Question

The manager of a cherry orchard wants to schedule the annual harvest If the cherries are picked now, the average yield per tree will be 100 ib, and the cherties can be sold for 50 cents per pound. Past experience shows that the yield per tree will increase about 8lb per woek, while the price wit decrease about 4 cents per pound por week. How many weeks should the managor wait to get an average revenue of $42,00 per tree?

Answer 1

The manager should wait a number of weeks determined by solving the quadratic equation – resulting from the increase in yield and decrease in price over time – to ensure the revenue per tree is $42.

Step-by-step explanation:

To determine how many weeks the manager of a cherry orchard should wait to get an average revenue of $42.00 per tree, we need to establish an equation that models the revenue per tree over time. Initially, we have a yield of 100 lb per tree, selling at 50 cents per pound, which gives us a revenue of $50 per tree.

Each week, the yield increases by 8 lb, and the price decreases by 4 cents per pound. If w represents the number of weeks waited, the yield becomes (100 + 8w) lb, and the price becomes (50 - 4w) cents per pound. The revenue R, in dollars, is calculated by the formula R = (yield) × (price), thus R = (100 + 8w) × (0.50 - 0.04w).

To find w when the revenue is $42 per tree, we set the revenue formula equal to 42 and solve for w:
42 = (100 + 8w) × (0.50 - 0.04w).

Expanding and simplifying the equation, we get:
42 = 50 - 0.04w + 0.50 × 8w - 0.04 × 8w²,

This is a quadratic equation that can be simplified to 0 = -0.32w² + 4w + 8. Using the quadratic formula or factoring, we solve for w to find the number of weeks the manager should wait.