Question

§9.2: solve the application.
18.1 a produce company has a 200 - acre farm on which it grows apples and oranges. each acre of oranges requires 300 hours of labor and each acre of apples requires 450 hours of labor. if 60,000 hours are available and if all land and labor resources are to be used, find the number of acres of each crop that should be planted.

Explanation:

Step1: Define Variables

Let \( x \) be the number of acres of oranges and \( y \) be the number of acres of apples.

Step2: Set Up Land Equation

The total land is 200 acres, so \( x + y = 200 \).

Step3: Set Up Labor Equation

Each acre of oranges needs 300 hours, apples need 450 hours, and total labor is 60,000 hours. So \( 300x + 450y = 60000 \). Simplify this equation by dividing by 150: \( 2x + 3y = 400 \).

Step4: Solve the System of Equations

From the land equation, \( x = 200 - y \). Substitute into the labor equation:
\( 2(200 - y) + 3y = 400 \)
\( 400 - 2y + 3y = 400 \)
\( y = 0 \)? Wait, no, wait—wait, let's check again. Wait, 2200 is 400, so 400 - 2y + 3y = 400 → \( y = 0 \)? That can't be. Wait, maybe I made a mistake. Wait, 60,000 divided by 150: 60,000 ÷ 150 = 400. Correct. Then \( x = 200 - y \). Substitute into 2x + 3y = 400: 2(200 - y) + 3y = 400 → 400 - 2y + 3y = 400 → y = 0. Then x = 200. Wait, that would mean all land is oranges. Let's check labor: 200 acres of oranges 300 hours = 60,000 hours. Correct. So apples: 0 acres, oranges: 200 acres. Wait, but maybe the problem is written correctly? Let me re-express.

Wait, maybe I mixed up x and y. Let's redefine: let x be apples, y be oranges. Then land: x + y = 200. Labor: 450x + 300y = 60000. Simplify labor: divide by 150: 3x + 2y = 400. Then from land: y = 200 - x. Substitute into labor: 3x + 2(200 - x) = 400 → 3x + 400 - 2x = 400 → x = 0. Then y = 200. So same result. So all land is oranges (200 acres), apples: 0 acres. Because 200*300=60,000 labor hours. Correct.

Answer:

Oranges: 200 acres, Apples: 0 acres.