9.7 Linear and Other Relationships
CA-251
2. The rocket scientists determined that another equation for the height of the rocket is
\[
h=2304-16(t-10)^{2}
\]
a. Describe the structure of the expression on the right side of the equal sign.
b. Reason about the structure of $2304-16(t-10)^{2}$ to determine the largest value it can have and to determine the value of $t$ at which this occurs.
To help your thinking, consider these questions:
-Why can $16(t-10)^{2}$ never be negative?
- How can you use the structure of $16(t-10)^{2}$ to determine for which value of $t$ it is 0 ?
c. Why would the rocket scientists want to know the largest value of the expression?
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Step 1 ：First, we understand the structure of the expression on the right side of the equation. It is a quadratic equation in the form of \(a(t-h)^2 + k\), where \(a = -16\), \(h = 10\), and \(k = 2304\). This form represents a parabola that opens downwards because \(a\) is negative.

Step 2 ：We know that \(16(t-10)^2\) can never be negative because it is a square of a real number and the square of any real number is always non-negative.

Step 3 ：The largest value of the expression \(2304-16(t-10)^2\) occurs when \(16(t-10)^2 = 0\), because subtracting anything from 2304 will make it smaller. And \(16(t-10)^2 = 0\) when \(t = 10\).

Step 4 ：So, the largest value of the expression \(2304-16(t-10)^2\) is \(2304\) and it occurs when \(t = 10\).

Step 5 ：The rocket scientists would want to know the largest value of the expression because it represents the maximum height the rocket can reach.