Question

A redwood tree in King’s Canyon National Park, California has a circumference of 220 inches. Tanner measured the circumference of some other redwoods that had been planted over the last 5 years. His data is recorded below. Write a rule for the data in the table.

Use your rule to predict the circumference of a tree after 50 years of growth.
Use your rule to estimate the age of the tree with a circumference of 221 inches.

Answer 1

Answer:

A). 101 inches

B). 110 years

Step-by-step explanation:

Years since planting           Circumference          Difference

             1                                        3                               -

             2                                       5                        (5 - 3) = 2

             3                                       7                         (7 - 5) = 2

             4                                       9                        (9 - 7) = 2

             5                                       11                        (11 - 9) = 2

             6                                       13                       (13 - 11) = 2

There is a difference of 2 in every successive term of the circumference.

Therefore, the given sequence is an arithmetic sequence.

Explicit formula of an arithmetic sequence is given by,

$A_n=a+(n-1)d$

Here a = first term

n = number of terms

d = common difference

From the given table,

$A_n=3+(n-1)2$

$A_n=2n+1$

Here $A_n$ = circumference after 'n' years

n = Age of the plant

By this formula,

Circumference of a tree after 50 years,

$A_{50}=2(50)+1$

      = 101 inches

Age of tree with the circumference = 221 inches,

221 = 2n + 1

2n = 220

n = 110 years