The production cost $ C of a watch is partly constant and partly inversely as the number of watches n produced in an hour . When 250 watches are produced per hour , the production cost of each watch is $ 90 ; and when 500 watches are produced per hour , the production cost of each watch is $ 80 .
a ) Express C in terms of n .
b ) Find the cost , if 1000 of watches were produced .
C = a + b/w
Now you know that
a + b/250 = 90
a + b/500 = 80
Solve for a and b, then find C(1000)
To express C in terms of n, we can use the information given in the problem.
Let's start by breaking down the information into equations.
When 250 watches are produced per hour, the production cost of each watch is $90:
C = k + (p/n), where k is the constant cost and p is the proportionality constant.
Substituting the given values, we have:
90 = k + (p/250) ----(equation 1)
Similarly, when 500 watches are produced per hour, the production cost of each watch is $80:
C = k + (p/n)
Substituting the given values:
80 = k + (p/500) ----(equation 2)
To find the value of k and p, we need to solve these two equations.
Let's rearrange equation 1 and 2 to solve them simultaneously:
Equation 1:
90 - k = p/250
250(90 - k) = p ----(equation 3)
Equation 2:
80 - k = p/500
500(80 - k) = p ----(equation 4)
Now, we can set equations 3 and 4 equal to each other to find the value of p:
250(90 - k) = 500(80 - k)
22500 - 250k = 40000 - 500k
Simplifying this equation, we get:
250k - 500k = 40000 - 22500
-250k = 17500
Dividing both sides by -250, we get:
k = -70
Plugging the value of k in equation 1 or 2, we can find the value of p:
90 = -70 + (p/250)
160 = p/250
p = 40000
Now, we have the value of k and p, so we can express C in terms of n:
C = -70 + (40000/n)
Now, to find the cost if 1000 watches were produced, we can substitute n = 1000 into the expression we found for C:
C = -70 + (40000/1000)
C = -70 + 40
C = -30
Therefore, the cost of producing 1000 watches would be -$30.
To express C in terms of n, we can use the given information to set up an equation.
Let's start by analyzing the information provided:
When 250 watches are produced per hour, the production cost of each watch is $90.
When 500 watches are produced per hour, the production cost of each watch is $80.
From this, we can deduce that as the number of watches produced per hour increases, the production cost per watch decreases. This suggests an inverse relationship between the number of watches produced and the production cost per watch.
Let's denote the constant part of the production cost as A. Since the constant part is the same regardless of the number of watches produced, we can express C, the production cost, as:
C = A + (B/n)
Where B represents the inverse proportionality between the production cost and the number of watches produced.
Now, let's use the given information to find the values of A and B. We have two equations:
When n = 250, C = 90
When n = 500, C = 80
Plugging these values into our equation, we get two equations:
90 = A + (B/250)
80 = A + (B/500)
To solve for A and B, we can subtract the second equation from the first equation:
90 - 80 = (A + (B/250)) - (A + (B/500))
10 = B/250 - B/500
10 = (2B - B) / 500
10 = B / 500
B = 5000
Now that we have the value of B, we can substitute it back into one of the equations to find A:
80 = A + (5000/500)
80 = A + 10
A = 70
So, we have found that A = 70 and B = 5000. Now, we can express C in terms of n:
C = 70 + (5000/n)
To find the cost when 1000 watches are produced (n = 1000), we substitute n = 1000 into our equation:
C = 70 + (5000/1000)
C = 70 + 5
C = $75
Therefore, the cost of producing 1000 watches would be $75.