How do I write a linear equation from given information?

[mathdad]

Imagine that you own a grove of orange trees, and suppose that from past experience 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.

I have trouple with problems requesting an equation from the given information in word problems. Can someone get me started? I know the equation required is written as y in terms of x but all those numbers in the application threw me into a loop.

 

[MarkFL]

Okay, let's look at the statement:

"from past experience you know that when 100 trees are planted, each tree will yield about 240 oranges per year."

This gives us a point on the line...$(100,240)$

Next, let's look at:

"you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted."

What does this tell us about the line? What I mean is, to which aspect of the line does this relate?

 

[mathdad]

Okay, let's look at the statement:

"from past experience you know that when 100 trees are planted, each tree will yield about 240 oranges per year."

This gives us a point on the line...$(100,240)$

Next, let's look at:

"you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted."

What does this tell us about the line? What I mean is, to which aspect of the line does this relate?

The line is getting smaller producing negative slope.

 

[MarkFL]

The line is getting smaller producing negative slope.

Yes, we are told that for every increase in $x$ of 1 ($\Delta x=1$), we get a decrease in $y$ of 20 ($\Delta y=-20$)...so what is the slope of the line?

\(\displaystyle m=\frac{\Delta y}{\Delta x}=?\)

 

[mathdad]

Yes, we are told that for every increase in $x$ of 1 ($\Delta x=1$), we get a decrease in $y$ of 20 ($\Delta y=-20$)...so what is the slope of the line?

\(\displaystyle m=\frac{\Delta y}{\Delta x}=?\)

m = -20/1 or -20.

- - - Updated - - -

I now plug -20 and the point into the point-slope formula to find the linear equation.

 

[mathdad]

y - 240 = -20(x - 100)

y - 240 = -20x + 2000

y = -20x + 2000 + 240

y = -20x + 2240

Correct?

 

[MarkFL]

y - 240 = -20(x - 100)

y - 240 = -20x + 2000

y = -20x + 2000 + 240

y = -20x + 2240

Correct?

Looks good to me. Should we give a domain for the yield function?

 

[mathdad]

Is the domain all real numbers?

 

[MarkFL]

Is the domain all real numbers?

Do we know what happens if less than 100 trees is planted? Can we have a negative yield?

 

[mathdad]

What happens if less than 100 trees is planted?

Less oranges are produced.

Can we have a negative yield?

It's a linear equation, and so, the domain can be any integer.

 

[MarkFL]

What happens if less than 100 trees is planted?

Less oranges are produced.

All we know is that when more than 100 trees are planted, the yield drops by 20 oranges per tree, but we don't know what happens if trees are remove, to a number than less than 100. It could be that yield increases, or it could be that 240 oranges is the maximum yield per tree that would be observed. We simply don't know, and so we should use 100 as the lower bound for $x$.

Can we have a negative yield?

It's a linear equation, and so, the domain can be any integer.

Suppose you plant a total of 120 trees...our yield function tells us to expect a yield of -160 oranges per tree...does this sound reasonable? We find that when we plant 112 trees, the yield per tree is 0, and so we should use 112 as the upper bound. In reality, we would likely find that the yield per tree would asymptotically approach some minimum value, but we don't have enough information to speculate about that. So, I would give the domain as:

$[100,112]$

 

[mathdad]

All we know is that when more than 100 trees are planted, the yield drops by 20 oranges per tree, but we don't know what happens if trees are remove, to a number than less than 100. It could be that yield increases, or it could be that 240 oranges is the maximum yield per tree that would be observed. We simply don't know, and so we should use 100 as the lower bound for $x$.
Suppose you plant a total of 120 trees...our yield function tells us to expect a yield of -160 oranges per tree...does this sound reasonable? We find that when we plant 112 trees, the yield per tree is 0, and so we should use 112 as the upper bound. In reality, we would likely find that the yield per tree would asymptotically approach some minimum value, but we don't have enough information to speculate about that. So, I would give the domain as:

$[100,112]$

Mark,

When you say "approach some minimum value," are you talking about limits as taught in calculus 1? Can we apply the idea of limits to this precalculus question?

 

[MarkFL]

Mark,

When you say "approach some minimum value," are you talking about limits as taught in calculus 1? Can we apply the idea of limits to this precalculus question?

I was just speaking of what we would likely find in the real world...there is likely some limiting maximum and minimum yields we would find. But, we aren't given any information regarding that...we simply want to restrict the domain to what we are given, and to what is reasonable. We don't know anything about the yield per tree when there are fewer than 100 trees, and we know a negative yield would be meaningless, so that's why we should restrict our domain. :)

 

[mathdad]

I was just speaking of what we would likely find in the real world...there is likely some limiting maximum and minimum yields we would find. But, we aren't given any information regarding that...we simply want to restrict the domain to what we are given, and to what is reasonable. We don't know anything about the yield per tree when there are fewer than 100 trees, and we know a negative yield would be meaningless, so that's why we should restrict our domain. :)

This question is from the David Cohen book. Finding the linear equation is PART A of this question. I will send the page number for this question. There are several parts in the textbook.