Question

A football is kicked into the air from an initial height of 3 feet. The height, in feet, of the football above the ground is given by s left-parenthesis t right-parenthesis equals negative 16 t squared plus 35 t plus 3, where t is time in seconds and t greater than or equal to 0. Which is closest to the time when the football will be 20 feet above the ground?

Answer 1

To find when a football will be 20 feet above the ground, use the quadratic formula on the height function s(t) = -16t^2 + 35t + 3 set equal to 20. The calculation yields two times, 0.54 seconds and 3.79 seconds, with the former being the closest time from the initial kick.

Step-by-step explanation:

To determine the time when the football will be 20 feet above the ground, we need to solve the equation s(t) = -16t2 + 35t + 3 for t, when s(t) = 20.

Setting the equation equal to 20 feet gives us: 20 = -16t2 + 35t + 3.

To solve for t, we need to rearrange the equation to 0 = -16t2 + 35t - 17. This is a quadratic equation in the standard form of at2 + bt + c = 0, where a = -16, b = 35, and c = -17.

Applying the quadratic formula, t = (-b ± √(b2 - 4ac)) / (2a), we find two potential values for t. Since time can't be negative in this context, we only consider the positive result.

Use of the quadratic formula yields t = 3.79 s and t = 0.54 s. Because the football reaches 20 feet twice during its trajectory, once on the way up and once on the way down, we acknowledge both times. However, to determine the closest time from the initial kick when the football will be at 20 feet, we would take the smaller value of t that is positive, which is t = 0.54 seconds.