Questions: Emilio throws a marshmallow into the air from his balcony. The height of the marshmallow (in feet) is represented by the equation h=-16t^2+8t+48, where t is the time (in seconds) after he throws the marshmallow.
How long does it take the marshmallow to hit the ground, where h=0?
0.5 seconds
3 seconds
2 seconds
1.5 seconds
Transcript text: Quiz Instructions
Question 18
4 pts
Emilio throws a marshmallow into the air from his balcony. The height of the marshmallow (in feet) is represented by the equation $h=-16 t^{2}+8 t+48$, where $t$ is the time (in seconds) after he throws the marshmallow.
How long does it take the marshmallow to hit the ground, where $\mathrm{h}=0$ ?
0.5 seconds
3 seconds
2 seconds
1.5 seconds
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Solution
Solution Steps
Step 1: Set the height equation to zero
Set the given height equation \( h = -16t^2 + 8t + 48 \) to zero to find the time when the marshmallow hits the ground:
\[ 0 = -16t^2 + 8t + 48 \]
Step 2: Solve the quadratic equation
Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( t \), where \( a = -16 \), \( b = 8 \), and \( c = 48 \):
\[ t = \frac{-8 \pm \sqrt{8^2 - 4(-16)(48)}}{2(-16)} \]
Step 3: Simplify the expression
Calculate the discriminant and simplify the expression:
\[ t = \frac{-8 \pm \sqrt{64 + 3072}}{-32} \]
\[ t = \frac{-8 \pm \sqrt{3136}}{-32} \]
\[ t = \frac{-8 \pm 56}{-32} \]
Step 4: Find the possible values of \( t \)
Calculate the two possible values for \( t \):
\[ t = \frac{-8 + 56}{-32} = \frac{48}{-32} = -1.5 \]
\[ t = \frac{-8 - 56}{-32} = \frac{-64}{-32} = 2 \]
Step 5: Select the positive value
Since time cannot be negative, select the positive value:
\[ t = 2 \]
Thus, it takes 2 seconds for the marshmallow to hit the ground.