Question

Two palm trees grow next to each other in Luke's backyard. One of the trees gets sick, so Luke cuts off the top 3 feet. The other tree, however, is healthy and grows 2 feet. How tall are the two trees if the healthy tree is now 4 feet taller than the sick tree, and together they are 28 feet tall?

• A. 8 and 20 feet
• B. 11 and 17 feet
• C. 12 and 16 feet
• D. cannot be determined
• E. 14 and 14 feet

Hint:

Read word problems carefully to distinguish between past information and information describing the current situation. Here, the equations should only model the "now" state of the trees. The past changes explain how they got to this state but are not directly used in the final system of equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This word problem describes the current state of two trees and can be solved by setting up a system of linear equations based on the information provided. The information about the past changes (cutting and growing) is background context but not needed for the equations about the current heights.
Step 2: Key Formula or Approach:
Let \(h\) be the current height of the healthy tree and \(s\) be the current height of the sick tree. Translate the sentences about the *current* state into equations: 1. "the healthy tree is now 4 feet taller than the sick tree": \( h = s + 4 \) 2. "together they are 28 feet tall": \( h + s = 28 \) We need to solve this system for \(h\) and \(s\).
Step 3: Detailed Explanation:
We have the system: \[ h = s + 4 \] \[ h + s = 28 \] We can use the substitution method. Substitute the first equation into the second equation: \[ (s + 4) + s = 28 \] Combine the s-terms: \[ 2s + 4 = 28 \] Subtract 4 from both sides: \[ 2s = 24 \] Solve for s: \[ s = 12 \] So, the current height of the sick tree is 12 feet.
Now find the current height of the healthy tree using the first equation: \[ h = s + 4 = 12 + 4 = 16 \] The current height of the healthy tree is 16 feet.
Step 4: Final Answer:
The heights of the two trees are 12 and 16 feet.