Questions: Solve for (t). [ d=-16 t^2+15 t ] What is the answer? ( t=frac-15 pm sqrt225-64 d-32 )

Solution Steps

To solve for $t$ in the equation $d = -16t^2 + 15t$, we recognize it as a quadratic equation in the form $at^2 + bt + c = 0$. We can use the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the values of $t$. Here, $a = -16$, $b = 15$, and $c = -d$.

The given equation is $d = -16t^2 + 15t$. This is a quadratic equation in the standard form $at^2 + bt + c = 0$, where $a = -16$, $b = 15$, and $c = -d$.

To solve for $t$, we use the quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substituting the values of $a$, $b$, and $c$, we have: $$t = \frac{-15 \pm \sqrt{15^2 - 4(-16)(-d)}}{2(-16)}$$

The discriminant is given by: $$b^2 - 4ac = 15^2 - 4(-16)(-d) = 225 - 64d$$ For $d = 0$, the discriminant simplifies to: $$225$$

Substitute the discriminant back into the quadratic formula: $$t = \frac{-15 \pm \sqrt{225}}{-32}$$ Calculate the two possible values for $t$: $$t_1 = \frac{-15 + 15}{-32} = 0$$ $$t_2 = \frac{-15 - 15}{-32} = 0.9375$$

Final Answer