Question

Takumi plants a tree in his backyard and studies how the number of branches grows over

He predicts that the relationship between N, the number of branches on the tree, and t
years, since the tree was planted can be modeled by the following equation.
N = 5. 100.3t
According to Takumi's model, in how many years will the tree have 100 branches?
Give an exact answer expressed as a base-10 logarithm.

Answer 1

According to Takumi's model, the tree will have 100 branches in approximately 1.464 years.

Step-by-step explanation:

To find the number of years it will take for the tree to have 100 branches according to Takumi's model, we can substitute N = 100 into the equation N = 5 * 100 * 3^t and solve for t. Let's solve it step-by-step:

1. Substitute N = 100 into the equation: 100 = 5 * 100 * 3^t
2. Cancel out the common factor of 100: 1 = 5 * 3^t
3. Divide both sides by 5: 1/5 = 3^t
4. Take the logarithm of both sides using base 10: log 1/5 = log 3^t
5. Apply the logarithm property log a^b = b * log a: log 1/5 = t * log 3
6. Use the logarithm property log(1/a) = -log a: -log 5 = t * log 3
7. Divide both sides by log 3: t = (-log 5) / log 3
8. Calculate the value using a calculator: t ≈ -0.69897 / 0.47712125 ≈ -1.464

Since years cannot be negative, we take the absolute value of t to get the positive value. Therefore, the tree will have 100 branches in approximately 1.464 years (rounded to three decimal places).

Answer 2

t (years)= (10log 20/log 1000) in base 10

Step-by-step explanation:

N = 5 × 10^0.3t

N = 100 branches

100 = 5 × 10^0.3t

100/5 = 10^0.3t

20 = 10^0.3t

Taking log of both sides

Log 20 base 10 = 0.3t

Log 20 base 10 = (3/10)t

Multiply both sides by 10

10log 20 base 10 = 3t

Recall, log 1000 base 10 = 3

10 Log 20 base 10 = (log 1000 base 10)×t

t = (10Log 20 base 10)/(log 1000 base 10)

t (years)= (10log 20/log 1000) in base 10

The number of years it will take for the tree to have 100branches is (10log 20/log 1000) in base 10