A tree that is 4 ft tall is growing at a rate of 1 ft each year a tree that is 12 feet tall is growing at a rate of 1/2 ft each year how many years will it take the two trees to reach the same height

16 years

Step-by-step explanation:

Given data

For the first tree

let the expression for the height be

y=4+x1

where y= the total height

          4= the initial height

          x= the number of years

For the second tree, the expression is

y=12+0.5x2

Equate 1 and 2

4+x=12+0.5x

x-0.5x=12-4

0.5x= 8

x= 8/0.5

x=16

Hence it will take 16 years for both trees to have the same height

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt

cross multiplying, we get :

⇒ (3) × (5) = (9) × (x)

⇒ 15 = 9x