Question

When Gavin moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 38 inches tall and Tree B was 22 inches tall. Each year thereafter, Tree A grew by 4 inches per year and Tree B grew by 8 inches per year. Let A represent the height of Tree A t years after being planted and let B represent the height of Tree B t years after being planted. Write an equation for each situation, in terms of t, and determine the number of years after the trees were planted when both trees have an equal height.

Answer 1

A. A(t) = 38" + t*(4"/yr) and B(t) = 22" + t*(8"/yr)

B. 4 years

Explanation:

Let Ta and Tb be the initial heights of trees A and B, respectively. Let t be the years after the initial plantings.

Ta = 38"

Tb = 22"

We can write Ta(t) and Tb(t) to represent the trees' heights after x years

We learn that the trees grow at different rates:

Ta rate = 4"/yr

Tb rate = 8"/yr

Let A and B represent each tree's height t years after being planted.

Equations that will predict each tree's height t years after planting can be written as:

A. The two equations

A(t) = Ta + t*(4"/yr) or A(t) = 38" + t*(4"/yr) [The initial height plus additional growth t years after planting at 4" per year] A(t) means the height will depend on (is a function of . .) t.

B(t) = Tb + t*(8"/yr) or B(t) = 22" + t*(8"/yr) [Same idea]

B. Years until same height:

"both trees have equal height" can be written as:

A(t) = B(t)

So we can write:

38" + t*(4"/yr) = 22" + t*(8"/yr)

t*(4"/yr) - t*(8"/yr) = 22" - 38"

-(4"/yr)t = - 16"

t = 4 years