Answer:
1.5 feet
Explanation:
Let’s analyze the situation. The height of the blades at any time t is given by the function h(t)=2.5t2−3t+2.7
We need to find out if there’s a time t when the height of the blades is less than or equal to 1.5 feet. To do this, we can set h(t) equal to 1.5 and solve for t:
2.5t2−3t+2.7=1.5
2.5t2−3t+1.2=0
This is a quadratic equation of the form at^2 + bt + c = 0, where a = 2.5, b = -3, and c = 1.2. The solutions for t can be found using the quadratic formula:
t=2a−b±b2−4ac
Substituting a, b, and c into the formula gives:
t=2∗2.53±(−3)2−4∗2.5∗1.2
t=53±9−12
Since the term under the square root is negative, there are no real solutions for t. This means that the height of the blades is always greater than 1.5 feet.
Therefore, Illinois Joan cannot navigate through the hallway safely by staying as low as 1.5 feet. To find out how much lower she needs to go, we need to find the minimum value of h(t).
The minimum value of a parabola y=at2+bt+c
occurs at t=−2ab
Substituting a = 2.5 and b = -3 gives:
t=−2∗2.5−3=0.6
Substituting t = 0.6 into h(t) gives the minimum height of the blades:
h(0.6)=2.5∗(0.6)2−3∗0.6+2.7=1.98
So, the blades reach a minimum height of 1.98 feet off the ground. This means that Illinois Joan needs to go at least 1.98 - 1.5 = 0.48 feet lower to navigate through the hallway safely. So, she needs to crawl lower than 1.5 feet to avoid the blades. Specifically, she needs to stay below 1.5 feet from the ground.
If you have any other questions or need further assistance, feel free to ask.