Megan has been planting young trees in her garden. One maple tree is 66 centimeters tall and is growing 8 centimeters per month. One oak tree is 70 centimeters tall and is growing... Megan has been planting young trees in her garden. One maple tree is 66 centimeters tall and is growing 8 centimeters per month. One oak tree is 70 centimeters tall and is growing 7 centimeters per month. In a few months, the two trees will be the same height. What will that height be? How long will that take?
Understand the Problem
The question is asking how long it will take for two trees, one maple and one oak, to reach the same height given their current height and growth rates. This involves setting up a system of equations based on their growth.
Answer
The two trees will both be $98$ centimeters tall in $4$ months.
Answer for screen readers
The two trees will both be 98 centimeters tall in 4 months.
Steps to Solve
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Identify the initial heights and growth rates The maple tree is currently 66 cm tall and grows 8 cm per month. The oak tree is currently 70 cm tall and grows 7 cm per month.
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Set up the equations for their heights Let $t$ be the number of months it takes for the trees to reach the same height. The height of the maple tree after $t$ months can be expressed as: $$ H_m = 66 + 8t $$ The height of the oak tree after $t$ months can be expressed as: $$ H_o = 70 + 7t $$
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Set the two height equations equal to each other To find out when the two trees will be the same height, set the equations from step 2 equal to each other: $$ 66 + 8t = 70 + 7t $$
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Solve for $t$ Rearranging the equation to isolate $t$: $$ 8t - 7t = 70 - 66 $$ $$ t = 4 $$
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Calculate the height of the trees at $t = 4$ months Substituting $t = 4$ back into either height equation: For the maple tree: $$ H_m = 66 + 8(4) = 66 + 32 = 98 $$ For the oak tree: $$ H_o = 70 + 7(4) = 70 + 28 = 98 $$
The two trees will both be 98 centimeters tall in 4 months.
More Information
Both trees will reach the same height of 98 cm after 4 months of growth. This demonstrates how to set up and solve a system of equations to determine when two variables will become equal.
Tips
- Forgetting to set the height equations equal to each other.
- Mixing up the growth rates or initial heights of the trees.
- Incorrectly simplifying the equation while solving for $t$.
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