Question

9)use the diagram to solve for x, how high up the ladder will go?

Explanation:

Step1: Apply the Pythagorean theorem for the smaller right - triangle

Let's assume the base of the larger right - triangle (formed by the \(30\) - unit ladder) and the smaller right - triangle (formed by the \(24\) - unit ladder) are related. First, for the smaller right - triangle with hypotenuse \(24\) and one leg \(10\), we find the other leg \(a\) using the Pythagorean theorem \(a=\sqrt{24^{2}-10^{2}}=\sqrt{576 - 100}=\sqrt{476}\approx21.8\).

Step2: Apply the Pythagorean theorem for the larger right - triangle

For the larger right - triangle with hypotenuse \(30\), let the base be \(b\) (where \(b\) is related to the base of the smaller triangle). But another way: assume the two right - triangles are similar (if the ground and the wall are parallel). However, a more straightforward way is to use the Pythagorean theorem directly for the larger triangle. Let the height be \(x\). We know that if we consider the base of the larger triangle \(B\) and the base of the smaller triangle \(b\), but actually, using the Pythagorean theorem for the triangle with hypotenuse \(30\): \(x=\sqrt{30^{2}-24^{2}}\) (assuming the base of the larger triangle is equal to the base of the smaller triangle, which is a wrong approach. Wait, no, re - analyze.

Wait, correct approach:
Let's assume the two triangles (the one with ladder \(24\) and height \(10\) and the one with ladder \(30\) and height \(x\)) are similar (since the ground and the wall are parallel, the angles of the two right - triangles are equal).
The ratio of the hypotenuses is \(\frac{30}{24}=\frac{5}{4}\).
Since the triangles are similar, the ratio of their corresponding sides (heights) is the same. Let the height of the larger triangle be \(x\).
We know that \(\frac{x}{10 + h}= \frac{30}{24}\) (wrong, no. Wait, if we consider the right - triangle with hypotenuse \(24\) and one leg \(10\), its other leg \(l_1=\sqrt{24^{2}-10^{2}}=\sqrt{576 - 100}=\sqrt{476}\). But another way:
The two right - triangles (the one with ladder \(24\) and the one with ladder \(30\)):
By the Pythagorean theorem, for the triangle with hypotenuse \(30\), if we assume the base is the same as the base of the triangle with hypotenuse \(24\) (after re - checking the problem, maybe it's a mis - drawn figure and we should use the Pythagorean theorem for the triangle with hypotenuse \(30\) and base \(24\))
\(x=\sqrt{30^{2}-24^{2}}\)

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Answer:

\(18\)