Find y to the nearest hundredths place. A 11.59 ft B 12.43 ft C 15.85 ft D 24.93 ft Tools Question 7: Find y to the nearest hundredths place. 17 ft 430 5 RA 11.59 ft RB 12.43 ft RC 15.85 ft RD 24.93 ft
The Correct Answer and Explanation is:
Correct Answer: B) 12.43 ft
Explanation:
This problem requires the use of trigonometry to find the length of an unknown side in a right-angled triangle. The triangle is labeled QRS, with a right angle at R. We are given an angle at vertex Q, which is 43 degrees, and the length of the hypotenuse (the side opposite the right angle), which is 17 ft. The goal is to find the length of side ‘y’, which is the side QR.
To solve this, we can use the trigonometric ratios, often remembered by the mnemonic SOH CAH TOA.
- SOH: Sine(θ) = Opposite / Hypotenuse
- CAH: Cosine(θ) = Adjacent / Hypotenuse
- TOA: Tangent(θ) = Opposite / Adjacent
First, we need to identify the sides of the triangle in relation to the given angle of 43 degrees.
- The hypotenuse is the longest side and is opposite the right angle. In this triangle, the hypotenuse is the side QS, which has a length of 17 ft.
- The opposite side is the side across from the 43-degree angle. This is side RS. Its length is unknown.
- The adjacent side is the side next to the 43-degree angle that is not the hypotenuse. This is side QR, which is labeled ‘y’.
We need to find the length of the adjacent side (‘y’) and we know the length of the hypotenuse (17 ft). The trigonometric ratio that connects the adjacent side and the hypotenuse is the cosine function (CAH).
The formula for cosine is:
cos(angle) = Adjacent / Hypotenuse
Now, we can substitute the values from the problem into this formula:
cos(43°) = y / 17
To solve for ‘y’, we need to isolate it. We can do this by multiplying both sides of the equation by 17:
y = 17 * cos(43°)
Next, we use a calculator to find the value of cos(43°):
cos(43°) ≈ 0.7313537
Now, we multiply this value by 17:
y ≈ 17 * 0.7313537
y ≈ 12.4320129
The question asks us to round the answer to the nearest hundredths place, which means we need to have two digits after the decimal point. To do this, we look at the third digit after the decimal, which is 2. Since 2 is less than 5, we round down, meaning we keep the hundredths digit as it is and drop the rest.
So, y ≈ 12.43.
The calculated value is 12.43 ft, which matches option B.
