The time taken for a
committee meeting is partly
constant and partly varies as the
square of the number of members
present. If there are fifteen
members present, the meeting
lasts only 45 minutes, but with
twenty-five it takes exactly 2hrs
15 minutes. How long will it last if
there are thirty members there?
A. 3hrs
B. 3hrs 17mins
C. 2hrs 19mins
D. 1 hr 18mins
Let's assume the constant time for the committee meeting is x minutes and the time that varies with the square of the number of members present is y minutes.
From the given information, we can form two equations:
Equation 1: x + 15^2 * y = 45
Equation 2: x + 25^2 * y = 135
Simplify equation 1:
x + 225y = 45
Simplify equation 2:
x + 625y = 135
Subtract equation 1 from equation 2 to eliminate x:
625y - 225y = 135 - 45
400y = 90
y = 0.225
Substitute y value back into equation 1 to find x:
x + 225 * 0.225 = 45
x + 50.625 = 45
x = 45 - 50.625
x = -5.625
Now, we have the values of x = -5.625 and y = 0.225. We can use these values to find the time for 30 members.
Equation 3: -5.625 + 30^2 * 0.225 = t
-5.625 + 900 * 0.225 = t
-5.625 + 202.5 = t
t = 196.875
Therefore, the meeting will last approximately 196.875 minutes, which is equal to 3 hours and 16.875 minutes. Rounding to the nearest minute, the answer is:
B. 3 hours 17 minutes
To solve this problem, we need to find the relationship between the number of members present and the time taken for the meeting.
Let's denote the constant part of the meeting time as C and the variable part as k * n^2, where k is a constant and n is the number of members present.
We are given two data points:
- When 15 members are present, the meeting lasts 45 minutes, so we have the equation 45 = C + k * 15^2.
- When 25 members are present, the meeting lasts 2 hours 15 minutes, which is equivalent to 135 minutes. So we have the equation 135 = C + k * 25^2.
First, let's solve the first equation for C:
45 = C + k * 225
C = 45 - k * 225
Substitute this value of C into the second equation:
135 = (45 - k * 225) + k * 625
135 = 45 - k * 225 + k * 625
90 = k * 625 - k * 225
90 = k * (625 - 225)
90 = 400k
k = 90 / 400
k = 0.225
Now that we have the value of k, we can find the constant part C:
C = 45 - k * 225
C = 45 - 0.225 * 225
C = 45 - 50.625
C = -5.625
Now, we can find the meeting time for 30 members:
Time = C + k * 30^2
Time = -5.625 + 0.225 * 900
Time = -5.625 + 202.5
Time = 196.875
Since 196.875 minutes is equivalent to 3 hours 17 minutes, the answer is:
B. 3hrs 17mins