To solve this problem, we need to find a relationship between the fixed cost of hiring the conference facility and the variable cost that depends on the number of participants.
Let's assume:
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The fixed cost (constant) is \( C_f \).
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The variable cost per participant is \( C_v \).
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The total cost \( C(n) \) for \( n \) participants is given by the equation:
\[ C(n) = C_f + C_v \times n \]
We have two scenarios given in the problem:
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For 100 participants, the cost is Kshs 45000: \[ C(100) = C_f + C_v \times 100 = 45000 \tag{1} \]
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For 60 participants, the cost is Kshs 40000: \[ C(60) = C_f + C_v \times 60 = 40000 \tag{2} \]
We can create a system of equations from these two scenarios. First, let's subtract equation (2) from equation (1):
\[ (C_f + C_v \times 100) - (C_f + C_v \times 60) = 45000 - 40000 \]
This simplifies to:
\[ C_v \times 40 = 5000 \]
From this, we can find \( C_v \):
\[ C_v = \frac{5000}{40} = 125 \]
Now we can substitute \( C_v \) back into either equation to solve for \( C_f \). Using equation (2):
\[ C_f + 125 \times 60 = 40000 \]
Calculating \( 125 \times 60 \):
\[ C_f + 7500 = 40000 \]
Now, solving for \( C_f \):
\[ C_f = 40000 - 7500 = 32500 \]
Now we have both costs:
- Fixed cost \( C_f = 32500 \)
- Variable cost \( C_v = 125 \)
Now, we can use these to find the number of participants \( n \) if Kshs 63000 is used:
\[ C(n) = C_f + C_v \times n \]
Setting \( C(n) = 63000 \):
\[ 63000 = 32500 + 125 \times n \]
Solving for \( n \):
\[ 63000 - 32500 = 125 \times n \]
This simplifies to:
\[ 30500 = 125 \times n \]
Now divide both sides by 125:
\[ n = \frac{30500}{125} = 244 \]
Therefore, the number of participants \( n \) if Kshs 63000 is used to hire the facility is 244 participants.