Find x round to the nearest tenth.
The Correct Answer and Explanation is:
The correct answer is 26.9.
To solve for the value of x, we need to use trigonometry because the image shows a right-angled triangle with a known angle and a known side length. The trigonometric functions sine, cosine, and tangent relate the angles of a right triangle to the ratios of its side lengths. The mnemonic SOH CAH TOA helps us remember these relationships:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
First, we must identify the sides of the triangle relative to the given angle of 52 degrees.
- The side labeled x is opposite the 52-degree angle.
- The side with the length 21 is adjacent to the 52-degree angle. It is the leg of the right triangle that forms the angle, but it is not the hypotenuse.
- The hypotenuse is the side opposite the right angle, and its length is not given.
Since we know the length of the adjacent side and we want to find the length of the opposite side, the tangent function (TOA) is the correct choice. The formula is:
tan(angle) = Opposite / Adjacent
Now, we can substitute the values from the triangle into this formula:
tan(52°) = x / 21
To solve for x, we need to isolate it. We can do this by multiplying both sides of the equation by 21:
x = 21 * tan(52°)
Using a calculator, we find the value of the tangent of 52 degrees:
tan(52°) ≈ 1.27994
Now, we multiply this value by 21 to find x:
x ≈ 21 * 1.27994
x ≈ 26.87874
The problem asks us to round the answer to the nearest tenth. The digit in the tenths place is 8, and the digit in the hundredths place is 7. Since 7 is 5 or greater, we round up the tenths digit from 8 to 9.
Therefore, x rounded to the nearest tenth is 26.9.
