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.