A ball is thrown upward from the top of a 20-meter-high building. Its height
h
h in meters after
t
t seconds is given by:
h
(
t
)
=
−
5
t
2
+
15
t
+
20
h(t)=−5t
2
+15t+20
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Step-by-Step Solution
Step 1: Set height to zero
When the ball hits the ground, its height h(t) will be 0 meters. We set the given height equation equal to 0 to find the time t when this occurs.
Step 2: Formulate the quadratic equation
Substitute h(t)=0 into the given equation to form a quadratic equation in terms of t. This equation will allow us to solve for the time t.
Step 3: Simplify the quadratic equation
To simplify the equation, we can divide the entire equation by −5. This makes the coefficients smaller and easier to work with, without changing the roots of the equation.
Step 4: Factor the quadratic equation
We factor the quadratic equation t2−3t−4=0. We need two numbers that multiply to −4 and add up to −3. These numbers are −4 and 1.
Step 5: Solve for t
From the factored equation (t−4)(t+1)=0, we set each factor equal to zero to find the possible values for t. This gives us t=4 and t=−1.
Step 6: Select the valid time
Since time cannot be negative in this physical context, we discard t=−1 seconds. Therefore, the ball hits the ground after 4 seconds.