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2. Elaine shoots an arrow upward at a speed of 32 feet per second from a bridge that is 28 feet high.
The height of the arrow is given by the function h(t) = -16t^2 + 32t + 28, where t is the time in seconds.
a. What is the maximum height that the arrow reaches?
b. How long does it take the arrow to reach its maximum height?
c. How long would it take before the arrow reached the ground? Round your answer to the hundredths place.
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To solve the problem, it is better to change the sequence of questions from a), b), and c) to b), a), and c). So, we start from b).
b. How long does it take the arrow to reach its maximum height?
The time "t" when the arrow reaches its maximum height is that "t" where the parabola (the quadratic function h(t) = )
reaches the maximum. It is the parabola vertex.
From Algebra you know that this "t" is t = = = 1.
Here I refer to the formula for the vertex of the parabola of the general form h(t) = . In your case a = -16, b = 32.
So, your answer for the time "t" is t = 1 seconds.
The arrow reaches its maximum height in 1 second.
After 1 second it will start its movement down.
So, b) is answered.
a. What is the maximum height that the arrow reaches?
Above you just found that the maximum height is reached at t = 1 seconds.
So, to find the maximum height, simply substitute t=1 into the parabola equation:
= = = -16 + 32 + 28 = 44.
44 feet above the ground level. It is the maximum height of the arrow.
Did I calculate it correctly? Check it please.
If "Yes", then a) is answered.
c. How long would it take before the arrow reached the ground?
The arrow reaches the ground when its height above the ground becomes zero: h(t) = 0.
It means that your quadratic function get the zero value = at that t.
So, you need to find the roots of the quadratic equation
= .
Do it by applying the quadratic formula.
The quadratic formula will give you two values.
Choose the value which is positive. Only positive "t" can be the solution.
I got the value t = 3.89 (after rounding).
What is your value?
If it is the same, then the question c) is answered too and the entire problem is solved.
Do the other problems in the same manner. They all are very similar to that.