A gardener is planting two types of trees:

Type A is 8 feet tall and grows at a rate of 6 inches per year.

Type B is 10 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.

12 years

Step-by-step explanation:

8 feet = 96 inches

10 feet = 120 inches

96 + 6t = 120 + 4t

2t = 24

t = 12

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4