Question

when vlad moved to his new home a few years ago, there was a young oak tree in his backyard. he measured it once a year and found that it grew by 26 centimeters each year. 4.5 years after he moved into the house, the tree was 292 centimeters tall. how tall was the tree when vlad moved into the house? centimeters how many years passed from the time vlad moved in until the tree was 357 centimeters tall? years

Explanation:

Step1: Define variables and formula

Let \( h_0 \) be the initial height (when Vlad moved in), \( r = 26 \) cm/year be the growth rate, \( t \) be time in years, and \( h(t) \) be the height at time \( t \). The formula is \( h(t)=h_0 + r\times t \).

Step2: Find initial height (\( h_0 \))

We know at \( t = 4.5 \) years, \( h(4.5)=292 \) cm. Substitute into the formula:
\( 292 = h_0+26\times4.5 \)
Calculate \( 26\times4.5 = 117 \). Then:
\( h_0=292 - 117 = 175 \) cm.

Step3: Find time for \( h(t)=357 \) cm

Use \( h(t)=h_0 + r\times t \), substitute \( h(t)=357 \), \( h_0 = 175 \), \( r = 26 \):
\( 357=175 + 26t \)
Subtract 175: \( 26t=357 - 175 = 182 \)
Divide by 26: \( t=\frac{182}{26}=7 \) years.

Answer:

First blank: \( 175 \)
Second blank: \( 7 \)