Question 387764
{{{h(t) = -9.8t^2 - 10t + 350}}}
Height is measured from the ground up. So when the stone hits the ground, its height will be zero. So the equation to solve is:
{{{0 = -9.8t^2 - 10t + 350}}}
This is a quadratic equation. We already have a zero on one side so we proceed to the next step: Factor or use the Quadratic Formula. The expression on the right side will not factor so we must use the Quadratic Formula:
{{{t = (-(-10) +- sqrt((-10)^2 - 4(-9.8)(350)))/2(-9.8)}}}
which simplifies as follows:
{{{t = (-(-10) +- sqrt(100 - 4(-9.8)(350)))/2(-9.8)}}}
{{{t = (-(-10) +- sqrt(100 + 13720))/2(-9.8)}}}
{{{t = (-(-10) +- sqrt(13820))/2(-9.8)}}}
{{{t = (10 +- sqrt(13820))/(-19.6)}}}
{{{t = (10 +- sqrt(4*3455))/(-19.6)}}}
{{{t = (10 +- sqrt(4)*sqrt(3455))/(-19.6)}}}
{{{t = (10 +- 2*sqrt(3455))/(-19.6)}}}
{{{t = (100 +- 20*sqrt(3455))/(-196)}}}
{{{t = (4(25 +- 5*sqrt(3455)))/(4*(-49))}}}
{{{t = (cross(4)(25 +- 5*sqrt(3455)))/(cross(4)*(-49))}}}
{{{t = (25 +- 5*sqrt(3455))/(-49)}}}
In long form this is:
{{{t = (25 + 5*sqrt(3455))/(-49)}}} or {{{t = (25 - 5*sqrt(3455))/(-49)}}}
The first equation gives us a negative value for t (because the numerator will be positive and the denominator is negative). Since time cannot be negative, we will reject the solution from that equation. So the only meaningful solution is:
{{{t = (25 - 5*sqrt(3455))/(-49)}}}
This is an exact expression for the solution. If you want a decimal approximation then get out your calculator, find the square root and simplify the expression.