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 growing at 8 centimeters per month and the other at 7 centimeters per month, to reach the same height. It requires solving a system of equations based on their growth rates and initial heights.
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
- Define the Variables
Let ( x ) represent the number of months it takes for the trees to be the same height.
- Set Up the Height Equations
The height of the maple tree after ( x ) months can be expressed as: $$ \text{Height of maple tree} = 66 + 8x $$
The height of the oak tree after ( x ) months can be expressed as: $$ \text{Height of oak tree} = 70 + 7x $$
- Set the Heights Equal
To find when the two trees are the same height, set the two equations equal to each other: $$ 66 + 8x = 70 + 7x $$
- Solve for ( x )
First, simplify the equation: $$ 8x - 7x = 70 - 66 $$
This simplifies to: $$ x = 4 $$
- Determine the Height After 4 Months
Substituting ( x = 4 ) back into one of the height equations, we can find the height: $$ \text{Height} = 66 + 8(4) = 66 + 32 = 98 \text{ centimeters} $$
The two trees will both be 98 centimeters tall in 4 months.
More Information
After 4 months, both trees will reach the same height of 98 centimeters. This type of problem illustrates how to use equations to compare growing quantities over time.
Tips
- Forgetting to combine like terms when setting the heights equal.
- Not clearly defining variables which can lead to confusion in calculations.
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