I am struggling with this question and need a help!
In a science experiment, a student is studying the relationship between the temperature and the growth of a plant. She develops the following quadratic model: $h = -0.0625t^2 + 3t - 20$, where h is the height of the plant, in cm, and t is the growing temperature, in °C.
Determine the temperatures that result in a plant height of zero.
Use your result from part a) to find the temperature that will maximize the height of the plant.
Use the equation to determine the height of the plant grown at a temperature of $5$ °C. Comment on the domain of the mathematical model.
Use your results above to construct a graph for this relation.
1. Temperatures that result in zero height:
In this case, let $h=0$ and solve the quadratic equation for $t$.
We have: $$0=-0.0625t^2+3t-20$$
Multiply both sides by $-16$ to get:$$0=t^2-48t+320$$
And then you could factor and solve:
$$0=t^2-48t+320$$ $$0=(t-40)(t-8)$$
Now, we know that $0=t-40$ or $0=t-8$, so $t = 8, 40$.
So, the plant will have zero height at $8$ and $40$ degrees.
2. Temperature that maximizes plant height:
All you need to do is find where the vertex of the equation $h=-0.0625t^2+3t-20$ is when you graph it.
The $t$-value of the vertex will be the average of the two solutions you found to part 1.
3. Height at $5$ degrees C:
Plug in $5$ for $t$ into the model and simplify.
$h=-0.0625(5)^2+3(5)-20$
(the rest is just numerical calculations)
4. Graph:
I can't really help you with this one, but you should find that the graph is an upside-down "U" shape.