On July 1, 2017, a certain tree was 128 centimeters tall.

On July 1st, 2017 the tree was 128 or \(2^7\) centimeters tall. Every year, the tree's height increase by 50%.

When there is a quantity increase of 50%, it becomes \(\frac{3}{2}\)th of the original quantity.

On July 1st, 2018 the tree is \(2^7 * \frac{3}{2}\) or \(2^6 * 3\) centimeter tall.
On July 1st, 2019 the tree is \(2^6 * 3 * \frac{3}{2}\) or \(2^5 * 3^2\) centimeter tall.

3 years later, On July 1st, 2022 the tree is \(2^3 * 3^4 * \frac{3}{2}\) or \(2^2 * 3^5\) centimeter tall.
4 years later, On July 1st, 2023 the tree is \(2^2 * 3^5 * \frac{3}{2}\) or \(2 * 3^6\) centimeter tall.

The tree's height difference is \(2 * 3^6 - 2^2 * 3^5 = 2 * 3^5(3 - 2) = 2 * 3^5\)

Therefore, the tree will be \(2 * 3^5\) centimeters(Option C) greater on July 1, 2023 than July 1, 2022.

Let's create a growth table and look for a pattern

year | height in cm
2017: 128
2018: 128(1.5)
2019: 128(1.5)^2
2020: 128(1.5)^3
2021: 128(1.5)^4
2022: 128(1.5)^5
2023: 128(1.5)^6

The tree's height on July 1, 2023 will be how many centimeters greater than the tree's height on July 1, 2022?
Difference = 128(1.5)^6 - 128(1.5)^5
Factor out 128(1.5^5) to get: difference = 128(1.5^5)[1.5 - 1]
Simplify: difference = 128(1.5^5)[0.5]
Rewrite with fractions: difference = (2^7)(3/2)^5)(1/2)
Expand: difference = (2^7)(3^5)/(2^6)
Simplify: difference = (2)(3^5)

Answer: C

Cheers,
Brent

I wonder how much time does it take to find the correct approach. it appeared too hard for me

Formula for EXPONENTIAL GROWTH:
final amount = original * (multiplier)^(number of changes)

Here:
original = \(128 = 2^7\)
Since the height increases each year by 50%, the multiplier \(= \frac{3}{2}\)

From 2017 to 2022 -- a period of 5 years -- there are total of 5 changes.
Thus:
final amount \(= 2^7 * (\frac{3}{2})^5 = 2^23^5\)

The tree's height on July 1, 2023 will be how many centimeters greater than the tree's height on July 1, 2022?
Since the height increases by 50% each year, the increase from 2022 to 2023 is equal to half the height in 2022:
\(\frac{1}{2} * 2^23^5 = 2*3^5\)

it depends on the approach you take. I solved using Compound interest formulae it hardy take 1 minute to solve
Compound Interest=principal∗(1+interestC)time∗C

If C=1, meaning that interest is compounded once a year

July 1, 2017, a certain tree was 128 centimeters tall: 128 = \(\frac{2^7 [}{fraction]\)

50% increase: \([fraction]150 / 100}\) = \(\frac{3}{2}\)

The common ratio is 3/2. And this forms a G.P with the first term as a = a * \(2^7\) and r = \(\frac{3}{2}\)

July 1, 2022 (5 years): a * \(r^5\)

July 1, 2023 (6 years): a * \(r^6\)

We have to find a difference between both: a * \(r^6\) - a * \(r^5\)

=> a * \(r^6\) - a * \(r^5\) = a * \(r^5\) ( r-1)

=> a * \(r^6\) - a * \(r^5\) = a * \(r^5\) ( \(\frac{3}{2}\) - 1)

=> a * \(r^6\) - a * \(r^5\) = a * \(r^5\) (\( \frac{1}{2}\))

=> a * \(r^5\) = \(2^7\) * \((\frac{3}{2})^5\) * \(\frac{1 }{ 2}\)

=> a * \(r^5\) = 2 * \((3)^5\)

Answer C

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