A manufacturer of souvenir items makes key rings which sell for $5.80 each. The cost of manufacturing is $1250 to set up the process, and then $0.80 per key ring.
1) Write a rule for the income in dollars, S, from selling n of these key rings.
2) Write a rule for the profit in dollars, P, from selling n of these key rings.
3) Find the number of key rings that must be sold to break even.
1. Y = 5.80n.
2. P = 5.80n - 0.8n - 1250,
P = 5n - 1250.
3. P = 5n - 1250 = 0,
n = 250.
1) The income, S, from selling n key rings can be calculated by multiplying the selling price, $5.80, by the number of key rings sold, n.
Therefore, the rule for the income in dollars is:
S = 5.80n
2) The profit, P, from selling n key rings can be calculated by subtracting the total cost from the income. The total cost includes the setup cost of $1250 and the variable manufacturing cost of $0.80 per key ring.
Therefore, the rule for the profit in dollars is:
P = S - (1250 + 0.80n)
= 5.80n - (1250 + 0.80n)
= 5n - 1250
3) To break even, the profit P must be equal to zero.
Setting P = 0, we can solve for n:
5n - 1250 = 0
5n = 1250
n = 1250/5
n = 250
Therefore, the number of key rings that must be sold to break even is 250.
1) The rule for income, S, from selling n key rings can be written as:
S = 5.80n
Explanation: To find the income from selling n key rings, you need to multiply the selling price ($5.80) by the number of key rings sold (n).
2) The rule for profit, P, from selling n key rings can be written as:
P = (5.80n) - (1250 + 0.80n)
Explanation: To find the profit from selling n key rings, you need to subtract the total cost of manufacturing (setup cost + cost per key ring) from the income. The total cost of manufacturing is given by $1250 as the setup cost and $0.80 per key ring.
3) To find the number of key rings that must be sold to break even, we need to set the profit (P) to zero and solve for n.
0 = (5.80n) - (1250 + 0.80n)
Simplifying the equation:
0.80n - 5.80n = -1250
Combining like terms:
-5n = -1250
Dividing both sides by -5:
n = 250
Explanation: To break even, the profit (P) needs to be zero. Setting the profit equation equal to zero and solving for n gives us the number of key rings that must be sold to break even. In this case, 250 key rings must be sold to cover the setup cost and break even.