What is the Linear Equation for the Yield of an Orange Grove?

[mathdad]

Imagine that you own a grove of orange trees, and suppose that from past experiences you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted.

Let y denote the yield per tree when x trees are planted. Find a linear equation relating x and y.

Can someone get me started?

 

[MarkFL]

Imagine that you own a grove of orange trees, and suppose that from past experiences you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted.

Let y denote the yield per tree when x trees are planted. Find a linear equation relating x and y.

Can someone get me started?

You are given a point on the line, and the slope...:D

 

[HallsofIvy]

Any linear function can be written in the form $$f(x)= ax+ b$$ for constants a and b.

"you know that when 100 trees are planted, each tree will yield about 240 oranges per year."
So when x= 100, f(x)= 240: $$240= a(100)+ b$$.

"Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted."
So if x= 101, that is, if one more tree is planted, the yield per tree decreases from 240 to 240- 20= 220. \(\displaystyle 220= a(101)+ b\).

That gives you two equations to solve for a and b. (Notice that subtracting the second equation from the first immediately eliminates b.)

 

[mathdad]

You are given a point on the line, and the slope...:D

Is the point (240, 100) and slope 20?

I then use the point-slope formula.

Yes?

 

[mathdad]

Any linear function can be written in the form $$f(x)= ax+ b$$ for constants a and b.

"you know that when 100 trees are planted, each tree will yield about 240 oranges per year."
So when x= 100, f(x)= 240: $$240= a(100)+ b$$.

"Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted."
So if x= 101, that is, if one more tree is planted, the yield per tree decreases from 240 to 240- 20= 220. \(\displaystyle 220= a(101)+ b\).

That gives you two equations to solve for a and b. (Notice that subtracting the second equation from the first immediately eliminates b.)

I like the way you created a system of linear equations in two variables from the given information in the word problem. I have been struggling with this for years. It is challenging for me to create an equation(s) given an application.

I believe this to be one of the best skills to learn in math. I recall taking a test at Bank One in 2006 for a Financial Advisor position that I really wanted at the time. I could not pass the test that involved more than 10 questions requiring the skill of converting word problems to equations leading to the right answer.

Good paying jobs, especially working with numbers and money, will require applicants to know basic word problems. I was told that my score was one of the lowest in the history of that particular bank. However, I did not give up on math. There is something about math that I really enjoy. I believe this to be the challenge of looking at a basic word problem like the sample with oranges and creating an equation that easily allows me to solve for x.

 

[MarkFL]

Is the point (240, 100) and slope 20?

I then use the point-slope formula.

Yes?

No, the given point is (100, 240) and the slope is -20. :D

 

[mathdad]

y - y_1 = m(x - x_1)

y - 240 = -20(x - 100)

y - 240 = -20x + 2000

y = -20x + 2000 + 240

y = -20x + 2240

Correct?