1. Electricity users pay a 15$ standing charge, plus 0.01$ for each unit of electricity used Q. a) Write an expression for the total cost TC in terms of Q. b) Write an expression f... 1. Electricity users pay a 15$ standing charge, plus 0.01$ for each unit of electricity used Q. a) Write an expression for the total cost TC in terms of Q. b) Write an expression for the average cost per unit used. 2. You run a small store selling books. If the rent is 1,000$ and the unit selling price is 50$ per book, and if this store has purchased Q books this month to sell it for 60$. a) Write down the associated: (i) total cost TC, (ii) total revenue TR, (iii) profit π and (iv) average cost AC as functions of the quantity Q. b) What profit (or loss) results from the sale of 200 books? c) How many books should be sold in order to break even? d) How many books should be sold to get an average cost of 55$ per book? Read more

Steps to Solve

1. Cost Expression for Electricity Usage

To find the total cost ( TC ) for electricity, you add the standing charge and the cost per unit of electricity used. The expression is:

$$ TC = 15 + 0.01Q $$

Where $15$ is the standing charge, and $0.01Q$ is the cost per unit.

2. Average Cost per Unit Calculation

To find the average cost per unit of electricity, you divide the total cost ( TC ) by the quantity ( Q ):

$$ AC = \frac{TC}{Q} = \frac{15 + 0.01Q}{Q} = \frac{15}{Q} + 0.01 $$

3. Total Cost for Selling Books

For selling books, total cost ( TC ) includes rent and the cost of purchased books. The expression is:

$$ TC = 1000 + Q \cdot C $$

Where ( C ) is the cost per book (assumed constant).

4. Total Revenue Expression

The total revenue ( TR ) from selling Q books is given by:

$$ TR = 60Q $$

Where ( 60 ) is the selling price per book.

5. Profit Expression

Profit ( \pi ) is calculated by subtracting total cost from total revenue:

$$ \pi = TR - TC = 60Q - (1000 + Q \cdot C) $$

6. Average Cost Expression for Books

The average cost ( AC ) of selling Q books is given by:

$$ AC = \frac{TC}{Q} = \frac{1000 + QC}{Q} = \frac{1000}{Q} + C $$

7. Profit from Selling 200 Books

To find profit from selling 200 books:

First calculate ( TR ) and ( TC ) when ( Q = 200 ):

$$ TR = 60 \cdot 200 = 12000 $$

$$ TC = 1000 + 200 \cdot C $$

Now, find profit:

$$ \pi = 12000 - (1000 + 200C) $$

8. Break-even Point Calculation

To find the break-even point, set profit ( \pi ) to zero and solve for ( Q ):

$$ 0 = 60Q - (1000 + QC) $$

Rearranging gives:

$$ QC = 60Q - 1000 \implies Q(60 - C) = 1000 \implies Q = \frac{1000}{60 - C} $$

9. Average Cost of 55$ per Book

To find how many books ( Q ) should be sold to have an average cost of $55 per book:

Setting ( AC = 55 ):

$$ 55 = \frac{1000 + QC}{Q} $$

Rearranging gives:

$$ 55Q = 1000 + QC $$

Thus,

$$ Q(55 - C) = 1000 \implies Q = \frac{1000}{55 - C} $$