Question

A ball is thrown upward from the top of a building. Its height \(h,\) in feet, after \(t\) seconds is given by the equation \(h=-16 t^{2}+96 t+200 .\) How long will it take for the ball to be \(88 \mathrm{ft}\) above the ground?

Short answer

The ball will be 88 feet above the ground after 7 seconds.

Step-by-step solution

Set up the equation

Given the height equation of the ball is \[h = -16t^2 + 96t + 200\] We are to find the time, \(t\), when the height, \(h\), is 88 feet. So, set \(h = 88\) and form the equation: \[-16t^2 + 96t + 200 = 88\]

Move all terms to one side of the equation

Subtract 88 from both sides to set the equation to zero: \[-16t^2 + 96t + 200 - 88 = 0\] Simplify to get: \[-16t^2 + 96t + 112 = 0\]

Simplify the equation

Divide every term by -16 to simplify the equation: \[t^2 - 6t - 7 = 0\]

Factor the quadratic equation

Factor the quadratic equation \[t^2 - 6t - 7 = 0\] The factors of -7 that add up to -6 are -7 and 1. So, the equation factors to: \[(t - 7)(t + 1) = 0\]

Solve for t

Set each factor equal to zero: \[t - 7 = 0\] \[t + 1 = 0\] Therefore, \(t = 7\) or \(t = -1\). Since time cannot be negative, we discard \(t = -1\).

Conclusion

The time when the ball is 88 feet above the ground is \(t = 7\) seconds.

Key concepts

solving quadratic equations

Quadratic equations are equations of the form \[-ax^2 + bx + c = 0\].
In the given exercise, the height of a ball thrown upward is modeled by a quadratic equation.
To solve these types of equations, we typically bring all terms to one side so the equation equals zero.
After this, different methods like factoring, completing the square, or using the quadratic formula can be applied.
In our case, after setting the height \[h = 88\] feet, we form the equation
\[-16t^2 + 96t + 200 = 88\].
From there, we move all terms to one side of the equation:
\[-16t^2 + 96t + 112 = 0\].
We can use factoring, making the equation easier to solve.
Quadratic equations often have two solutions, but only solutions that make sense in the given context (e.g., time can't be negative) are considered valid.

factoring

Factoring is a method for solving quadratic equations where we express the quadratic as a product of two binomials.
Consider the equation we simplified earlier:
\[t^2 - 6t - 7 = 0\].
Factoring involves finding two numbers that multiply to give the constant term \[-7\], and add to give the coefficient of the linear term \[-6\].
In this case, \[-7\] and \[1\] do the trick:
\[(t - 7)(t + 1) = 0\].
Setting each factor to zero gives us the potential solutions \[t - 7 = 0\] and \[t + 1 = 0\].
Solving these, we find \[t = 7\] and \[t = -1\].
Since time can't be negative, we discard \[t = -1\] and keep \[t = 7\] as our solution.
Factoring is especially useful when the quadratic can be easily broken down into integer factors.

physics-based problems

Quadratic equations often appear in physics, modeling various real-life phenomena.
In the given exercise, we deal with the motion of a ball thrown upwards.
The ball’s height is modeled by the quadratic equation
\[h = -16t^2 + 96t + 200\].
Here, the coefficients represent physical concepts:

• \[-16\] represents the effect of gravity (in ft/s²)
• \[96\] is the initial velocity (in ft/s)
• \[200\] is the initial height (in ft)

We are finding the time it takes for the ball to reach 88 feet.
By substituting and solving the quadratic equation, we determine that it takes \[t = 7\] seconds.
These kinds of problems help students understand the application of mathematics in real-world scenarios.