t + 1200.
In this function, the coefficient of t^2 (-4.9) represents the acceleration due to gravity, which is negative since it acts in the opposite direction of motion. The coefficient of t (10) represents the initial velocity of the package, and the constant term (1200) represents the initial distance from which the package is thrown.
To find the distance the package travels, we need to evaluate the function for a specific value of t. Let's say we want to find the distance after 5 seconds. We substitute t = 5 into the function:
d = -4.9(5)^2 + 10(5) + 1200
d = -4.9(25) + 50 + 1200
d = -122.5 + 50 + 1200
d = 1127.5 meters
So, the package travels a distance of 1127.5 meters after 5 seconds.
To determine the time it takes for the package to reach the water, we need to set the function equal to zero. This is because when the package reaches the water, its distance will be zero.
-4.9t^2 + 10t + 1200 = 0
To solve this quadratic equation, you can use various methods. One common method is the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
Using the quadratic formula, we can find the values of t when the package reaches the water. Plugging in the values for a, b, and c:
a = -4.9, b = 10, c = 1200
t = (-10 ± sqrt(10^2 - 4(-4.9)(1200))) / (2(-4.9))
After solving this equation, you will get two values for t. Typically, one value will be positive, representing the time it takes for the package to reach the water, and the other value will be negative and can be ignored.
By substituting the positive value of t into the equation, you can find the distance the package travels to reach the water.
Please note that this explanation assumes a few things, such as the absence of air resistance and the direction of the package's motion. It is also important to consider the limitations of the model and its application to a real-world scenario.