Hugo lies on top of a building, throwing pennies straight down to the street below. The formula for the height in meters, H, that a penny falls is `H=Vt+5t^(2)`, where V is the original velocity of the penny (how fast Hugo throws it as it leaves his hand in meters per second) and t is equal to the time it takes to hit the ground in seconds. The building is 60 meters high, and Hugo throws the penny down at an initial speed of 20 meters per second. How long does it take for the penny to hit the ground?

Answer

To solve the problem, we will follow these steps:

### Step 1: Write down the given formula and values
We have the formula for the height \( H \) of the penny as it falls:
\[ H = Vt + 5t^2 \]
Where:
- \( H \) = height (60 meters)
- \( V \) = initial velocity (20 meters per second)
- \( t \) = time in seconds (unknown)

### Step 2: Substitute the known values into the equation
We substitute \( H = 60 \) and \( V = 20 \) into the equation:
\[ 60 = 20t + 5t^2 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ 5t^2 + 20t - 60 = 0 \]

### Step 4: Simplify the equation
We can simplify the equation by dividing all terms by 5:
\[ t^2 + 4t - 12 = 0 \]

### Step 5: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to \(-12\) (the constant term) and add to \(4\) (the coefficient of \(t\)). The numbers are \(6\) and \(-2\). Thus, we can write:
\[ (t + 6)(t - 2) = 0 \]

### Step 6: Solve for \( t \)
Setting each factor equal to zero gives us:
1. \( t + 6 = 0 \) → \( t = -6 \) (not valid since time cannot be negative)
2. \( t - 2 = 0 \) → \( t = 2 \)

### Conclusion
The time it takes for the penny to hit the ground is:
\[ t = 2 \text{ seconds} \]