Question

find the length x to the nearest whole number.

Explanation:

Step1: Find the two segments of x

We can split the length \( x \) into two parts. Let the upper part be \( x_1 \) and the lower part be \( x_2 \). Using the tangent function, for the upper triangle with angle \( 36^\circ \) and adjacent side 450, we have \( \tan(36^\circ)=\frac{x_1}{450} \), so \( x_1 = 450\times\tan(36^\circ) \). For the lower triangle with angle \( 28^\circ \) and adjacent side 450, we have \( \tan(28^\circ)=\frac{x_2}{450} \), so \( x_2 = 450\times\tan(28^\circ) \).

Step2: Calculate \( x_1 \) and \( x_2 \)

First, calculate \( \tan(36^\circ)\approx0.7265 \), so \( x_1 = 450\times0.7265 = 326.925 \). Then, calculate \( \tan(28^\circ)\approx0.5317 \), so \( x_2 = 450\times0.5317 = 239.265 \).

Step3: Find the total length \( x \)

The total length \( x=x_1 + x_2 \). So \( x=326.925+239.265 = 566.19 \). Rounding to the nearest whole number, we get \( x\approx566 \).

Answer:

566