Questions: A projectile is fired vertically, and its height (in feet) after t seconds is given by s(t)=-3 t^2+90 t+525. -Step 2 of 3: At what time will the projectile hit the ground? Round your final answer to the nearest hundredth. Answer seconds

A projectile is fired vertically, and its height (in feet) after t seconds is given by s(t)=-3 t^2+90 t+525.
-Step 2 of 3: At what time will the projectile hit the ground? Round your final answer to the nearest hundredth.

Answer
seconds

Transcript text: A projectile is fired vertically, and its height (in feet) after $t$ seconds is given by $s(t)=-3 t^{2}+90 t+525$. -Step 2 of 3: At what time will the projectile hit the ground? Round your final answer to the nearest hundredth. AnswerHow to enter your answer (opens in new window) 2 Points $\square$ seconds

Solution

Solution Steps

Step 1: Set the Height Equation to Zero

To find when the projectile hits the ground, set the height equation $ s(t) = -3t^2 + 90t + 525 $ equal to zero, since the height at ground level is zero.

\[ -3t^2 + 90t + 525 = 0 \]

Step 2: Solve the Quadratic Equation

Use the quadratic formula to solve for $ t $. The quadratic formula is given by:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation $ -3t^2 + 90t + 525 = 0 $, identify the coefficients: $ a = -3 $, $ b = 90 $, and $ c = 525 $.

Step 3: Calculate the Discriminant

Calculate the discriminant $ b^2 - 4ac $:

\[ b^2 - 4ac = 90^2 - 4(-3)(525) \]

\[ = 8100 + 6300 \]

\[ = 14400 \]

Step 4: Apply the Quadratic Formula

Substitute the values into the quadratic formula:

\[ t = \frac{-90 \pm \sqrt{14400}}{2(-3)} \]

\[ t = \frac{-90 \pm 120}{-6} \]

Step 5: Calculate the Possible Values of $ t $

Calculate the two possible values for $ t $:

$ t = \frac{-90 + 120}{-6} = \frac{30}{-6} = -5 $ (not physically meaningful as time cannot be negative)
$ t = \frac{-90 - 120}{-6} = \frac{-210}{-6} = 35 $

Step 6: Round the Final Answer

Round the physically meaningful solution to the nearest hundredth:

\[ t = 35.00 \]

The projectile will hit the ground at $ 35.00 $ seconds.