yvonne12 wrote:when a certain tree was first planted, it was 4 feet tall and the height of the tree increased by a constnt amount each year for the next 6 years. At the end of the 6th year the tree was 1/5 taller than it was at the end of the 4the year. By how many feet did the height fo the tree increase each year?

When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. Since we know that the growth is by a constant amount, we have a linear growth problem. Thus, we can let x = the yearly growth amount in feet:

Starting height = 4

Height after year one = 4 + x

Height after year two = 4 + 2x

Height after year three = 4 + 3x

Height after year four = 4 + 4x

Height after year five = 4 + 5x

Height after year six = 4 + 6x

We are also given that at the end of the 6th year the tree was 1/5 taller than it was at the end of the 4th year. This means the height of the tree at the end of the 6th year is 6/5 times as tall as its height at the end of the 4th year. Thus, we can create the following equation:

(6/5)(4 + 4x) = 4 + 6x

To eliminate the fraction 6/5, we multiply the entire equation by 5:

6(4 + 4x) = 20 + 30x

24 + 24x = 20 + 30x

6x = 4

x = 4/6 = 2/3 feet

Answer: 2/3

Here's the complete question (with answer choices):

When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of 6th year, the tree was 1/5 taller than it was at the end of 4th year. By how many feet the height of the tree increase each year?

1) 3/10
2) 2/5
3) 1/2
4) 2/3
5) 6/5

Height of tree on day 0 = 4
Let d = the height increase each year
Height of tree at the end of the 1st year = 4+d
Height of tree at the end of the 2nd year = 4+d+d = 4 + 2d
Height of tree at the end of the 3rd year = 4+d+d+d = 4 + 3d
Height of tree at the end of the 4th year = 4+d+d+d+d = 4 + 4d
Height of tree at the end of the 5th year = 4+d+d+d+d+d = 4 + 5d
Height of tree at the end of the 6th year = 4+d+d+d+d+d+d = 4 + 6d

At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year
In other words, 6th year height = 4th year height + 1/5(4th year height)
Or we can write 4 + 6d = (4 + 4d) + 1/5(4 + 4d)
Simplify: 4 + 6d = 6/5(4 + 4d)
Multiply both sides by 5 to get: 5(4 + 6d) = 6(4 + 4d)
Expand: 20 + 30d = 24 + 24d
Simplify: 6d = 4
Solve: d = 4/6 = 2/3

Answer: D

Cheers,
Brent