For some word problems, nothing more will be required of you than to substitute a given value into a formula, which is either given to you or is readily available. The most difficult part of these problems will be to decide which variable the given quantity will be. For example, the formula might look like R = 8q and the value given to you is 440. Is R = 440 or is q = 440? The answer lies in the way the variables are described In R ¼ 8q, it might be that R represents revenue (in dollars) and q represents quantity (in units) sold of some item. ‘‘If 440 units were sold, what is the revenue?’’ Here 440 is q. You would then solve R = 8(440). ‘‘If the revenue is $440, how many units were sold?’’ Here 440 is R, and you would solve 440 = 8q.
Examples
The cost
formula for a manufacturer’s product is C ¼ 5000 þ 2x, where C is the cost (in
dollars) and x is the number of units manufactured.
(a) If
no units are produced, what is the cost?
(b) If
the manufacturer produces 3000 units, what is the cost?
(c) If
the manufacturer has spent $16,000 on production, how many units were
manufactured?
Answer these questions by substituting the
numbers into the formula.
(a) If
no units are produced, then x = 0, and C = 5000 + 2x becomes C = 5000 + 2(0) = 5000.
The cost is $5,000.
(b) If
the manufacturer produces 3000 units, then x = 3000, and C = 5000 + 2x becomes C
= 5000 + 2(3000) = 5000 + 6000 = 11.000. The manufacturer’s cost would be
$11,000.
(c) The
manufacturer’s cost is $16,000, so C = 16,000. Substitute C = 16,000 into C = 5000
+ 2x to get 16.000 = 5000 + 2x.
There
were 5500 units produced.
The profit
formula for a manufacturer’s product is P = 2x – 4000 where x is the number of
units sold and P is the profit (in dollars).
(a) What
is the profit when 12,000 units were sold?
(b) What
is the loss when 1500 units were sold?
(c) How
many units must be sold for the manufacturer to have a profit of $3000?
(d) How
many units must be sold for the manufacturer to break even?
(This
question could have been phrased, ‘‘How many units must be sold in order for
the manufacturer to cover its costs?’’)
(a) If
12,000 units are sold, then x = 12,000. The profit equation then becomes P = 2(12.000)
– 4000 = 24.000 – 4000 = 20,000. The profit is $20,000.
(b) Think
of a loss as a negative profit. When 1500 units are sold, P ¼ 2x _ 4000 becomes
P = 2(1500) – 4000 = 3000 – 4000 = –1000. The manufacturer loses $1000 when
1500 units are sold.
(c) If
the profit is $3000, then P = 3000. P = 2x – 4000 becomes 3000 = 2x – 4000.
A
total of 3500 units were sold.
(d) The
break-even point occurs when the profit is zero, that is when P = 0. Then P = 2x
– 4000 becomes 0 = 2x – 4000.
The
manufacturer must sell 2000 units in order to break even.
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