Step 1. Given:
The Christmas tree is supported by a wire, and this wire is 2 feet longer than the height of the tree.
The wire is anchored at a distance from the base 34 ft shorter than the height of the tree.
Required: Find the height of the tree.
Step 2. Make a diagram of the situation:
-the green line represents the tree
-the black line represents the wire
-the red line represents the base.
Also, let h be the height of the tree:
Step 3. To solve this problem and find h, we will use the Pythagorean theorem:
In our case:
[tex]\begin{gathered} c=h+1 \\ a=h \\ b=h-34 \end{gathered}[/tex]Substituting these values into the Pythagorean theorem:
[tex]\begin{gathered} a^2+b^2=c^2 \\ \downarrow\downarrow \\ h^2+(h-34)^2=(h+2)^2 \end{gathered}[/tex]Step 4. Use the formula for the square of a binomial:
[tex](x\pm y)^2=x^2\pm2xy+y^2[/tex]and apply it to the two binomial squared expressions:
[tex]\begin{gathered} h^2+(h-34)^2=(h+2)^2 \\ \downarrow\downarrow \\ h^2+h^2-68h+1,156=h^2+4h+4 \end{gathered}[/tex]Step 5. Combine the like terms:
[tex]2h^2-68h+1,156=h^2+4h+4[/tex]Move the terms on the right-hand side, to the left side of the equation with the opposite sign:
[tex]2h^2-h^2-68h-4h+1,156-4=0[/tex]Combine the like terms again:
[tex]h^2-72h+1,152=0[/tex]Step 6. Factor the expression:
[tex]\begin{gathered} h^2-72h+1,152=0 \\ \downarrow\downarrow \\ (h-24)(h-48)=0 \end{gathered}[/tex]Find the solutions by making the expression on each parenthesis equal to 0:
[tex]\begin{gathered} h-24=0\longrightarrow h=24 \\ h-48=0\longrightarrow h=48 \end{gathered}[/tex]Since the length of the base has to be 34 feet shorter, with a height of 24 ft, the base will be 24-34=-10ft, and the length of the base cannot be a negative number. Thus, the only possible solution is 48 ft.
Answer: 48 feet
