The equation is y=80.4- 11 ln x, 100<x<1500 Which approximates the minmum required ventilation rate in terms of the air sapace per child in a public school class room. In the model, x is the air space per child in cubic feet and y is the ventilation rate per child in cubic ft. per minute.
A classroom is designed for 30 students. The air conditioning system in the room has the capacity of moving 450 cubic feet of air per minute.
A) Determine the ventilation rate per child, assuming that the room is filled to capacity.
B) Estimate the air space required per child
C) Determine the minimum number of square feet of floor space required for the room if the ceiling height is 30 feet.
To solve this problem, we'll use the given equation y = 80.4 - 11 ln(x), where x is the air space per child in cubic feet and y is the ventilation rate per child in cubic ft. per minute.
A) To determine the ventilation rate per child, assuming the room is filled to capacity with 30 students, we'll substitute x = 450/30 = 15 in the equation:
y = 80.4 - 11 ln(15)
Calculate the ventilation rate by evaluating this expression:
y ≈ 80.4 - 11 * 2.708
y ≈ 80.4 - 29.788
y ≈ 50.612 cubic ft. per minute
B) To estimate the air space required per child, we'll rearrange the equation for x:
y = 80.4 - 11 ln(x)
11 ln(x) = 80.4 - y
ln(x) = (80.4 - y)/11
x = e^((80.4 - y)/11)
Substitute y = 450/30 = 15 to find x:
x ≈ e^((80.4 - 15)/11)
Calculate the air space per child by evaluating this expression:
x ≈ e^(65.4/11)
x ≈ e^5.945
x ≈ 381.22 cubic ft.
C) To determine the minimum number of square feet of floor space required for the room if the ceiling height is 30 feet, we need to consider the volume that can be occupied by 30 students. Since the room has a standardized air conditioning capacity of 450 cubic ft. per minute, we can calculate the minimum floor space required as follows:
Volume of room = 30 * x * 30 (assuming the room is rectangular)
Set this volume equal to the air conditioning capacity:
30 * x * 30 = 450
Solve for x:
x = 450 / (30 * 30)
Calculate the floor space required by evaluating this expression:
x ≈ 450 / 900
x ≈ 0.5 square ft.
Therefore, the minimum number of square feet of floor space required for the room is approximately 0.5 square ft.