Question

A gardener is planting two types of trees: Type A is 8 feet tall and grows at a rate of 3 inches per year. Type B is 7 feet tall and grows at a rate of 4 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.

Answer 1

Exactly 12 years it will take for these trees to be the same height

Explanation:

Slope intercept form: An equation of line is in the form of
`y = mx+b` where m is the slope or unit rate and b is the y-intercepts.

Let x represents the time in years and y represents the height of the tree.

Use conversion:

1 ft = 12 inches

As per the given statement:

Type A is 8 feet tall and grows at a rate of 3 inches per year.

⇒unit rate per year = 3 inches =
`(1)/(4)` ft

Then, we have;

`y =(1)/(4)x + 8` ......[1]

Similarly for;

Type B is 7 feet tall and grows at a rate of 4 inches per year.

⇒unit rate per year = 4 inches =
`(1)/(3)` ft

then;

`y =(1)/(3)x + 7` .....[2]

To find after how many years it will take for these trees to be the same height.

Since, trees to be the same height;

⇒equate [1] and [2], to solve for x;

`(1)/(4)x + 8 = (1)/(3)x +7`

Subtract 7 from both sides we get;

`(1)/(4)x + 8-7= (1)/(3)x +7-7`

Simplify:

`(1)/(4)x + 1= (1)/(3)x`

Subtract
`(1)/(4)x` from both sides we get;

`1= (1)/(3)x-(1)/(4)x`

Simplify:

`1 = (x)/(12)`

Multiply both sides by 12 we get;

x = 12

Therefore, exactly it will take for these trees to be the same height is, 12 years