Question

A soccer player jumps up and heads the ball while it is 7 feet above the ground. If the function

y = -16t squared + 20t + 7 represents this situation, about how long will it take the ball to hit the ground?

Answer 1

3.3 seconds

Explanation:

To find the time it takes for the ball to hit the ground, we need to solve the equation y = 0, since the ball will be on the ground when its height (y) is zero.

The equation given is:

y = -16t^2 + 20t + 7

Setting y to zero:

0 = -16t^2 + 20t + 7

To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring may not be straightforward, so let's use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = -16, b = 20, and c = 7. Substituting these values into the quadratic formula:

t = (-20 ± √(20^2 - 4(-16)(7))) / (2(-16))

t = (-20 ± √(400 + 448)) / (-32)

t = (-20 ± √848) / (-32)

Simplifying further:

t = (-20 ± √(16 * 53)) / (-32)

t = (-20 ± 4√53) / (-32)

Now we have two possible solutions for t:

t1 = (-20 + 4√53) / (-32)

t2 = (-20 - 4√53) / (-32)

Since time cannot be negative in this context, we can disregard the negative solution.

t ≈ (-20 + 4√53) / (-32)

Calculating this value:

t ≈ 3.3 seconds (rounded to one decimal place)

Therefore, it will take approximately 3.3 seconds for the ball to hit the ground.