Question

Exit CertiryLesson: Chapter 13 ReviewISAIAH COVERTQuestion 5 of 21, Step 11 of 13/22CorrectA Christmas tree is supported by a wire that is 2 feet longer than the height of the tree. The wire isanchored at a point whose distance from the base of the tree is 34 feet shorter than the height of thetree. What is the height of the tree?AnswerKeypadKeyboard ShortcutsHeight =feet

Answer 1

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:

`\begin{gathered} c=h+1 \\ a=h \\ b=h-34 \end{gathered}`

Substituting these values into the Pythagorean theorem:

`\begin{gathered} a^2+b^2=c^2 \\ \downarrow\downarrow \\ h^2+(h-34)^2=(h+2)^2 \end{gathered}`

Step 4. Use the formula for the square of a binomial:

`(x\pm y)^2=x^2\pm2xy+y^2`

and apply it to the two binomial squared expressions:

`\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}`

Step 5. Combine the like terms:

`2h^2-68h+1,156=h^2+4h+4`

Move the terms on the right-hand side, to the left side of the equation with the opposite sign:

`2h^2-h^2-68h-4h+1,156-4=0`

Combine the like terms again:

`h^2-72h+1,152=0`

Step 6. Factor the expression:

`\begin{gathered} h^2-72h+1,152=0 \\ \downarrow\downarrow \\ (h-24)(h-48)=0 \end{gathered}`

Find the solutions by making the expression on each parenthesis equal to 0:

`\begin{gathered} h-24=0\longrightarrow h=24 \\ h-48=0\longrightarrow h=48 \end{gathered}`

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