An egg is dropped from the window of am building. The height, in feet, of the egg (t)
seconds after it is thrown is represented by d=16^t-7t+61 . How many seconds after the egg is thrown will it be 10 feet from the ground?
The Correct Answer and Explanation is:
To determine how many seconds after the egg is thrown it will be 10 feet from the ground, we start with the equation that represents the height of the egg at any time ( t ):
[
d(t) = -16t^2 – 7t + 61
]
We need to find the time ( t ) when the height ( d(t) ) equals 10 feet. Thus, we set up the equation:
[
-16t^2 – 7t + 61 = 10
]
Next, we simplify this equation:
[
-16t^2 – 7t + 61 – 10 = 0
]
[
-16t^2 – 7t + 51 = 0
]
To make the equation easier to work with, we can multiply through by -1:
[
16t^2 + 7t – 51 = 0
]
Now, we can apply the quadratic formula to solve for ( t ). The quadratic formula is given by:
[
t = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In our case, ( a = 16 ), ( b = 7 ), and ( c = -51 ). We first calculate the discriminant:
[
b^2 – 4ac = 7^2 – 4 \cdot 16 \cdot (-51) = 49 + 3264 = 3313
]
Now we can substitute the values into the quadratic formula:
[
t = \frac{-7 \pm \sqrt{3313}}{2 \cdot 16}
]
Calculating ( \sqrt{3313} ):
[
\sqrt{3313} \approx 57.6
]
Now we have:
[
t = \frac{-7 \pm 57.6}{32}
]
This results in two possible solutions for ( t ):
- ( t_1 = \frac{-7 + 57.6}{32} \approx \frac{50.6}{32} \approx 1.58 )
- ( t_2 = \frac{-7 – 57.6}{32} \approx \frac{-64.6}{32} \approx -2.02 )
Since time cannot be negative, we disregard ( t_2 ). Thus, the egg will be 10 feet from the ground approximately 1.58 seconds after it is thrown.
Explanation
This calculation demonstrates how we can model the motion of an object under the influence of gravity using a quadratic function. The height of the egg is expressed as a function of time, and we apply algebraic techniques, including setting the height to a specific value, rearranging the equation, and utilizing the quadratic formula to find the time when the egg reaches that height. By isolating the variable and employing the discriminant to determine the nature of the roots, we ensure that our solutions are valid within the context of the problem, which is critical in physical scenarios.