Solving Linear Systems: 14" Hemlock & 8" Blue Spruce

[jojonea]

I have a question on my homework, it is on Applications of Linear Systems, the question is : You plant a 14-inch hemlock tree in your backyard that grows at a rate of 4 inches per year and an 8-inch blue spruce tree that grows at a rate of 6 inches per year. In how many years after you plant the trees will the two trees be the same height? how tall will each tree be?

I know the answer: 3 years, both 26 inches, I don't know how to write out the systems, maybe I'm just an idiot, I don't know.

 

[AngelofMusic]

I'll try to help, but keep in mind - I don't have linear algebra until next year due to some curriculum changes - so I'll be working mostly from high school knowledge.

Basically, you should try to find an equation that represents the height of each tree. For example, let H be the height of the hemlock tree, and let S be the height of the spruce tree.

Let t represent the number of years that have passed.

Now, you just have to try and find a way to relate H and t and S and t. That is, find H and S as a function of t.

H = H(t)
S = S(t)

Then, set H and S equal to each other, and solve for t.

Keep in mind that when you're solving this problem, the t for both trees is the same as well as the height.

Does that help?

 

[HallsofIvy]

Hardly a matter of Linear Algebra!

The hemlock is initially 14 in and increases 4 in every year:
After 1 year 14+4 inches,after 2 years 14+ 4+ 4= 14+ 4(2), after 3 years, 14+ 4+ 4+ 4= 14+4(3), etc. Taking H to be the height of the hemlock and t the number of years, H= 14+ 4t.

The spruce is initially 8 inches and grows 6 inches each year: taking S to be the height of the spruce and t the number of years,
S= 8+ 6t.

They will be "equal height" when H= S. That is, when 14+ 4t= 8+ 6t.
Solve for t and then find the height for that t.

That's fairly basic algebra.

You could also do this by noting that, since the spruce grows 6 inches each year, while the hemlock grows only 4 inches, the spruce "catches up" 2 in per year. Since the hemlock is originally 14- 8= 6 inches higher, it will take the spruce 6/2= 3 years to catch up to the hemlock.

 

[acmoa]

A little help

To make answring this questions easier, you can just create a formula
(equation more accurately).

you have the following "knowns":
1- plants height 14, 8 inches
2- plants rate of growth 4(hemlock) and 6(spruce)
you don't have:
1- The number of years it will take for the plants to get to the same height(lets say Years = Y)

here is the formula: 4Y + 14 = 6Y + 8

Notice that getting Y(years) correctly by solving the equation will give you the answer, and that the left side would equal the right side (height of hemlock would equal height of the spurce in Y years)

lets solve it out:

4Y + 14 = 6Y + 8
4Y - 6Y = 8 - 14 (switching)
-2Y = -6 (subtracting)
-2 / -2Y = -6 / -2 (getting rid of -2 before the Y and making sure Y is not negative)

Y = 3 (the answer you have)

to get the height of the two plants you subtitute Y in the equation:

4Y + 14 = ? (hemlock)
4(3) + 14 = ?
12 + 14 = 26 inches


6Y + 8 = ? (spurce)
6(3) + 8 = ?
18 + 8 = 26 inches

simply that's it!