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
To solve this problem, we need to understand that the time taken for the committee meeting is partly constant and partly varies as the square of the number of members present.
Let's assume that the constant time taken for the committee meeting is "c" and the time taken that varies with the square of the number of members is "k * n^2", where "n" is the number of members present and "k" is a constant.
Given that with fifteen members, the meeting lasts 45 minutes, we can write the equation as:
c + k * 15^2 = 45
Simplifying this equation, we get:
c + k * 225 = 45
c + 225k = 45
Now, given that with twenty-five members, the meeting takes 2 hours 15 minutes (which can be written as 2*60 + 15 = 135 minutes), we can write the equation as:
c + k * 25^2 = 135
Simplifying this equation, we get:
c + k * 625 = 135
c + 625k = 135
We now have a system of two equations:
c + 225k = 45
c + 625k = 135
To solve this system, we can subtract the first equation from the second equation:
(c + 625k) - (c + 225k) = 135 - 45
400k = 90
Dividing both sides by 400, we get:
k = 9/40
Now that we have the value of "k", we can substitute it back into one of the original equations to find the value of "c". Let's use the first equation:
c + 225k = 45
c + 225*(9/40) = 45
c + (225*9)/40 = 45
c + 2025/40 = 45
Multiplying both sides by 40 to get rid of the fraction, we get:
40c + 2025 = 1800
40c = 1800 - 2025
40c = -225
Dividing both sides by 40, we get:
c = -225/40
c = -5.625
Now that we have the values of "c" and "k", we can find the time taken for the meeting with thirty members by using the equation:
Time = c + k * 30^2
Time = -5.625 + (9/40) * 30^2
Time = -5.625 + (9/40) * 900
Time = -5.625 + 2025/40
Time = -5.625 + 50.625
Time = 45
Therefore, the meeting will last 45 minutes if there are thirty members present.
The correct answer is not provided in the options, so it cannot be determined based on the given information.
To find the time taken for a committee meeting with thirty members, we can set up a system of equations using the given information.
Let's denote the constant time as "a" and the variable time as "b." We need to find the value of "b" when there are thirty members.
From the given information, we have two equations:
Equation 1: a + 15^2b = 45 minutes
Equation 2: a + 25^2b = 2 hours 15 minutes = 135 minutes
Now, let's substitute the value of "a" from Equation 1 into Equation 2:
(45 - 15^2b) + 25^2b = 135
Simplify the equation:
45 - 225b + 625b = 135
400b = 90
b = 90/400
b = 9/40
Now, substitute the value of "b" back into Equation 1 to find "a":
a + 15^2(9/40) = 45
(40a + 15^2(9)) / 40 = 45
(40a + 15^2(9)) = 1800
40a + 15^2(9) = 1800
40a + 15^2(9) = 1800
40a + 2025 = 1800
40a = 1800 - 2025
40a = -225
a = -225/40
a = -9/8
Now, we have the values of "a" and "b":
a = -9/8
b = 9/40
To find the time taken for a committee meeting with thirty members, substitute the values of "a" and "b" into the equation:
-9/8 + 30^2(9/40) = x
-9/8 + 900(9/40) = x
(-9/8) + (81/8) = x
72/8 = x
9 = x
Therefore, the meeting will last for 9 hours.
Answer: A. 9 hrs