5. You have inherited an orange grove. The more orange trees there are in the grove, the fewer oranges each tree produces. When there are 30 trees, each tree produces 84 kg of oranges. When there are 90 trees, each tree produces 74 kg of oranges. Assume that the amount of oranges each tree produces is a linear function of the number of trees.
1. Let x be the number of trees grown in the grove. In terms of x, what is the total number of kg of oranges produced on each tree?
2. In terms of x, what is the total number of kg of oranges produced in the grove?
3. What is the maximum number of kg of oranges that can be produced in the grove?
x y
30 84
90 74
So, every extra 60 trees reduces the yield by 10kg, so
y = -(10/60)x + c
Since y(30)=84,
(-1/6)(30)+c = 84
c=89
y = -x/6 + 89
The total yield is trees * avgyield, so
x(-x/6+89) = -x^2/6 + 89x
This is just a parabola, where the vertex is at x = 3*89
You can figure the total yield for that value of x.
To find the answers to these questions, we need to identify the linear function that represents the amount of oranges each tree produces based on the number of trees. To do that, we can use the given data points and apply linear regression.
1. Let's start by finding the slope of the linear function. We can use the formula for slope, which is (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the function. In this case, we can use the points (30, 84) and (90, 74) to calculate the slope:
slope = (74 - 84) / (90 - 30) = -10 / 60 = -1/6
Thus, the slope of the linear function is -1/6.
2. Next, we need to find the y-intercept of the linear function. We can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Using the point (30, 84) and the slope calculated earlier, we can rearrange the equation to solve for y:
y - 84 = (-1/6)(x - 30)
y - 84 = (-1/6)x + 5
y = (-1/6)x + 89
Now we have the equation that represents the amount of oranges produced on each tree in terms of the number of trees (x).
3. To find the maximum number of kg of oranges that can be produced in the grove, we need to determine the value of x that gives the maximum value of y (the total number of kg of oranges produced). In this case, since the more trees there are, the fewer oranges each tree produces, the maximum value will occur at the minimum number of trees.
To find the minimum number of trees, we can calculate the x-coordinate of the vertex of the linear function. The x-coordinate of the vertex can be calculated using the formula x = -b / (2a), where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, since we have a linear function, the quadratic term is 0.
So, x = -(-1/6) / (2*0) = 0 / 0, which is undefined.
Since the x-coordinate of the vertex is undefined, there is no maximum number of kg of oranges that can be produced in the grove. As the number of trees increases, the amount of oranges produced per tree decreases, so there is no limit to the number of trees that can be grown in the grove.