\begin{tabular}{l|l} A 11-foot ladder is leaning \ against a building, with the base \ of the ladder 3 feet from the \ building. How high up on the \ building will the top of the ladder \ reach? & $$ \begin{array}{l} ext { The ladder will reach } \ ext { ft high up on the building. } \ ext { (Simplify your answer. } \ ext { Type an exact answer, } \ ext { using radicals as } \ ext { needed.) }\end{array} $$\end{tabular}

Question

\begin{tabular}{l|l} A 11-foot ladder is leaning \ against a building, with the base \ of the ladder 3 feet from the \ building. How high up on the \ building will the top of the ladder \ reach? & $$ \begin{array}{l}	ext { The ladder will reach } \ 	ext { ft high up on the building. } \ 	ext { (Simplify your answer. } \ 	ext { Type an exact answer, } \ 	ext { using radicals as } \ 	ext { needed.) }\end{array} $$\end{tabular}

UpStudy AI · Accepted Answer

To find the height up on the building that the top of the ladder will reach, we can use the Pythagorean theorem since we have a right triangle formed by the ladder, the ground, and the building.

Let's denote:
- The height up on the building as $$ h $$ (in feet),
- The base of the ladder as 3 feet, and
- The length of the ladder as 11 feet.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the height up on the building and the base of the ladder).

So, we have:
$$ h^2 + 3^2 = 11^2 $$

Now, we can solve for $$ h $$ by rearranging the equation and taking the square root of both sides.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$h^{2}+3^{2}=11^{2}$$
- step1: Evaluate the power:
  $$h^{2}+9=11^{2}$$
- step2: Add or subtract both sides:
  $$h^{2}=112$$
- step3: Simplify the expression:
  $$h=\pm \sqrt{112}$$
- step4: Simplify the expression:
  $$h=\pm 4\sqrt{7}$$
- step5: Separate into possible cases:
  $$\begin{align}&h=4\sqrt{7}\&h=-4\sqrt{7}\end{align}$$
- step6: Rewrite:
  $$h_{1}=-4\sqrt{7},h_{2}=4\sqrt{7}$$
The solutions to the equation are $$ h = -4\sqrt{7} $$ and $$ h = 4\sqrt{7} $$. Since the height cannot be negative, the height up on the building that the top of the ladder will reach is $$ h = 4\sqrt{7} $$ feet.

Therefore, the ladder will reach $$ 4\sqrt{7} $$ feet high up on the building.
The ladder will reach $$ 4\sqrt{7} $$ feet high up on the building.

A 11-foot ladder is leaning

against a building, with the base

of the ladder 3 feet from the

building. How high up on the

building will the top of the ladder

reach?

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