Question

find the length x to the nearest whole number.

(do not round until the final answer. then round to the nearest whole number.)

triangle with angles 37° and 28°, adjacent side 490, right angle, and side x to find

Explanation:

Step1: Identify the two right triangles

We have two right triangles, both with adjacent side 490. Let the upper segment of \( x \) be \( x_1 \) and the lower segment be \( x_2 \). So \( x = x_1 + x_2 \).

Step2: Calculate \( x_1 \) using tangent

For the upper triangle with angle \( 37^\circ \), \( \tan(37^\circ)=\frac{x_1}{490} \), so \( x_1 = 490\times\tan(37^\circ) \).

Step3: Calculate \( x_2 \) using tangent

For the lower triangle with angle \( 28^\circ \), \( \tan(28^\circ)=\frac{x_2}{490} \), so \( x_2 = 490\times\tan(28^\circ) \).

Step4: Sum \( x_1 \) and \( x_2 \)

\( x = 490\times\tan(37^\circ)+490\times\tan(28^\circ) = 490\times(\tan(37^\circ)+\tan(28^\circ)) \).

Calculate \( \tan(37^\circ)\approx0.7536 \), \( \tan(28^\circ)\approx0.5317 \).

Then \( \tan(37^\circ)+\tan(28^\circ)\approx0.7536 + 0.5317=1.2853 \).

\( x\approx490\times1.2853 = 629.797 \).

Step5: Round to nearest whole number

Rounding 629.797 to the nearest whole number gives 630.

Answer:

\( \boxed{630} \)