Theo's flying disc got stuck in a tree 14 feet from the ground. Theo threw his shoe up at the disc to dislodge it. The height in feet h of the shoe is given by the equation h= -16^2+25t+6, where t is the time in seconds. Determine whether the shoe hit the disc. Use the discriminant to explain your answer.

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carlosego carlosego · Answer 1 · 2019-03-25T19:24:44+01:00

Answer:

The shoe hit the disc.

Step-by-step explanation:

Let [tex]ax ^ 2 + bx + c[/tex] be a quadratic function where a, b and c are the real coefficients of the function.

Then the discriminant of the function is:

[tex]b ^ 2 - 4ac[/tex]

The result of this expression gives us information about the roots of this equation.

* If the discriminant is > 0 then the equation has 2 real solutions

* If the discriminant is > 0 then the equation has 2 complex solutions

* If the discriminant is = 0 then the equation has 1 real solution.

For this case we have the equation:

[tex]h = -16t ^ 2 + 25t + 6[/tex]

If we assume that the shoe reached the disc, then [tex]h = 14\ ft[/tex]

So:

[tex]14 = -16t ^ 2 + 25t + 6[/tex]

[tex]-16t ^ 2 + 25t + 6 -14 = 0\\\\-16t ^ 2 + 25t -8 = 0\\\\a = -16\\\\b = 25\\\\c = -8[/tex]

Then the discriminant is:

[tex]25^2 -4(-16)(- 8) = 113[/tex]

The discriminant is greater than 0. Then the equation has 2 real solutions. [tex]t_1[/tex] and [tex]t_2[/tex].

This means that if there are values of t for which [tex]h = 14\ ft[/tex]. In other words, this means that the shoe reached the disc.

In fact if you use the quadratic formula to solve the equation you will get:

[tex]t_1 = 1.11\ s[/tex]

[tex]t_2 = 0.450\ s[/tex]