Final answer:
The artist can implement several $5 price increases each reducing the quantity sold by 10 units. To find the maximum number of increases while making at least $32,000 in profit, we solve the inequality 50p^2 + 1400p - 8000 ≥ 0 for the highest integer value of p.
Step-by-step explanation:
The artist initially sold 400 glass figurines at $60 each. With each $5 price increase, the number of figurines sold decreases by 10. We need to determine how many $5 price increases can be added while still making a total profit of at least $32,000.
To calculate total profit, we use the formula:
- Total Profit = (Price per figurine × Number of figurines sold) - Total costs
If the artist initially sells 400 figurines at $60, this means the total revenue without any price increase is:
$60 × 400 = $24,000
Assuming no costs, to achieve at least $32,000 in profit, we need additional revenue:
$32,000 - $24,000 = $8,000
Each $5 price increase will both increase the price per figurine and decrease the amount of figurines sold by 10 units.
Let's define:
- p = number of $5 price increases
- New price per figurine = $60 + $5p
- New number of figurines sold = 400 - 10p
The total profit with the new price and quantity sold is given by:
Profit = (New price per figurine) × (New number of figurines sold)
To calculate the maximum number of $5 price increments, the profit equation must satisfy:
($60 + $5p) × (400 - 10p) ≥ $32,000
The artist needs to find the highest integer value of p for which the inequality holds true. This requires solving for p:
($60 + $5p) × (400 - 10p) = $32,000
$24,000 + (400p × $5 - $60 × 10p + $5 × 10p × p) ≥ $32,000
$24,000 + (2000p - 600p + 50p^2) ≥ $32,000
$24,000 + 1400p + 50p^2 ≥ $32,000
50p^2 + 1400p - 8000 ≥ 0
This quadratic inequality must be solved to find the range of p values. The artist will choose the maximum integer that satisfies this inequality to maximize price without falling below the $32,000 profit threshold.