How high up the wall can a 12-foot ladder reach if its base is 4 feet from the wall? Round your answer to the nearest tenth of a foot if necessary.

Answer:

11.3 feet

Step-by-step explanation:

let x= answer

x²+4²=12²

x²=128

x=11.3137085

which rounds to

11.3

Answer:

≈ 11.3 ft.

Step-by-step explanation:

In order to solve this, we must use some trigonometry, in this case, SOH-CAH-TOA. If you don't know what it is, here's a quick explanation:

- SOH: Sin(θ) = Opposite / Hypotenuse

- CAH: Cos(θ) = Adjacent / Hypotenuse

- TOA: Tan(θ) = Opposite / Adjacent

(Remember, SOH-CAH-TOA can ONLY be used for right triangles. If the problem does not clearly show a right triangle, meaning that there's no box in the corner, you cannot use it)

When using SOH-CAH-TOA, we first must choose an angle, either the top one or the bottom one. We can't use the right angle as, well, you just can't :). First, we have to find the bottom angle, so circle it. Literally, just circle the angle, you'll thank me later. I'll explain why we don't solve the top angle later on.

After that, we just do some labeling on the sides of the triangle. First, label the shortest side that forms the circled angle, which is the floor (4 ft), as the Adjacent side. Then label the longest side of the triangle, which is the ladder (12 ft), as the Hypotenuse. Finally, label the side that doesn't form the circled angle as the Opposite side. This is why the circling comes in handy :)

Then, out of SOH, CAH, and TOA, we choose the one that has the sides that forms the circled angle. In this case, it's CAH, or cosine, as the adjacent and hypotenuse form the circled angle.

Next, we write out the formula and solve:

Cos(m∠Circled Angle) = Adjacent/Hypotenuse

Cos(m∠Circled Angle) = 4/12

Cos(m∠Circled Angle) x Cos^-1 = 1/3 x Cos^-1

m∠Circled Angle ≈ 70.5288 (Rounding to the nearest ten-thousandth)

Note: Pay attention to the 3rd line. 'Cos^-1' is basically the equivalent of '÷ Cos'.

The reason why we did the bottom angle first is because you cannot have two unknown variables when solving for one of them. That would've been the case if we did choose the top angle as we wouldn't know the measure of the wall nor the angle. Now that we have solved the bottom angle, we can move onto solving the length of the wall.

Looking at SOH-CAH-TOA, we can either use sine or tangent as we have the measures for both the adjacent and hypotenuse. I'll use sine cause why not? Anyways, let's move on!

Just like last time, we write out the formula and solve:

Sin(m∠Circled Angle) = Opposite/Hypotenuse

Sin(70.5288) = Opposite/12

0.9428 = Opposite/12

0.9428 x 12 = Opposite/12 x 12

11.3137 ≈ Opposite (Wall)

Wall ≈ 11.3 ft. (Rounding to the nearest tenth, as asked to do so by the problem)

Note: Be careful with SOH-CAH-TOA because I've messed up multiple times with my calculator while typing out my explanation, so just make sure you go over it several times before submitting your answer when solving trig problems :). But I'm confident with my answer, so just submit it without hesitation as I've already double-checked my work.

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11
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