Questions: Solve for (t).
[ d=-16 t^2+15 t ]
What is the answer?
( t=frac-15 pm sqrt225-64 d-32 )
Transcript text: Solve for $t$.
\[
d=-16 t^{2}+15 t
\]
What is the answer?
$t=\frac{-15 \pm \sqrt{225-64 d}}{-32}$
Solution
Solution Steps
To solve for t in the equation d=−16t2+15t, we recognize it as a quadratic equation in the form at2+bt+c=0. We can use the quadratic formula t=2a−b±b2−4ac to find the values of t. Here, a=−16, b=15, and c=−d.
Step 1: Identify the Quadratic Equation
The given equation is d=−16t2+15t. This is a quadratic equation in the standard form at2+bt+c=0, where a=−16, b=15, and c=−d.
Step 2: Apply the Quadratic Formula
To solve for t, we use the quadratic formula:
t=2a−b±b2−4ac
Substituting the values of a, b, and c, we have:
t=2(−16)−15±152−4(−16)(−d)
Step 3: Calculate the Discriminant
The discriminant is given by:
b2−4ac=152−4(−16)(−d)=225−64d
For d=0, the discriminant simplifies to:
225
Step 4: Solve for t
Substitute the discriminant back into the quadratic formula:
t=−32−15±225
Calculate the two possible values for t:
t1=−32−15+15=0t2=−32−15−15=0.9375