To determine the number of months [tex]\(t\)[/tex] in terms of the price [tex]\(p\)[/tex], we need to manipulate the given function [tex]\(p(t) = 600 - 2t^2\)[/tex] to express [tex]\(t\)[/tex] as a function of [tex]\(p\)[/tex].
1. Start with the given equation:
[tex]\[
p = 600 - 2t^2
\][/tex]
2. Isolate the term with [tex]\(t\)[/tex]:
[tex]\[
2t^2 = 600 - p
\][/tex]
3. Divide both sides by 2 to solve for [tex]\(t^2\)[/tex]:
[tex]\[
t^2 = \frac{600 - p}{2}
\][/tex]
4. Simplify the right-hand side:
[tex]\[
t^2 = 300 - 0.5p
\][/tex]
5. Take the square root of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[
t = \sqrt{300 - 0.5p}
\][/tex]
So, the correct expression for the number of months [tex]\(t\)[/tex] in terms of the price [tex]\(p\)[/tex] is:
[tex]\[
t = \sqrt{300 - 0.5p}
\][/tex]
Therefore, the correct option is:
C. [tex]\( \sqrt{300 - 0.5p} \)[/tex]
