Question 9·Hard·Ratios, Rates, Proportional Relationships, and Units
One method of estimating the age, in years, of a tree of a particular species is to multiply the tree’s diameter, in inches, by a constant called the growth factor for that species.
| Species of tree | Growth factor |
|---|---|
| Red maple | 4.5 |
| River birch | 3.5 |
| Cottonwood | 2.0 |
| Black walnut | 4.5 |
| White birch | 5.0 |
| American elm | 4.0 |
| Pin oak | 3.0 |
| Shagbark hickory | 7.5 |
If a white birch tree and a pin oak tree each now have a diameter of 1 foot, which of the following is closest to the difference, in inches, of their diameters 10 years from now? (1 foot = 12 inches)
For growth-factor problems, first make sure your units match the formula (here, convert any feet to inches). Use the given relationship to go from diameter to age, adjust for the time change (add or subtract years), and then invert the relationship (divide by the growth factor) to return from age to diameter. When both objects start at the same size, focus on how much each changes over the given time, and avoid rounding intermediate values too aggressively so that your final difference matches the closest answer choice accurately.
Hints
Match units to the formula
The formula uses diameter in inches. Convert 1 foot to inches before using the growth factors. How many inches are in 1 foot?
Go from diameter to age
Use age = diameter × growth factor to find how old each tree is now using the 12-inch diameter and the growth factors for white birch and pin oak.
Advance 10 years, then go back to diameters
After adding 10 years to each tree’s age, use the same relationship again, but this time divide by the growth factor to find the new diameters in inches.
Compare the future diameters
Once you have both future diameters, subtract one from the other and then choose the answer choice that is closest to that difference.
Desmos Guide
Compute the difference directly
After you’ve found that the white birch’s future diameter is 14 inches and the pin oak’s future diameter is inches, type abs(46/3 - 14) into Desmos and press Enter. The output is the positive difference in inches between the two future diameters; compare that decimal to the answer choices and pick the closest one.
Step-by-step Explanation
Use the formula to find each tree’s current age
The relationship given is: age (years) = diameter (inches) × growth factor.
Each tree now has a diameter of 1 foot, which is 12 inches.
- White birch growth factor: 5.0, so its current age is years.
- Pin oak growth factor: 3.0, so its current age is years.
Find each tree’s age 10 years from now
Add 10 years to each tree’s current age:
- White birch future age: years.
- Pin oak future age: years.
Convert future ages back to diameters
Use the same relationship, but solve for diameter: diameter (inches) = age (years) ÷ growth factor.
So 10 years from now:
- White birch diameter: inches.
- Pin oak diameter: inches, which is about inches.
Compute the difference in future diameters and match the choice
Now subtract the white birch diameter from the pin oak diameter:
The difference in diameters is about inches, which is closest to among the answer choices, so the correct answer is .